Research ArticleSolution of Some Types of Differential EquationsOperational Calculus and Inverse Differential Operators

K Zhukovsky

Faculty of Physics M V Lomonosov Moscow State University Leninskie Gory Moscow 119899 Russia

Correspondence should be addressed to K Zhukovsky zhukovskyphysicsmsuru

Received 23 January 2014 Accepted 23 February 2014 Published 30 April 2014

Academic Editors D Baleanu and H Jafari

Copyright copy 2014 K ZhukovskyThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We present a generalmethod of operational nature to analyze and obtain solutions for a variety of equations ofmathematical physicsand related mathematical problems We construct inverse differential operators and produce operational identities involvinginverse derivatives and families of generalised orthogonal polynomials such as Hermite and Laguerre polynomial families Wedevelop the methodology of inverse and exponential operators employing them for the study of partial differential equationsAdvantages of the operational technique combined with the use of integral transforms generating functions with exponentialsand their integrals for solving a wide class of partial derivative equations related to heat wave and transport problems aredemonstrated

1 Introduction

Most of physical systems can be described by appropriatesets of differential equations which are well suited as modelsfor systems Hence understanding of differential equationsand finding its solutions are of primary importance forpure mathematics as for physics With rapidly developingcomputer methods for the solutions of equations the ques-tion of understanding of the obtained solutions and theirapplication to real physical situations remains opened foranalytical study There are few types of differential equationsallowing explicit and straightforward analytical solutions Itis common knowledge that expansion into series of HermiteLaguerre and other relevant polynomials [1] is useful whensolvingmany physical problems (see eg [2 3]) Generalisedforms of these polynomials exist with many variables andindices [4 5] In what follows we develop an analyticalmethod to obtain solutions for various types of partial differ-ential equations on the base of operational identities employ-ing expansions in series of Hermite Laguerre polynomialsand their modified forms [1 6] The key for building thesesolutions will be an operational approach and developmentof the formalism of inverse functions and inverse differentialoperators already touched in [7 8] We will demonstratein what follows that when used properly and combined

in particular with integral transforms such an approachleads to elegant analytical solutions with transparent physicalmeaning without particularly cumbersome calculations

2 Inverse Derivative

For a common differential operator 119863 = 119889119889119909 we can definean inverse derivative such that upon the action on a function119891(119909) it gives another function 119865(119909)

119863minus1

119891 (119909) = 119865 (119909) (1)

whose derivative is 1198651015840(119909) = 119891(119909) Evidently the inversederivative 119863minus1 is executed by an integral operator being theinverse of differential operator acting on119891(119909) and its generalform is int119891(119909) = 119865(119909) + 119862 where 119862 is the constant ofintegration The action of its 119899th order

119863minus119899

119909

119891 (119909) =1

(119899 minus 1)int

119909

0

(119909 minus 120585)119899minus1

119891 (120585) 119889120585

(119899 isin 119873 = 1 2 3 )

(2)

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 454865 8 pageshttpdxdoiorg1011552014454865

2 The Scientific World Journal

can be complemented with the definition of the action of zeroorder derivative as follows

1198630

119909

119891 (119909) = 119891 (119909) (3)

so that evidently

119863minus119899

119909

1 = 119909119899

119899 (119899 isin 119873

0

= 119873 cup 0) (4)

In what follows we will appeal to various modifications of thefollowing equation

(1205732

minus 1198632

)]119865 (119909) = 119891 (119909) (5)

Thus it is important to construct the particular integral 119865(119909)with the help of the following operational identity (see eg[6])

119902minus]=1

Γ (])int

infin

0

exp (minus119902119905) 119905]minus1119889119905

min Re (119902) Re (]) gt 0(6)

For the operator 119902 = 1205732 minus 1198632 we have

(1205732

minus 1198632

)minus]119891 (119909)

=1

Γ (])int

infin

0

exp (minus1205732119905) 119905]minus1 exp (1199051198632) 119891 (119909) 119889119905(7)

We will also make explicit use of the generalized form ofthe Glaisher operational rule [9] action of the operator 119878 =exp(1199051198632

119909

) on the function 119891(119909) = exp(minus1199092) yields

119878119891 (119909) = exp(119910 1205972

1205971199092) exp (minus1199092)

=1

radic1 + 4119910exp(minus 119909

2

1 + 4119910)

(8)

Exponential operator mentioned above is closely related toHermite orthogonal polynomials

119867119899

(119909 119910) = 119899

[1198992]

sum

119903=0

119909119899minus2119903

119910119903

(119899 minus 2119903)119903 (9)

as demonstrated in [4 5 10] by operational relations

119867119899

(119909 119910) = exp(119910 1205972

1205971199092)119909119899

(10)

Moreover the following generating function for Hermitepolynomials exists

exp (119909119905 + 1199101199052) =infin

sum

119899=0

119905119899

119899119867119899

(119909 119910) (11)

Note also an easy to prove and useful relation [11]

119911119899

119867119899

(119909 119910) = 119867119899

(119909119911 1199101199112

) (12)

Laguerre polynomials of two variables [4]

119871119899

(119909 119910) = exp(minus119910 120597120597119909119909120597

120597119909)(minus119909)119899

119899= 119899

119899

sum

119903=0

(minus1)119903

119910119899minus119903

119909119903

(119899 minus 119903)(119903)2

(13)

are related to the following operator 120597119909

119909120597119909

[10]

119871

119863119909

=120597

120597119909119909120597

120597119909=

120597

120597119863minus1119909

(14)

sometimes called Laguerre derivative119871

119863119909

Note their non-commutative relation with the inverse derivative operator

[119871

119863119909

119863minus1

119909

] = minus1 ([119860 119861] = 119860119861 minus 119861119860) (15)

They represent solutions of the following partial differentialequation with proper initial conditions

120597119910

119871119899

(119909 119910) = minus (120597119909

119909120597119909

) 119871119899

(119909 119910) (16)

where

119871119899

(119909 119910) = 119899

119899

sum

119903=0

(minus1)119903

119910119899minus119903

119909119903

(119899 minus 119903)(119903)2

(17)

with proper initial conditions

119871119899

(119909 0) =(minus119909)119899

119899 (18)

In the following sections we will investigate the possi-bilities to solve some partial differential equations involvingthe differential operators studied above Now we just notehow the technique of inverse operator applied for derivativesof various orders and their combinations and combinedwith integral transforms allows for easy and straightforwardsolutions of various types of differential equations

3 Diffusion Type Heat Propagation TypeProblems and Inverse Derivatives

Heat propagation and diffusion type problems play a key rolein the theory of partial differential equations Combinationof exponential operator technique and inverse derivativetogether with the operational identities of the previoussection is useful for the solution of a broad spectrumof partialdifferential equations related to heat and diffusion processesSome of them have been already studied by operationalmethod (see eg [10 12]) Below we will focus on thegeneralities of the solution of the following problem

120597

120597119910119865 (119909 119910) = ( + ) 119865 (119909 119910) (19)

with initial conditions

119865 (119909 0) = 119902 (119909) (20)

The Scientific World Journal 3

We will employ operational approach combined with inte-gral transforms and exponential operator technique Formalsolution for our generic formulation reads as follows

119865 (119909 119910) = exp (119910 ( + )) 119902 (119909) (21)

We would like to underline that operators and may notcommute so dependently on the value of their commutatorwe will obtain different sequences of operators in (23)disentangling and [13 14] In the simplest case whenoperators and are multiplication and differentiationoperators respectively they can be easily disentangled in theexponent with account for

[ ] = 1 (22)

which yields

119865 (119909 119910) = exp (119910 ( + )) 119902 (119909)

= exp(minus1199102

2) exp (119910) exp (119910) 119902 (119909)

(23)

Explicit expressions for the action of these operators on theinitial condition function 119892(119909) can be obtained by integraltransforms series expansions and operational technique tobe evaluated in every special case This formulation simplein its essence nevertheless has wide application and allows usto frame either some of integrodifferential equations in thisscheme An elegant and interesting example is given by thefollowing equation

120597

120597119905119865 (119909 119905) = minus

120597

120597119909119909120597

120597119909119865 (119909 119905) + int

119909

0

119865 (120585 119905) 119889120585

119865 (119909 0) = 119892 (119909)

(24)

Its formal solution reads

119865 (119909 119905) = exp (119860 + 119861) 119892 (119909) (25)

where operators

119860 = minus119905119871

119863119909

= minus119905120597

120597119863minus1119909

119861 = 119905119863minus1

119909

(26)

do not commute

[119860 119861] = minus1199052

(27)

Evidently operators in the exponential disentangle

119865 (119909 119905) = exp (119860 + 119861) 119902 (119909)

= exp(1199052

2) exp (119905119863minus1

119909

) exp(minus119905 120597120597119863minus1119909

)119892 (119909)

(28)

Thus we have obtained the solution of the integrodifferen-tial equation (24) as a sequence of exponential operatorstransforming the initial condition 119892(119909) Our further steps

depend on the explicit form of this function In the mostgeneral case of 119892(119909) we may take advantage of the inversederivative technique First consider the action of the operatorexp(minus120597(120597119863minus1

119909

)) on 119892(119909)

119891 (119909 119905) = exp(minus119905 120597120597119863minus1119909

)119892 (119909) (29)

where119863minus1119909

is defined in (2) Equation (29) represents in factthe diffusion process and it is the solution of the followinginitial value problem

120597

120597119905119891 (119909 119905) = minus

120597

120597119909119909120597

120597119909119891 (119909 119905) 119891 (119909 0) = 119892 (119909) (30)

Relevant studies were performed in [10 12] The initialcondition function 119891(119909 0) = 119892(119909) can be written as follows

119892 (119909) = 120593 (119863minus1

119909

) 1 (31)

and the image 120593(119909) is explicitly given by the followingintegral

120593 (119909) = int

infin

0

exp (minus120577) 119892 (119909120577) 119889120577 (32)

which is supposed to converge Then the result of theLaguerre diffusion (29) appears in the form of the translationof the image function 120593

119891 (119909 119910) = 120593 (119863minus1

119909

minus 119905) 1 (33)

Consequently we have to apply the exponential operatorexp(119905119863minus1

119909

) which can be expanded in series

119865 (119909 119905) = exp(1199052

2)

infin

sum

119899=0

119905119899

119863minus119899

119909

119899119891 (119909 119905) (34)

The simplest example of the initial function 119892(119909) =exp(minus119909) demonstrates the technique sketched above result-ing in

120593 (119909) =1

(1 + 119909)when |119909| lt 1 (35)

and the Laguerre diffusion contribution (29) produces thefollowing function

119891 (119909 119905) =1

1 minus 119905exp (minus 1

1 minus 119905119909) (36)

The inverse derivative action on the exponent reads119863minus119899

119909

exp(120572119909) = exp(120572119909)120572119899 and eventually we obtain

119865 (119909 119905) = exp(1199052

2)1

1 minus 119905

infin

sum

119899=0

119905119899

119863minus119899

119909

119899exp (minus 1

1 minus 119905119909)

=1

1 minus 119905exp (minus 119909

1 minus 119905) exp(minus119905 (1 minus 119905) + 119905

2

2)

(37)

4 The Scientific World Journal

In conclusion of the present chapter we consider theexample of the solution of a heat propagation type equationby operational method involving the inverse derivativeoperator and exponential operator technique We recall thatthe common heat equation with initial condition problem

120597

120597119905119891 (119909 119905) = 120597

2

119909

119891 (119909 119905) 119891 (119909 0) = 119892 (119909) (38)

can be solved by Gauss transforms

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

exp(minus(119909 minus 120585)2

4119905) 119892 (120585) 119889120585 (39)

In complete analogy with the above statements we can solvethe heat-type equation with differential operator

119871

119863119909

(14)with for example the following initial condition

120597

120597119905119891 (119909 119905) =

119871

1198632

119909

119891 (119909 119905) 119891 (119909 0) = 119892 (119909) (40)

The Laguerre heat-type propagation problem (40) possessesthe following solution (see the Appendix)

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

119890minus120585

24119905

119867

1198620

(minus120585119909

2119905 minus1199092

4119905)120593 (120585) 119889120585

(41)

4 Operational Approach and Other Types ofDifferential Equations

Operational approach to solution of partial differential equa-tions demonstrated on the examples of diffusion-like andheat-like equations with 120597

119909

119909120597119909

derivatives can be furtherextended to other equation types Consider the followingexample of a rather complicated differential equation

1

120588

120597

120597119905119860 (119909 119905) = 119909

2

1205972

1205971199092119860 (119909 119905) + 120582119909

120597

120597119909119860 (119909 119905)

minus 120583119860 (119909 119905) 119892 (119909) = 119860 (119909 0)

(42)

where 120588 120582 and 120583 are some arbitrary constant coefficientsand function 119892(119909) = 119860(119909 119905 = 0) is the initial condition Byintroducing operator119863 = 119909120597

119909

and distinguishing the perfectsquare we rewrite (42)

1

120588

120597

120597 119905119860 (119909 119905) = ((119863 +

120582

2)

2

minus 120576)119860 (119909 119905) (43)

where

120576 = 120583 + (120582

2)

2

(44)

Thus the following exponential solution for (42) appears

119860 (119909 119905) = exp120588119905((119863 + 1205822)

2

minus 120576)119892 (119909) (45)

Now making use of the operational identity

exp (1199012) = 1

radic120587int

infin

minusinfin

exp (minus1205852 + 2120585119901) 119889120585 (46)

and applying exp(119886119863) according to

exp (119886119909120597119909

) 119891 (119909) = 119891 (119890119886

119909) (47)

we obtain the following compact expression for 119860(119909 119905)

119860 (119909 119905) =exp (minus120588120576119905)radic120587

times int

infin

minusinfin

exp [minus1205902 + 120590120572 1205822120588] 119892 (119909 exp (120590120572)) 119889120590

(48)

where

120572 = 120572 (119905) = 2radic120588119905 (49)

Consider two simple examples of initial condition functionsThe first one is

119892 (119909) = 119909119899

(50)

Then we immediately obtain the solution of (42) as follows

119860 (119909 119905) = 119909119899 exp 120588119905 (1198992 + 120582119899 minus 120583) (51)

The second example is given by the following initial conditionfunction

119892 (119909) = ln119909 (52)

Trivial computations yield the following solution

119860 (119909 119905) = (ln119909 + 120588119905120582) exp (minus120588119905120583) (53)

Thus operational technique combined with integraltransforms operational identities and extended forms oforthogonal polynomials represents powerful tool for findingsolutions of various classes of differential equations and initialvalue problems Note that within the framework of inversedifferential operators developed and described above theusage of the evolution operator method opens new possibili-ties which we will elucidate in what follows

Let us consider the following generalization of the heatequation

120597119905

119865 (119909 119905) = 1205721205972

119909

119865 (119909 119905) + 120573119909119865 (119909 119905) (54)

with the initial condition

119865 (119909 0) = 119891 (119909) (55)

The evolution type equation (54) contains linear coordinateterm in addition to the second order derivative Its formalsolution can be written via the evolution operator

119865 (119909 119905) = 119891 (119909) (56)

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where

= exp (119860 + 119861) 119860 = 1205721199051205972

119909

119861 = 120573119905119909 (57)

The exponential of the evolution operator in (56) is thesum of two noncommuting operators and it can be writtenas the ordered product of two exponential operators Indeedthe commutator of 119860 and 119861 has the following nonzero value

[119860 119861] = 11989811986012

= 21205721205731199052

120597119909

119898 = 212057311990532

12057212

(58)

Then we can apply the disentanglement operational identity

119890

119860+

119861

= 119890(119898

212)minus(1198982)

119860

12+

119860

119890

119861

(59)

and the following chain rule

119890119901120597

2

119909119890119902119909

119892 (119909) = 119890119901119902

2

119890119902119909

1198902119901119902120597

119909119890119901120597

2

119909119892 (119909) (60)

where119901 119902 are constant parametersWith their help we obtainthe evolution operator for (54)

= 119890Φ(119909119905120573)

Θ119878 (61)

Note that we have factorized two commuting operators theoperator of translation in space

Θ = 119890120572120573119905

2120597

119909 (62)

and operator

119878 = 119890120572119905120597

2

119909 (63)

which is in fact operator 119878 = exp(1199051198632119909

) The phase in iswritten as follows

Φ(119909 119905 120572 120573) =1

31205721205732

1199053

+ 120573119905119909 (64)

The action of on the initial condition function 119891(119909) yieldsthe following solution for our problem

119865 (119909 119905) = 119890Φ(119909119905120572120573)

Θ119878119891 (119909) (65)

Thus we conclude from the form of (65) that the problem(54) with the initial condition (55) can be solved by theconsequent application of commuting operators Θ (62) and 119878(63) to 119891(119909) apart from the factor 119890Φ(119909119905120572120573) Now the explicitform of the solution (65) can be obtained by recalling that Θacts as a translation operator

119890119904120597

119909119892 (119909) = 119892 (119909 + 119904) (66)

and that the action of 119878 on the function 119892(119909) yields thesolution of the ordinary heat equation through Gauss-Weierstrass transform (A1) Accordingly we denote

119891 (119909 119905) equiv 119878119891 (119909) equiv 119890120572119905120597

2

119909119891 (119909) (67)

and we write

Θ119891 (119909 119905) = 119891 (119909 + 1205721205731199052

119905) (68)

Thus (54)with initial condition (55) has the following explicitsolution119865 (119909 119905)

= 119890Φ(119909119905120572120573)

1

2radic120587120572119905int

infin

minusinfin

119890minus(119909+120572120573119905

2minus120585)

2

4119905120572

119891 (120585) 119889120585

(69)

provided that the integral converges Summarizing the aboveoutlined procedure we conclude that a solution for (54)consists in finding a Gauss transformed function 119891 with ashifted argument

119865 (119909 119905) = 119890Φ(119909119905120572120573)

119891 (119909 + 1205721205731199052

119905) (70)

Moreover this is a general observation for this type ofequation valid for any function 119891(119909) (provided the integralconverges) In other words we have obtained the solution ofFokker plank equation as a consequent action of operator 119878of heat diffusion and operator Θ of translation on the initialcondition functionNote that119891(119909 119905) is the solution of the heatequation representing a natural propagation phenomenon

The effect produced by the translation operator Θ and theoperator 119878 is best illustrated with the example of Gaussian119891(119909) = exp(minus1199092) evolution when (69) becomes

119865(119909 119905)|119891(119909)=exp(minus1199092) =

exp (Φ (119909 119905))radic1 + 4119905

exp(minus(119909 + 120573119905

2

)2

1 + 4119905)

119905 ge 0

(71)

The above result is exactly the generalization of Gleisherrule (8) considered earlier in the context of the heat equationThus (71) is the solution of the ordinary heat equation with120573 = 0 when the initial function is Gaussian

Another interesting example of solving (54) appearswhenthe initial function119891(119909) allows the expansion in the followingseries

119891 (119909) = sum

119896

119888119896

119909119896

119890120574119909

(72)

In this case we refer to the identity

exp (1199101198632119909

) 119909119896

119890120574 119909

= 119890(120574 119909+120574

2119910)

119867119896

(119909 + 2120574119910 119910) (73)

which arises from

exp(119910 120597119898

120597119909119898)119891 (119909) = 119891(119909 + 119898119910

120597119898minus1

120597119909119898minus1) 1 (74)

with account for the generating function of Hermite polyno-mials (11) Then the action (67) of operator 119878 on the initialfunction (72) produces

119891 (119909 119905) = sum

119896

119888119896

119891119896

(119909 119905)

119891119896

(119909 119905) = 119890120574(119909+120574119886)

119867119896

(119909 + 2120574119886 119886) 119886 = 120572119905

(75)

6 The Scientific World Journal

In addition the translation operated by Θ shifts the argu-ment 119891(119909 119905) rarr 119891(119909 + 119886119887 119905) 119887 = 120573119905 Thus we obtain thesolution in a form of Hermite polynomial as follows

119865 (119909 119905) = sum

119896

119888119896

119890Φ+Φ

1119867119896

(119909 +1205721205742

120573(2119905120573

120574+ 1199052

1205732

1205742) 120572119905)

(76)

where Φ is defined by (64) and

Φ1

= 120574(119909 +1205721205742

120573(119905120573

120574+ (119905

120573

120574)

2

)) (77)

It is now evident that for a short time when 119905 ≪ 120574120573 thesolution will be spanning in space off the initial functionmodulated by 119867

119896

(119909 120572119905) and exponentially depending ontime

119865(119909 119905)|119905≪120574120573

asymp sum

119896

119888119896

exp (1205721205742119905 + 120574119909)119867119896

(119909 120572119905)

119865(119909 119905)|119905≪120574120573119905≪1

asymp sum

119896

119888119896

119890120572120574

2119905

119890120574119909

119909119896

(78)

Note that for extended times 119905 ≫ 120574120573 we have dominanceof time variable Φ ≫ Φ

1

and the solution asymptoticallybehaves as exp(120573119905119909) while119909 playsminor role in119867

119896

(119909+2120574120572119905+

1205721205731199052

120572119905)The same operational technique as employed for the

treatise of (54) can be easily adopted for the solution of theSchrodinger equation

119894ℏ120597119905

Ψ (119909 119905) = minusℏ2

21198981205972

119909

Ψ (119909 119905) + 119865119909Ψ (119909 119905)

Ψ (119909 0) = 119891 (119909)

(79)

where 119865 is constant (119865 has the dimension of force) Indeedrescaling variables in (79) we obtain the form of equationsimilar to (54)

119894120597120591

Ψ (119909 120591) = minus1205972

119909

Ψ (119909 120591) + 119887119909Ψ (119909 120591) (80)

where

120591 =ℏ119905

2119898 119887 =

2119865119898

ℏ2 (81)

Following the operational methodology developed for (54)we write the following solution of (80) in the form (56)

Ψ (119909 120591) =119880119891 (119909) (82)

where119880 = exp (119894120591) = 120597

2

119909

minus 119887119909 (83)

Then on account of the substitution 119905 rarr 119894120591 120573 rarr minus119887operators

Θ = exp (1198871205912120597

119909

) Θ119891 (119909 120591) = 119891 (119909 + 119887120591

2

120591) (84)

119878 = exp (1198941205911205972

119909

) 119878119891 (119909) = 119891 (119909 119894120591) (85)

ariseThus the solution of the Schrodinger equation is a resultof consequent action of the operator 119878 and further action ofΘ on the initial condition function

Ψ (119909 119905) = exp (minus119894Φ (119909 120591 119887)) Θ 119878119891 (119909) (86)

where Φ is defined by (64) The integral form of the solutionthen is written as follows

Ψ (119909 119905) = exp (minus119894Φ (119909 120591 119887)) 1

2radic119894120587120591

times int

infin

minusinfin

exp(minus(119909 + 119887120591

2

minus 120585)2

4119894120591)119891 (120585) 119889120585

(87)

Again as well as in (70) without any assumption onthe nature of the initial condition function 119891(119909) of theSchrodinger equation its solution

Ψ (119909 120591) = exp (minus119894Φ (119909 120591 119887)) 119891 (119909 + 1198871205912 119894120591) (88)

where 119891(119909 119905) is given by (85) and is expressed in terms ofthe function of two variables obtained by the consequentapplication of the heat propagation and translation operators(85) and (84) to119891(119909) So far we have demonstrated asGauss-Weierstrass transform (A1) describe the action of 119878on the initial probability amplitude and how the shift (84)finally yields the explicit form of the solution of Schrodingerequation It means that the result of the action of evolutionoperator (83) on 119891(119909) is the product of combined actionof translation operator Θ and heat propagation operator 119878representing the evolution operator of the free particle

Now let us consider another Fokker-Plank type equationthat is the following example

120597119905

119865 (119909 119905) = 1205721205972

119909

119865 (119909 119905) + 120573119909120597119909

119865 (119909 119905) (89)

with initial condition (55) Proceeding along the aboveoutlined scheme of the solution of (54) we write the solutionof (89) in general form (56)

119865 (119909 119905) = 119891 (119909) = exp (119860 + 119861) (90)

where operators 119860 and 119861 are defined as follows

119860 = 1199051205721205972

119909

119861 = 119905120573119909120597119909

(91)

Quantities 119860 and 119861 evidently do not commute

[119860 119861] = 21205721205731199052

1205972

119909

= 119898119860 119898 = 2119905120573 (92)

It allows disentanglement of the operators in the exponentialaccording to the following rule

exp (119860 + 119861) = exp(1 minus exp (minus119898)119898

119860) exp (119861) (93)

The Scientific World Journal 7

Thus the evolution operator action on 119891(119909) given below

119891 (119909) = exp (1205901205972119909

) exp (120573119905119909120597119909

) 119891 (119909)

= exp (1205901205972119909

) 119891 (119890120573119905

119909)

(94)

simply reduces to a Gauss transforms where the parameter120590 = 120590(119905) reads as follows

120590 (119905) =(1 minus exp (minus2120573119905)) 120572

120573 (95)

Upon the trivial change of variables we obtain

119891 (119909) = 119878119891 (119910) 119910 = 119909 exp (119887) 119878 = exp(120588

21205972

119910

)

(96)

where

120588 (119905) =120572

120573(1198902120573119905

minus 1) (97)

and eventually we end up with the following simple solutionof (89)

119865 (119909 119905) =1

radic2120587120588int

infin

minusinfin

119890minus(exp(120573119905)119909minus120585)22120588

119891 (120585) 119889120585 (98)

The same example of the initial Gaussian function 119891(119909) =exp(minus1199092) as in the case of (54) yields (compare with (71))

119865(119909 119905)|119891(119909)=exp(minus1199092) =

1

radic1 + 2120588 (119905)exp(minus 119890

2120573119905

1199092

1 + 2120588 (119905))

(99)

where 120588 is defined in (97) Note that we can meet thefollowing modified form of Fokker-Plank type equation (89)

120597119905

119865 (119909 119905) = 1205721205972

119909

119865 (119909 119905) + 120573120597119909

119909119865 (119909 119905) (100)

in problems related to propagation of electron beams inaccelerators Its solution arises from (99) immediately anddiffers from it just by a factor exp(120573119905) as written below fora Gaussian 119891(119909)

119865(119909 119905)|119891(119909)=exp(minus1199092) =

119890120573119905

radic1 + 2120588 (119905)exp(minus 119890

2120573119905

1199092

1 + 2120588 (119905))

=1

radic120578 (119905)exp(minus 119909

2

120578 (119905))

120578 (119905) = 2120572

120573(1 minus 119890

minus2120573119905

+120573

2120572119890minus2120573119905

)

(101)

However differently from the solution of (54) where we hadconsequent transforms of the initial condition function 119891(119909)by operators of translation and heat diffusion 119878 (63) (see also[15]) here we have just the action of 119878 alone with much morecomplicated dependence of the solution on time

5 Conclusions

Operational method is fast and universal mathematical toolfor obtaining solutions of differential equations Combina-tion of operational method integral transforms and theoryof special functions together with orthogonal polynomi-als closely related to them provides a powerful analyti-cal instrument for solving a wide spectrum of differentialequations and relevant physical problems The technique ofinverse operator applied for derivatives of various ordersand combined with integral transforms allows for easyand straightforward solutions of various types of differentialequations With operational approach we developed themethodology of inverse differential operators and deriveda number of operational identities with them We havedemonstrated that using the technique of inverse derivativesand inverse differential operators combinedwith exponentialoperator integral transforms and special functions we canmake significant progress in solution of variousmathematicalproblems and relevant physical applications described bydifferential equations

Appendix

In complete analogy with the heat equation solution byGauss-Weierstrass transform [16]

exp (1198871205972119909

) 119892 (119909) =1

2radic120587119887

int

infin

minusinfin

exp(minus(119909 minus 120585)2

4119887)119892 (120585) 119889120585

(A1)

accounting for noncommutative relation for operators ofinverse derivative 119863minus1

119909

and119871

119863119909

defined through the opera-tional relation (14) and accounting for (31) and (32) we writethe solution of (40) in the following form

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

exp(minus(119863minus1

119909

minus 120585)2

4119905) 1120593 (120585) 119889120585 (A2)

where 120593 is the image (32) of the initial condition function119892 The kernel of the integral in the above formula can beexpanded into series of two-variable Hermite polynomials119867119899

(119909 119910)

exp(minus(119863minus1

119909

minus 120585)2

4119905) 1

= 119890minus120585

24119905

infin

sum

119899=0

119863minus119899

119909

119899119867119899

(120585

2119905 minus1

4119905)

= 119890minus120585

24119905

infin

sum

119899=0

119909119899

(119899)2

119867119899

(120585

2119905 minus1

4119905)

(A3)

Taking into account formula (12) for 119909119899119867119899

(1205852119905 minus14119905) inthe operational identity above which can be viewed asa generating function in terms of inverse derivative for

8 The Scientific World Journal

119867119899

(119909 119910) we obtain the following expansion for the kernel ofthe integral

exp(minus(119863minus1

119909

minus 120585)

4119905) 1 = 119890minus120585

24119905

infin

sum

119899=0

1

(119899)2

119867119899

(119909120585

2119905 minus1199092

4119905)

(A4)

where the series of Hermite polynomials of two arguments119867119899

(119909 119910) can be expressed in terms of Hermite-Bessel-Tricomi functions

119867

119862119899

(119909 119910)mdashgeneralization of Bessel-Tricomi functionsmdashand related to Bessel-Wright functionsand to common Bessel functions [17]

119867

119862119899

(119909 119910) =

infin

sum

119898=0

119867119898

(minus119909 119910)

119903 (119899 + 119898) 119899 isin 119873

0

(A5)

In particular for 119899 = 0 we immediately find our series

119867

1198620

(119909 119910) =

infin

sum

119898=0

119867119898

(minus119909 119910)

(119898)2

(A6)

which obviously leads to the solution of the heat-typepropagation problem (40) by the following appropriate Gausstransform

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

119890minus120585

24119905

119867

1198620

(minus120585119909

2119905 minus1199092

4119905)120593 (120585) 119889120585

(A7)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] A Appel and J Kampe de Feriet Fonctions Hypergeometriqueset Hyperspheriques Polynomes drsquoHermite Gauthier-VillarsParis France 1926

[2] D T Haimo and C Markett ldquoA representation theory for solu-tions of a higher order heat equation Irdquo Journal ofMathematicalAnalysis and Applications vol 168 no 1 pp 89ndash107 1992

[3] D T Haimo and C Markett ldquoA representation theory forsolutions of a higher order heat equation IIrdquo Journal ofMathematical Analysis and Applications vol 168 no 2 pp 289ndash305 1992

[4] G Dattoli ldquoGeneralized polynomials operational identitiesand their applicationsrdquo Journal of Computational and AppliedMathematics vol 118 no 1-2 pp 111ndash123 2000

[5] G Dattoli H M Srivastava and K Zhukovsky ldquoOrthogonalityproperties of the Hermite and related polynomialsrdquo Journal ofComputational andAppliedMathematics vol 182 no 1 pp 165ndash172 2005

[6] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol 2 McGraw-Hill BookCompany New York NY USA 1953

[7] G Dattoli B GermanoM RMartinelli and P E Ricci ldquoNega-tive derivatives and special functionsrdquoAppliedMathematics andComputation vol 217 no 8 pp 3924ndash3928 2010

[8] G Dattoli M Migliorati and S Khan ldquoSolutions of integro-differential equations and operational methodsrdquo Applied Math-ematics and Computation vol 186 no 1 pp 302ndash308 2007

[9] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Mathematics and its Applications (Ellis HorwoodLtd) Halsted Press JohnWiley and Sons Chichester UK 1984

[10] GDattoli HM Srivastava andK Zhukovsky ldquoA new family ofintegral transforms and their applicationsrdquo Integral Transformsand Special Functions vol 17 no 1 pp 31ndash37 2006

[11] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 no 1 pp 1ndash174 1962

[12] G Dattoli H M Srivastava and K Zhukovsky ldquoOperationalmethods and differential equations with applications to initial-value problemsrdquo Applied Mathematics and Computation vol184 no 2 pp 979ndash1001 2007

[13] G Dattoli and K V Zhukovsky ldquoAppel polynomial seriesexpansionrdquo InternationalMathematical Forum vol 5 no 14 pp649ndash6662 2010

[14] G Dattoli M Migliorati and K Zhukovsky ldquoSummationformulae and stirling numbersrdquo International MathematicalForum vol 4 no 41 pp 2017ndash22040 2009

[15] J E Avron and I W Herbst ldquoSpectral and scattering theory ofSchrodinger operators related to the stark effectrdquo Communica-tions in Mathematical Physics vol 52 no 3 pp 239ndash254 1977

[16] K B Wolf Integral Transforms in Science and EngineeringPlenum Press New York NY USA 1979

[17] G N Watson A Treatise on the Theory of Bessel FunctionsCambridgeUniversity Press CambridgeUK 2nd edition 1944

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 The Scientific World Journal

can be complemented with the definition of the action of zeroorder derivative as follows

1198630

119909

119891 (119909) = 119891 (119909) (3)

so that evidently

119863minus119899

119909

1 = 119909119899

119899 (119899 isin 119873

0

= 119873 cup 0) (4)

In what follows we will appeal to various modifications of thefollowing equation

(1205732

minus 1198632

)]119865 (119909) = 119891 (119909) (5)

Thus it is important to construct the particular integral 119865(119909)with the help of the following operational identity (see eg[6])

119902minus]=1

Γ (])int

infin

0

exp (minus119902119905) 119905]minus1119889119905

min Re (119902) Re (]) gt 0(6)

For the operator 119902 = 1205732 minus 1198632 we have

(1205732

minus 1198632

)minus]119891 (119909)

=1

Γ (])int

infin

0

exp (minus1205732119905) 119905]minus1 exp (1199051198632) 119891 (119909) 119889119905(7)

We will also make explicit use of the generalized form ofthe Glaisher operational rule [9] action of the operator 119878 =exp(1199051198632

119909

) on the function 119891(119909) = exp(minus1199092) yields

119878119891 (119909) = exp(119910 1205972

1205971199092) exp (minus1199092)

=1

radic1 + 4119910exp(minus 119909

2

1 + 4119910)

(8)

Exponential operator mentioned above is closely related toHermite orthogonal polynomials

119867119899

(119909 119910) = 119899

[1198992]

sum

119903=0

119909119899minus2119903

119910119903

(119899 minus 2119903)119903 (9)

as demonstrated in [4 5 10] by operational relations

119867119899

(119909 119910) = exp(119910 1205972

1205971199092)119909119899

(10)

Moreover the following generating function for Hermitepolynomials exists

exp (119909119905 + 1199101199052) =infin

sum

119899=0

119905119899

119899119867119899

(119909 119910) (11)

Note also an easy to prove and useful relation [11]

119911119899

119867119899

(119909 119910) = 119867119899

(119909119911 1199101199112

) (12)

Laguerre polynomials of two variables [4]

119871119899

(119909 119910) = exp(minus119910 120597120597119909119909120597

120597119909)(minus119909)119899

119899= 119899

119899

sum

119903=0

(minus1)119903

119910119899minus119903

119909119903

(119899 minus 119903)(119903)2

(13)

are related to the following operator 120597119909

119909120597119909

[10]

119871

119863119909

=120597

120597119909119909120597

120597119909=

120597

120597119863minus1119909

(14)

sometimes called Laguerre derivative119871

119863119909

Note their non-commutative relation with the inverse derivative operator

[119871

119863119909

119863minus1

119909

] = minus1 ([119860 119861] = 119860119861 minus 119861119860) (15)

They represent solutions of the following partial differentialequation with proper initial conditions

120597119910

119871119899

(119909 119910) = minus (120597119909

119909120597119909

) 119871119899

(119909 119910) (16)

where

119871119899

(119909 119910) = 119899

119899

sum

119903=0

(minus1)119903

119910119899minus119903

119909119903

(119899 minus 119903)(119903)2

(17)

with proper initial conditions

119871119899

(119909 0) =(minus119909)119899

119899 (18)

In the following sections we will investigate the possi-bilities to solve some partial differential equations involvingthe differential operators studied above Now we just notehow the technique of inverse operator applied for derivativesof various orders and their combinations and combinedwith integral transforms allows for easy and straightforwardsolutions of various types of differential equations

3 Diffusion Type Heat Propagation TypeProblems and Inverse Derivatives

Heat propagation and diffusion type problems play a key rolein the theory of partial differential equations Combinationof exponential operator technique and inverse derivativetogether with the operational identities of the previoussection is useful for the solution of a broad spectrumof partialdifferential equations related to heat and diffusion processesSome of them have been already studied by operationalmethod (see eg [10 12]) Below we will focus on thegeneralities of the solution of the following problem

120597

120597119910119865 (119909 119910) = ( + ) 119865 (119909 119910) (19)

with initial conditions

119865 (119909 0) = 119902 (119909) (20)

The Scientific World Journal 3

We will employ operational approach combined with inte-gral transforms and exponential operator technique Formalsolution for our generic formulation reads as follows

119865 (119909 119910) = exp (119910 ( + )) 119902 (119909) (21)

We would like to underline that operators and may notcommute so dependently on the value of their commutatorwe will obtain different sequences of operators in (23)disentangling and [13 14] In the simplest case whenoperators and are multiplication and differentiationoperators respectively they can be easily disentangled in theexponent with account for

[ ] = 1 (22)

which yields

119865 (119909 119910) = exp (119910 ( + )) 119902 (119909)

= exp(minus1199102

2) exp (119910) exp (119910) 119902 (119909)

(23)

Explicit expressions for the action of these operators on theinitial condition function 119892(119909) can be obtained by integraltransforms series expansions and operational technique tobe evaluated in every special case This formulation simplein its essence nevertheless has wide application and allows usto frame either some of integrodifferential equations in thisscheme An elegant and interesting example is given by thefollowing equation

120597

120597119905119865 (119909 119905) = minus

120597

120597119909119909120597

120597119909119865 (119909 119905) + int

119909

0

119865 (120585 119905) 119889120585

119865 (119909 0) = 119892 (119909)

(24)

Its formal solution reads

119865 (119909 119905) = exp (119860 + 119861) 119892 (119909) (25)

where operators

119860 = minus119905119871

119863119909

= minus119905120597

120597119863minus1119909

119861 = 119905119863minus1

119909

(26)

do not commute

[119860 119861] = minus1199052

(27)

Evidently operators in the exponential disentangle

119865 (119909 119905) = exp (119860 + 119861) 119902 (119909)

= exp(1199052

2) exp (119905119863minus1

119909

) exp(minus119905 120597120597119863minus1119909

)119892 (119909)

(28)

Thus we have obtained the solution of the integrodifferen-tial equation (24) as a sequence of exponential operatorstransforming the initial condition 119892(119909) Our further steps

depend on the explicit form of this function In the mostgeneral case of 119892(119909) we may take advantage of the inversederivative technique First consider the action of the operatorexp(minus120597(120597119863minus1

119909

)) on 119892(119909)

119891 (119909 119905) = exp(minus119905 120597120597119863minus1119909

)119892 (119909) (29)

where119863minus1119909

is defined in (2) Equation (29) represents in factthe diffusion process and it is the solution of the followinginitial value problem

120597

120597119905119891 (119909 119905) = minus

120597

120597119909119909120597

120597119909119891 (119909 119905) 119891 (119909 0) = 119892 (119909) (30)

Relevant studies were performed in [10 12] The initialcondition function 119891(119909 0) = 119892(119909) can be written as follows

119892 (119909) = 120593 (119863minus1

119909

) 1 (31)

and the image 120593(119909) is explicitly given by the followingintegral

120593 (119909) = int

infin

0

exp (minus120577) 119892 (119909120577) 119889120577 (32)

which is supposed to converge Then the result of theLaguerre diffusion (29) appears in the form of the translationof the image function 120593

119891 (119909 119910) = 120593 (119863minus1

119909

minus 119905) 1 (33)

Consequently we have to apply the exponential operatorexp(119905119863minus1

119909

) which can be expanded in series

119865 (119909 119905) = exp(1199052

2)

infin

sum

119899=0

119905119899

119863minus119899

119909

119899119891 (119909 119905) (34)

The simplest example of the initial function 119892(119909) =exp(minus119909) demonstrates the technique sketched above result-ing in

120593 (119909) =1

(1 + 119909)when |119909| lt 1 (35)

and the Laguerre diffusion contribution (29) produces thefollowing function

119891 (119909 119905) =1

1 minus 119905exp (minus 1

1 minus 119905119909) (36)

The inverse derivative action on the exponent reads119863minus119899

119909

exp(120572119909) = exp(120572119909)120572119899 and eventually we obtain

119865 (119909 119905) = exp(1199052

2)1

1 minus 119905

infin

sum

119899=0

119905119899

119863minus119899

119909

119899exp (minus 1

1 minus 119905119909)

=1

1 minus 119905exp (minus 119909

1 minus 119905) exp(minus119905 (1 minus 119905) + 119905

2

2)

(37)

4 The Scientific World Journal

In conclusion of the present chapter we consider theexample of the solution of a heat propagation type equationby operational method involving the inverse derivativeoperator and exponential operator technique We recall thatthe common heat equation with initial condition problem

120597

120597119905119891 (119909 119905) = 120597

2

119909

119891 (119909 119905) 119891 (119909 0) = 119892 (119909) (38)

can be solved by Gauss transforms

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

exp(minus(119909 minus 120585)2

4119905) 119892 (120585) 119889120585 (39)

In complete analogy with the above statements we can solvethe heat-type equation with differential operator

119871

119863119909

(14)with for example the following initial condition

120597

120597119905119891 (119909 119905) =

119871

1198632

119909

119891 (119909 119905) 119891 (119909 0) = 119892 (119909) (40)

The Laguerre heat-type propagation problem (40) possessesthe following solution (see the Appendix)

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

119890minus120585

24119905

119867

1198620

(minus120585119909

2119905 minus1199092

4119905)120593 (120585) 119889120585

(41)

4 Operational Approach and Other Types ofDifferential Equations

Operational approach to solution of partial differential equa-tions demonstrated on the examples of diffusion-like andheat-like equations with 120597

119909

119909120597119909

derivatives can be furtherextended to other equation types Consider the followingexample of a rather complicated differential equation

1

120588

120597

120597119905119860 (119909 119905) = 119909

2

1205972

1205971199092119860 (119909 119905) + 120582119909

120597

120597119909119860 (119909 119905)

minus 120583119860 (119909 119905) 119892 (119909) = 119860 (119909 0)

(42)

where 120588 120582 and 120583 are some arbitrary constant coefficientsand function 119892(119909) = 119860(119909 119905 = 0) is the initial condition Byintroducing operator119863 = 119909120597

119909

and distinguishing the perfectsquare we rewrite (42)

1

120588

120597

120597 119905119860 (119909 119905) = ((119863 +

120582

2)

2

minus 120576)119860 (119909 119905) (43)

where

120576 = 120583 + (120582

2)

2

(44)

Thus the following exponential solution for (42) appears

119860 (119909 119905) = exp120588119905((119863 + 1205822)

2

minus 120576)119892 (119909) (45)

Now making use of the operational identity

exp (1199012) = 1

radic120587int

infin

minusinfin

exp (minus1205852 + 2120585119901) 119889120585 (46)

and applying exp(119886119863) according to

exp (119886119909120597119909

) 119891 (119909) = 119891 (119890119886

119909) (47)

we obtain the following compact expression for 119860(119909 119905)

119860 (119909 119905) =exp (minus120588120576119905)radic120587

times int

infin

minusinfin

exp [minus1205902 + 120590120572 1205822120588] 119892 (119909 exp (120590120572)) 119889120590

(48)

where

120572 = 120572 (119905) = 2radic120588119905 (49)

Consider two simple examples of initial condition functionsThe first one is

119892 (119909) = 119909119899

(50)

Then we immediately obtain the solution of (42) as follows

119860 (119909 119905) = 119909119899 exp 120588119905 (1198992 + 120582119899 minus 120583) (51)

The second example is given by the following initial conditionfunction

119892 (119909) = ln119909 (52)

Trivial computations yield the following solution

119860 (119909 119905) = (ln119909 + 120588119905120582) exp (minus120588119905120583) (53)

Thus operational technique combined with integraltransforms operational identities and extended forms oforthogonal polynomials represents powerful tool for findingsolutions of various classes of differential equations and initialvalue problems Note that within the framework of inversedifferential operators developed and described above theusage of the evolution operator method opens new possibili-ties which we will elucidate in what follows

Let us consider the following generalization of the heatequation

120597119905

119865 (119909 119905) = 1205721205972

119909

119865 (119909 119905) + 120573119909119865 (119909 119905) (54)

with the initial condition

119865 (119909 0) = 119891 (119909) (55)

The evolution type equation (54) contains linear coordinateterm in addition to the second order derivative Its formalsolution can be written via the evolution operator

119865 (119909 119905) = 119891 (119909) (56)

The Scientific World Journal 5

where

= exp (119860 + 119861) 119860 = 1205721199051205972

119909

119861 = 120573119905119909 (57)

The exponential of the evolution operator in (56) is thesum of two noncommuting operators and it can be writtenas the ordered product of two exponential operators Indeedthe commutator of 119860 and 119861 has the following nonzero value

[119860 119861] = 11989811986012

= 21205721205731199052

120597119909

119898 = 212057311990532

12057212

(58)

Then we can apply the disentanglement operational identity

119890

119860+

119861

= 119890(119898

212)minus(1198982)

119860

12+

119860

119890

119861

(59)

and the following chain rule

119890119901120597

2

119909119890119902119909

119892 (119909) = 119890119901119902

2

119890119902119909

1198902119901119902120597

119909119890119901120597

2

119909119892 (119909) (60)

where119901 119902 are constant parametersWith their help we obtainthe evolution operator for (54)

= 119890Φ(119909119905120573)

Θ119878 (61)

Note that we have factorized two commuting operators theoperator of translation in space

Θ = 119890120572120573119905

2120597

119909 (62)

and operator

119878 = 119890120572119905120597

2

119909 (63)

which is in fact operator 119878 = exp(1199051198632119909

) The phase in iswritten as follows

Φ(119909 119905 120572 120573) =1

31205721205732

1199053

+ 120573119905119909 (64)

The action of on the initial condition function 119891(119909) yieldsthe following solution for our problem

119865 (119909 119905) = 119890Φ(119909119905120572120573)

Θ119878119891 (119909) (65)

Thus we conclude from the form of (65) that the problem(54) with the initial condition (55) can be solved by theconsequent application of commuting operators Θ (62) and 119878(63) to 119891(119909) apart from the factor 119890Φ(119909119905120572120573) Now the explicitform of the solution (65) can be obtained by recalling that Θacts as a translation operator

119890119904120597

119909119892 (119909) = 119892 (119909 + 119904) (66)

and that the action of 119878 on the function 119892(119909) yields thesolution of the ordinary heat equation through Gauss-Weierstrass transform (A1) Accordingly we denote

119891 (119909 119905) equiv 119878119891 (119909) equiv 119890120572119905120597

2

119909119891 (119909) (67)

and we write

Θ119891 (119909 119905) = 119891 (119909 + 1205721205731199052

119905) (68)

Thus (54)with initial condition (55) has the following explicitsolution119865 (119909 119905)

= 119890Φ(119909119905120572120573)

1

2radic120587120572119905int

infin

minusinfin

119890minus(119909+120572120573119905

2minus120585)

2

4119905120572

119891 (120585) 119889120585

(69)

provided that the integral converges Summarizing the aboveoutlined procedure we conclude that a solution for (54)consists in finding a Gauss transformed function 119891 with ashifted argument

119865 (119909 119905) = 119890Φ(119909119905120572120573)

119891 (119909 + 1205721205731199052

119905) (70)

Moreover this is a general observation for this type ofequation valid for any function 119891(119909) (provided the integralconverges) In other words we have obtained the solution ofFokker plank equation as a consequent action of operator 119878of heat diffusion and operator Θ of translation on the initialcondition functionNote that119891(119909 119905) is the solution of the heatequation representing a natural propagation phenomenon

The effect produced by the translation operator Θ and theoperator 119878 is best illustrated with the example of Gaussian119891(119909) = exp(minus1199092) evolution when (69) becomes

119865(119909 119905)|119891(119909)=exp(minus1199092) =

exp (Φ (119909 119905))radic1 + 4119905

exp(minus(119909 + 120573119905

2

)2

1 + 4119905)

119905 ge 0

(71)

The above result is exactly the generalization of Gleisherrule (8) considered earlier in the context of the heat equationThus (71) is the solution of the ordinary heat equation with120573 = 0 when the initial function is Gaussian

Another interesting example of solving (54) appearswhenthe initial function119891(119909) allows the expansion in the followingseries

119891 (119909) = sum

119896

119888119896

119909119896

119890120574119909

(72)

In this case we refer to the identity

exp (1199101198632119909

) 119909119896

119890120574 119909

= 119890(120574 119909+120574

2119910)

119867119896

(119909 + 2120574119910 119910) (73)

which arises from

exp(119910 120597119898

120597119909119898)119891 (119909) = 119891(119909 + 119898119910

120597119898minus1

120597119909119898minus1) 1 (74)

with account for the generating function of Hermite polyno-mials (11) Then the action (67) of operator 119878 on the initialfunction (72) produces

119891 (119909 119905) = sum

119896

119888119896

119891119896

(119909 119905)

119891119896

(119909 119905) = 119890120574(119909+120574119886)

119867119896

(119909 + 2120574119886 119886) 119886 = 120572119905

(75)

6 The Scientific World Journal

In addition the translation operated by Θ shifts the argu-ment 119891(119909 119905) rarr 119891(119909 + 119886119887 119905) 119887 = 120573119905 Thus we obtain thesolution in a form of Hermite polynomial as follows

119865 (119909 119905) = sum

119896

119888119896

119890Φ+Φ

1119867119896

(119909 +1205721205742

120573(2119905120573

120574+ 1199052

1205732

1205742) 120572119905)

(76)

where Φ is defined by (64) and

Φ1

= 120574(119909 +1205721205742

120573(119905120573

120574+ (119905

120573

120574)

2

)) (77)

It is now evident that for a short time when 119905 ≪ 120574120573 thesolution will be spanning in space off the initial functionmodulated by 119867

119896

(119909 120572119905) and exponentially depending ontime

119865(119909 119905)|119905≪120574120573

asymp sum

119896

119888119896

exp (1205721205742119905 + 120574119909)119867119896

(119909 120572119905)

119865(119909 119905)|119905≪120574120573119905≪1

asymp sum

119896

119888119896

119890120572120574

2119905

119890120574119909

119909119896

(78)

Note that for extended times 119905 ≫ 120574120573 we have dominanceof time variable Φ ≫ Φ

1

and the solution asymptoticallybehaves as exp(120573119905119909) while119909 playsminor role in119867

119896

(119909+2120574120572119905+

1205721205731199052

120572119905)The same operational technique as employed for the

treatise of (54) can be easily adopted for the solution of theSchrodinger equation

119894ℏ120597119905

Ψ (119909 119905) = minusℏ2

21198981205972

119909

Ψ (119909 119905) + 119865119909Ψ (119909 119905)

Ψ (119909 0) = 119891 (119909)

(79)

where 119865 is constant (119865 has the dimension of force) Indeedrescaling variables in (79) we obtain the form of equationsimilar to (54)

119894120597120591

Ψ (119909 120591) = minus1205972

119909

Ψ (119909 120591) + 119887119909Ψ (119909 120591) (80)

where

120591 =ℏ119905

2119898 119887 =

2119865119898

ℏ2 (81)

Following the operational methodology developed for (54)we write the following solution of (80) in the form (56)

Ψ (119909 120591) =119880119891 (119909) (82)

where119880 = exp (119894120591) = 120597

2

119909

minus 119887119909 (83)

Then on account of the substitution 119905 rarr 119894120591 120573 rarr minus119887operators

Θ = exp (1198871205912120597

119909

) Θ119891 (119909 120591) = 119891 (119909 + 119887120591

2

120591) (84)

119878 = exp (1198941205911205972

119909

) 119878119891 (119909) = 119891 (119909 119894120591) (85)

ariseThus the solution of the Schrodinger equation is a resultof consequent action of the operator 119878 and further action ofΘ on the initial condition function

Ψ (119909 119905) = exp (minus119894Φ (119909 120591 119887)) Θ 119878119891 (119909) (86)

where Φ is defined by (64) The integral form of the solutionthen is written as follows

Ψ (119909 119905) = exp (minus119894Φ (119909 120591 119887)) 1

2radic119894120587120591

times int

infin

minusinfin

exp(minus(119909 + 119887120591

2

minus 120585)2

4119894120591)119891 (120585) 119889120585

(87)

Again as well as in (70) without any assumption onthe nature of the initial condition function 119891(119909) of theSchrodinger equation its solution

Ψ (119909 120591) = exp (minus119894Φ (119909 120591 119887)) 119891 (119909 + 1198871205912 119894120591) (88)

where 119891(119909 119905) is given by (85) and is expressed in terms ofthe function of two variables obtained by the consequentapplication of the heat propagation and translation operators(85) and (84) to119891(119909) So far we have demonstrated asGauss-Weierstrass transform (A1) describe the action of 119878on the initial probability amplitude and how the shift (84)finally yields the explicit form of the solution of Schrodingerequation It means that the result of the action of evolutionoperator (83) on 119891(119909) is the product of combined actionof translation operator Θ and heat propagation operator 119878representing the evolution operator of the free particle

Now let us consider another Fokker-Plank type equationthat is the following example

120597119905

119865 (119909 119905) = 1205721205972

119909

119865 (119909 119905) + 120573119909120597119909

119865 (119909 119905) (89)

with initial condition (55) Proceeding along the aboveoutlined scheme of the solution of (54) we write the solutionof (89) in general form (56)

119865 (119909 119905) = 119891 (119909) = exp (119860 + 119861) (90)

where operators 119860 and 119861 are defined as follows

119860 = 1199051205721205972

119909

119861 = 119905120573119909120597119909

(91)

Quantities 119860 and 119861 evidently do not commute

[119860 119861] = 21205721205731199052

1205972

119909

= 119898119860 119898 = 2119905120573 (92)

It allows disentanglement of the operators in the exponentialaccording to the following rule

exp (119860 + 119861) = exp(1 minus exp (minus119898)119898

119860) exp (119861) (93)

The Scientific World Journal 7

Thus the evolution operator action on 119891(119909) given below

119891 (119909) = exp (1205901205972119909

) exp (120573119905119909120597119909

) 119891 (119909)

= exp (1205901205972119909

) 119891 (119890120573119905

119909)

(94)

simply reduces to a Gauss transforms where the parameter120590 = 120590(119905) reads as follows

120590 (119905) =(1 minus exp (minus2120573119905)) 120572

120573 (95)

Upon the trivial change of variables we obtain

119891 (119909) = 119878119891 (119910) 119910 = 119909 exp (119887) 119878 = exp(120588

21205972

119910

)

(96)

where

120588 (119905) =120572

120573(1198902120573119905

minus 1) (97)

and eventually we end up with the following simple solutionof (89)

119865 (119909 119905) =1

radic2120587120588int

infin

minusinfin

119890minus(exp(120573119905)119909minus120585)22120588

119891 (120585) 119889120585 (98)

The same example of the initial Gaussian function 119891(119909) =exp(minus1199092) as in the case of (54) yields (compare with (71))

119865(119909 119905)|119891(119909)=exp(minus1199092) =

1

radic1 + 2120588 (119905)exp(minus 119890

2120573119905

1199092

1 + 2120588 (119905))

(99)

where 120588 is defined in (97) Note that we can meet thefollowing modified form of Fokker-Plank type equation (89)

120597119905

119865 (119909 119905) = 1205721205972

119909

119865 (119909 119905) + 120573120597119909

119909119865 (119909 119905) (100)

in problems related to propagation of electron beams inaccelerators Its solution arises from (99) immediately anddiffers from it just by a factor exp(120573119905) as written below fora Gaussian 119891(119909)

119865(119909 119905)|119891(119909)=exp(minus1199092) =

119890120573119905

radic1 + 2120588 (119905)exp(minus 119890

2120573119905

1199092

1 + 2120588 (119905))

=1

radic120578 (119905)exp(minus 119909

2

120578 (119905))

120578 (119905) = 2120572

120573(1 minus 119890

minus2120573119905

+120573

2120572119890minus2120573119905

)

(101)

However differently from the solution of (54) where we hadconsequent transforms of the initial condition function 119891(119909)by operators of translation and heat diffusion 119878 (63) (see also[15]) here we have just the action of 119878 alone with much morecomplicated dependence of the solution on time

5 Conclusions

Operational method is fast and universal mathematical toolfor obtaining solutions of differential equations Combina-tion of operational method integral transforms and theoryof special functions together with orthogonal polynomi-als closely related to them provides a powerful analyti-cal instrument for solving a wide spectrum of differentialequations and relevant physical problems The technique ofinverse operator applied for derivatives of various ordersand combined with integral transforms allows for easyand straightforward solutions of various types of differentialequations With operational approach we developed themethodology of inverse differential operators and deriveda number of operational identities with them We havedemonstrated that using the technique of inverse derivativesand inverse differential operators combinedwith exponentialoperator integral transforms and special functions we canmake significant progress in solution of variousmathematicalproblems and relevant physical applications described bydifferential equations

Appendix

In complete analogy with the heat equation solution byGauss-Weierstrass transform [16]

exp (1198871205972119909

) 119892 (119909) =1

2radic120587119887

int

infin

minusinfin

exp(minus(119909 minus 120585)2

4119887)119892 (120585) 119889120585

(A1)

accounting for noncommutative relation for operators ofinverse derivative 119863minus1

119909

and119871

119863119909

defined through the opera-tional relation (14) and accounting for (31) and (32) we writethe solution of (40) in the following form

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

exp(minus(119863minus1

119909

minus 120585)2

4119905) 1120593 (120585) 119889120585 (A2)

where 120593 is the image (32) of the initial condition function119892 The kernel of the integral in the above formula can beexpanded into series of two-variable Hermite polynomials119867119899

(119909 119910)

exp(minus(119863minus1

119909

minus 120585)2

4119905) 1

= 119890minus120585

24119905

infin

sum

119899=0

119863minus119899

119909

119899119867119899

(120585

2119905 minus1

4119905)

= 119890minus120585

24119905

infin

sum

119899=0

119909119899

(119899)2

119867119899

(120585

2119905 minus1

4119905)

(A3)

Taking into account formula (12) for 119909119899119867119899

(1205852119905 minus14119905) inthe operational identity above which can be viewed asa generating function in terms of inverse derivative for

8 The Scientific World Journal

119867119899

(119909 119910) we obtain the following expansion for the kernel ofthe integral

exp(minus(119863minus1

119909

minus 120585)

4119905) 1 = 119890minus120585

24119905

infin

sum

119899=0

1

(119899)2

119867119899

(119909120585

2119905 minus1199092

4119905)

(A4)

where the series of Hermite polynomials of two arguments119867119899

(119909 119910) can be expressed in terms of Hermite-Bessel-Tricomi functions

119867

119862119899

(119909 119910)mdashgeneralization of Bessel-Tricomi functionsmdashand related to Bessel-Wright functionsand to common Bessel functions [17]

119867

119862119899

(119909 119910) =

infin

sum

119898=0

119867119898

(minus119909 119910)

119903 (119899 + 119898) 119899 isin 119873

0

(A5)

In particular for 119899 = 0 we immediately find our series

119867

1198620

(119909 119910) =

infin

sum

119898=0

119867119898

(minus119909 119910)

(119898)2

(A6)

which obviously leads to the solution of the heat-typepropagation problem (40) by the following appropriate Gausstransform

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

119890minus120585

24119905

119867

1198620

(minus120585119909

2119905 minus1199092

4119905)120593 (120585) 119889120585

(A7)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] A Appel and J Kampe de Feriet Fonctions Hypergeometriqueset Hyperspheriques Polynomes drsquoHermite Gauthier-VillarsParis France 1926

[2] D T Haimo and C Markett ldquoA representation theory for solu-tions of a higher order heat equation Irdquo Journal ofMathematicalAnalysis and Applications vol 168 no 1 pp 89ndash107 1992

[3] D T Haimo and C Markett ldquoA representation theory forsolutions of a higher order heat equation IIrdquo Journal ofMathematical Analysis and Applications vol 168 no 2 pp 289ndash305 1992

[4] G Dattoli ldquoGeneralized polynomials operational identitiesand their applicationsrdquo Journal of Computational and AppliedMathematics vol 118 no 1-2 pp 111ndash123 2000

[5] G Dattoli H M Srivastava and K Zhukovsky ldquoOrthogonalityproperties of the Hermite and related polynomialsrdquo Journal ofComputational andAppliedMathematics vol 182 no 1 pp 165ndash172 2005

[6] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol 2 McGraw-Hill BookCompany New York NY USA 1953

[7] G Dattoli B GermanoM RMartinelli and P E Ricci ldquoNega-tive derivatives and special functionsrdquoAppliedMathematics andComputation vol 217 no 8 pp 3924ndash3928 2010

[8] G Dattoli M Migliorati and S Khan ldquoSolutions of integro-differential equations and operational methodsrdquo Applied Math-ematics and Computation vol 186 no 1 pp 302ndash308 2007

[9] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Mathematics and its Applications (Ellis HorwoodLtd) Halsted Press JohnWiley and Sons Chichester UK 1984

[10] GDattoli HM Srivastava andK Zhukovsky ldquoA new family ofintegral transforms and their applicationsrdquo Integral Transformsand Special Functions vol 17 no 1 pp 31ndash37 2006

[11] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 no 1 pp 1ndash174 1962

[12] G Dattoli H M Srivastava and K Zhukovsky ldquoOperationalmethods and differential equations with applications to initial-value problemsrdquo Applied Mathematics and Computation vol184 no 2 pp 979ndash1001 2007

[13] G Dattoli and K V Zhukovsky ldquoAppel polynomial seriesexpansionrdquo InternationalMathematical Forum vol 5 no 14 pp649ndash6662 2010

[14] G Dattoli M Migliorati and K Zhukovsky ldquoSummationformulae and stirling numbersrdquo International MathematicalForum vol 4 no 41 pp 2017ndash22040 2009

[15] J E Avron and I W Herbst ldquoSpectral and scattering theory ofSchrodinger operators related to the stark effectrdquo Communica-tions in Mathematical Physics vol 52 no 3 pp 239ndash254 1977

[16] K B Wolf Integral Transforms in Science and EngineeringPlenum Press New York NY USA 1979

[17] G N Watson A Treatise on the Theory of Bessel FunctionsCambridgeUniversity Press CambridgeUK 2nd edition 1944

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Stochastic AnalysisInternational Journal of

The Scientific World Journal 3

We will employ operational approach combined with inte-gral transforms and exponential operator technique Formalsolution for our generic formulation reads as follows

119865 (119909 119910) = exp (119910 ( + )) 119902 (119909) (21)

We would like to underline that operators and may notcommute so dependently on the value of their commutatorwe will obtain different sequences of operators in (23)disentangling and [13 14] In the simplest case whenoperators and are multiplication and differentiationoperators respectively they can be easily disentangled in theexponent with account for

[ ] = 1 (22)

which yields

119865 (119909 119910) = exp (119910 ( + )) 119902 (119909)

= exp(minus1199102

2) exp (119910) exp (119910) 119902 (119909)

(23)

Explicit expressions for the action of these operators on theinitial condition function 119892(119909) can be obtained by integraltransforms series expansions and operational technique tobe evaluated in every special case This formulation simplein its essence nevertheless has wide application and allows usto frame either some of integrodifferential equations in thisscheme An elegant and interesting example is given by thefollowing equation

120597

120597119905119865 (119909 119905) = minus

120597

120597119909119909120597

120597119909119865 (119909 119905) + int

119909

0

119865 (120585 119905) 119889120585

119865 (119909 0) = 119892 (119909)

(24)

Its formal solution reads

119865 (119909 119905) = exp (119860 + 119861) 119892 (119909) (25)

where operators

119860 = minus119905119871

119863119909

= minus119905120597

120597119863minus1119909

119861 = 119905119863minus1

119909

(26)

do not commute

[119860 119861] = minus1199052

(27)

Evidently operators in the exponential disentangle

119865 (119909 119905) = exp (119860 + 119861) 119902 (119909)

= exp(1199052

2) exp (119905119863minus1

119909

) exp(minus119905 120597120597119863minus1119909

)119892 (119909)

(28)

Thus we have obtained the solution of the integrodifferen-tial equation (24) as a sequence of exponential operatorstransforming the initial condition 119892(119909) Our further steps

depend on the explicit form of this function In the mostgeneral case of 119892(119909) we may take advantage of the inversederivative technique First consider the action of the operatorexp(minus120597(120597119863minus1

119909

)) on 119892(119909)

119891 (119909 119905) = exp(minus119905 120597120597119863minus1119909

)119892 (119909) (29)

where119863minus1119909

is defined in (2) Equation (29) represents in factthe diffusion process and it is the solution of the followinginitial value problem

120597

120597119905119891 (119909 119905) = minus

120597

120597119909119909120597

120597119909119891 (119909 119905) 119891 (119909 0) = 119892 (119909) (30)

Relevant studies were performed in [10 12] The initialcondition function 119891(119909 0) = 119892(119909) can be written as follows

119892 (119909) = 120593 (119863minus1

119909

) 1 (31)

and the image 120593(119909) is explicitly given by the followingintegral

120593 (119909) = int

infin

0

exp (minus120577) 119892 (119909120577) 119889120577 (32)

which is supposed to converge Then the result of theLaguerre diffusion (29) appears in the form of the translationof the image function 120593

119891 (119909 119910) = 120593 (119863minus1

119909

minus 119905) 1 (33)

Consequently we have to apply the exponential operatorexp(119905119863minus1

119909

) which can be expanded in series

119865 (119909 119905) = exp(1199052

2)

infin

sum

119899=0

119905119899

119863minus119899

119909

119899119891 (119909 119905) (34)

The simplest example of the initial function 119892(119909) =exp(minus119909) demonstrates the technique sketched above result-ing in

120593 (119909) =1

(1 + 119909)when |119909| lt 1 (35)

and the Laguerre diffusion contribution (29) produces thefollowing function

119891 (119909 119905) =1

1 minus 119905exp (minus 1

1 minus 119905119909) (36)

The inverse derivative action on the exponent reads119863minus119899

119909

exp(120572119909) = exp(120572119909)120572119899 and eventually we obtain

119865 (119909 119905) = exp(1199052

2)1

1 minus 119905

infin

sum

119899=0

119905119899

119863minus119899

119909

119899exp (minus 1

1 minus 119905119909)

=1

1 minus 119905exp (minus 119909

1 minus 119905) exp(minus119905 (1 minus 119905) + 119905

2

2)

(37)

4 The Scientific World Journal

In conclusion of the present chapter we consider theexample of the solution of a heat propagation type equationby operational method involving the inverse derivativeoperator and exponential operator technique We recall thatthe common heat equation with initial condition problem

120597

120597119905119891 (119909 119905) = 120597

2

119909

119891 (119909 119905) 119891 (119909 0) = 119892 (119909) (38)

can be solved by Gauss transforms

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

exp(minus(119909 minus 120585)2

4119905) 119892 (120585) 119889120585 (39)

In complete analogy with the above statements we can solvethe heat-type equation with differential operator

119871

119863119909

(14)with for example the following initial condition

120597

120597119905119891 (119909 119905) =

119871

1198632

119909

119891 (119909 119905) 119891 (119909 0) = 119892 (119909) (40)

The Laguerre heat-type propagation problem (40) possessesthe following solution (see the Appendix)

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

119890minus120585

24119905

119867

1198620

(minus120585119909

2119905 minus1199092

4119905)120593 (120585) 119889120585

(41)

4 Operational Approach and Other Types ofDifferential Equations

Operational approach to solution of partial differential equa-tions demonstrated on the examples of diffusion-like andheat-like equations with 120597

119909

119909120597119909

derivatives can be furtherextended to other equation types Consider the followingexample of a rather complicated differential equation

1

120588

120597

120597119905119860 (119909 119905) = 119909

2

1205972

1205971199092119860 (119909 119905) + 120582119909

120597

120597119909119860 (119909 119905)

minus 120583119860 (119909 119905) 119892 (119909) = 119860 (119909 0)

(42)

where 120588 120582 and 120583 are some arbitrary constant coefficientsand function 119892(119909) = 119860(119909 119905 = 0) is the initial condition Byintroducing operator119863 = 119909120597

119909

and distinguishing the perfectsquare we rewrite (42)

1

120588

120597

120597 119905119860 (119909 119905) = ((119863 +

120582

2)

2

minus 120576)119860 (119909 119905) (43)

where

120576 = 120583 + (120582

2)

2

(44)

Thus the following exponential solution for (42) appears

119860 (119909 119905) = exp120588119905((119863 + 1205822)

2

minus 120576)119892 (119909) (45)

Now making use of the operational identity

exp (1199012) = 1

radic120587int

infin

minusinfin

exp (minus1205852 + 2120585119901) 119889120585 (46)

and applying exp(119886119863) according to

exp (119886119909120597119909

) 119891 (119909) = 119891 (119890119886

119909) (47)

we obtain the following compact expression for 119860(119909 119905)

119860 (119909 119905) =exp (minus120588120576119905)radic120587

times int

infin

minusinfin

exp [minus1205902 + 120590120572 1205822120588] 119892 (119909 exp (120590120572)) 119889120590

(48)

where

120572 = 120572 (119905) = 2radic120588119905 (49)

Consider two simple examples of initial condition functionsThe first one is

119892 (119909) = 119909119899

(50)

Then we immediately obtain the solution of (42) as follows

119860 (119909 119905) = 119909119899 exp 120588119905 (1198992 + 120582119899 minus 120583) (51)

The second example is given by the following initial conditionfunction

119892 (119909) = ln119909 (52)

Trivial computations yield the following solution

119860 (119909 119905) = (ln119909 + 120588119905120582) exp (minus120588119905120583) (53)

Thus operational technique combined with integraltransforms operational identities and extended forms oforthogonal polynomials represents powerful tool for findingsolutions of various classes of differential equations and initialvalue problems Note that within the framework of inversedifferential operators developed and described above theusage of the evolution operator method opens new possibili-ties which we will elucidate in what follows

Let us consider the following generalization of the heatequation

120597119905

119865 (119909 119905) = 1205721205972

119909

119865 (119909 119905) + 120573119909119865 (119909 119905) (54)

with the initial condition

119865 (119909 0) = 119891 (119909) (55)

The evolution type equation (54) contains linear coordinateterm in addition to the second order derivative Its formalsolution can be written via the evolution operator

119865 (119909 119905) = 119891 (119909) (56)

The Scientific World Journal 5

where

= exp (119860 + 119861) 119860 = 1205721199051205972

119909

119861 = 120573119905119909 (57)

The exponential of the evolution operator in (56) is thesum of two noncommuting operators and it can be writtenas the ordered product of two exponential operators Indeedthe commutator of 119860 and 119861 has the following nonzero value

[119860 119861] = 11989811986012

= 21205721205731199052

120597119909

119898 = 212057311990532

12057212

(58)

Then we can apply the disentanglement operational identity

119890

119860+

119861

= 119890(119898

212)minus(1198982)

119860

12+

119860

119890

119861

(59)

and the following chain rule

119890119901120597

2

119909119890119902119909

119892 (119909) = 119890119901119902

2

119890119902119909

1198902119901119902120597

119909119890119901120597

2

119909119892 (119909) (60)

where119901 119902 are constant parametersWith their help we obtainthe evolution operator for (54)

= 119890Φ(119909119905120573)

Θ119878 (61)

Note that we have factorized two commuting operators theoperator of translation in space

Θ = 119890120572120573119905

2120597

119909 (62)

and operator

119878 = 119890120572119905120597

2

119909 (63)

which is in fact operator 119878 = exp(1199051198632119909

) The phase in iswritten as follows

Φ(119909 119905 120572 120573) =1

31205721205732

1199053

+ 120573119905119909 (64)

The action of on the initial condition function 119891(119909) yieldsthe following solution for our problem

119865 (119909 119905) = 119890Φ(119909119905120572120573)

Θ119878119891 (119909) (65)

Thus we conclude from the form of (65) that the problem(54) with the initial condition (55) can be solved by theconsequent application of commuting operators Θ (62) and 119878(63) to 119891(119909) apart from the factor 119890Φ(119909119905120572120573) Now the explicitform of the solution (65) can be obtained by recalling that Θacts as a translation operator

119890119904120597

119909119892 (119909) = 119892 (119909 + 119904) (66)

and that the action of 119878 on the function 119892(119909) yields thesolution of the ordinary heat equation through Gauss-Weierstrass transform (A1) Accordingly we denote

119891 (119909 119905) equiv 119878119891 (119909) equiv 119890120572119905120597

2

119909119891 (119909) (67)

and we write

Θ119891 (119909 119905) = 119891 (119909 + 1205721205731199052

119905) (68)

Thus (54)with initial condition (55) has the following explicitsolution119865 (119909 119905)

= 119890Φ(119909119905120572120573)

1

2radic120587120572119905int

infin

minusinfin

119890minus(119909+120572120573119905

2minus120585)

2

4119905120572

119891 (120585) 119889120585

(69)

provided that the integral converges Summarizing the aboveoutlined procedure we conclude that a solution for (54)consists in finding a Gauss transformed function 119891 with ashifted argument

119865 (119909 119905) = 119890Φ(119909119905120572120573)

119891 (119909 + 1205721205731199052

119905) (70)

Moreover this is a general observation for this type ofequation valid for any function 119891(119909) (provided the integralconverges) In other words we have obtained the solution ofFokker plank equation as a consequent action of operator 119878of heat diffusion and operator Θ of translation on the initialcondition functionNote that119891(119909 119905) is the solution of the heatequation representing a natural propagation phenomenon

The effect produced by the translation operator Θ and theoperator 119878 is best illustrated with the example of Gaussian119891(119909) = exp(minus1199092) evolution when (69) becomes

119865(119909 119905)|119891(119909)=exp(minus1199092) =

exp (Φ (119909 119905))radic1 + 4119905

exp(minus(119909 + 120573119905

2

)2

1 + 4119905)

119905 ge 0

(71)

The above result is exactly the generalization of Gleisherrule (8) considered earlier in the context of the heat equationThus (71) is the solution of the ordinary heat equation with120573 = 0 when the initial function is Gaussian

Another interesting example of solving (54) appearswhenthe initial function119891(119909) allows the expansion in the followingseries

119891 (119909) = sum

119896

119888119896

119909119896

119890120574119909

(72)

In this case we refer to the identity

exp (1199101198632119909

) 119909119896

119890120574 119909

= 119890(120574 119909+120574

2119910)

119867119896

(119909 + 2120574119910 119910) (73)

which arises from

exp(119910 120597119898

120597119909119898)119891 (119909) = 119891(119909 + 119898119910

120597119898minus1

120597119909119898minus1) 1 (74)

with account for the generating function of Hermite polyno-mials (11) Then the action (67) of operator 119878 on the initialfunction (72) produces

119891 (119909 119905) = sum

119896

119888119896

119891119896

(119909 119905)

119891119896

(119909 119905) = 119890120574(119909+120574119886)

119867119896

(119909 + 2120574119886 119886) 119886 = 120572119905

(75)

6 The Scientific World Journal

In addition the translation operated by Θ shifts the argu-ment 119891(119909 119905) rarr 119891(119909 + 119886119887 119905) 119887 = 120573119905 Thus we obtain thesolution in a form of Hermite polynomial as follows

119865 (119909 119905) = sum

119896

119888119896

119890Φ+Φ

1119867119896

(119909 +1205721205742

120573(2119905120573

120574+ 1199052

1205732

1205742) 120572119905)

(76)

where Φ is defined by (64) and

Φ1

= 120574(119909 +1205721205742

120573(119905120573

120574+ (119905

120573

120574)

2

)) (77)

It is now evident that for a short time when 119905 ≪ 120574120573 thesolution will be spanning in space off the initial functionmodulated by 119867

119896

(119909 120572119905) and exponentially depending ontime

119865(119909 119905)|119905≪120574120573

asymp sum

119896

119888119896

exp (1205721205742119905 + 120574119909)119867119896

(119909 120572119905)

119865(119909 119905)|119905≪120574120573119905≪1

asymp sum

119896

119888119896

119890120572120574

2119905

119890120574119909

119909119896

(78)

Note that for extended times 119905 ≫ 120574120573 we have dominanceof time variable Φ ≫ Φ

1

and the solution asymptoticallybehaves as exp(120573119905119909) while119909 playsminor role in119867

119896

(119909+2120574120572119905+

1205721205731199052

120572119905)The same operational technique as employed for the

treatise of (54) can be easily adopted for the solution of theSchrodinger equation

119894ℏ120597119905

Ψ (119909 119905) = minusℏ2

21198981205972

119909

Ψ (119909 119905) + 119865119909Ψ (119909 119905)

Ψ (119909 0) = 119891 (119909)

(79)

where 119865 is constant (119865 has the dimension of force) Indeedrescaling variables in (79) we obtain the form of equationsimilar to (54)

119894120597120591

Ψ (119909 120591) = minus1205972

119909

Ψ (119909 120591) + 119887119909Ψ (119909 120591) (80)

where

120591 =ℏ119905

2119898 119887 =

2119865119898

ℏ2 (81)

Following the operational methodology developed for (54)we write the following solution of (80) in the form (56)

Ψ (119909 120591) =119880119891 (119909) (82)

where119880 = exp (119894120591) = 120597

2

119909

minus 119887119909 (83)

Then on account of the substitution 119905 rarr 119894120591 120573 rarr minus119887operators

Θ = exp (1198871205912120597

119909

) Θ119891 (119909 120591) = 119891 (119909 + 119887120591

2

120591) (84)

119878 = exp (1198941205911205972

119909

) 119878119891 (119909) = 119891 (119909 119894120591) (85)

ariseThus the solution of the Schrodinger equation is a resultof consequent action of the operator 119878 and further action ofΘ on the initial condition function

Ψ (119909 119905) = exp (minus119894Φ (119909 120591 119887)) Θ 119878119891 (119909) (86)

where Φ is defined by (64) The integral form of the solutionthen is written as follows

Ψ (119909 119905) = exp (minus119894Φ (119909 120591 119887)) 1

2radic119894120587120591

times int

infin

minusinfin

exp(minus(119909 + 119887120591

2

minus 120585)2

4119894120591)119891 (120585) 119889120585

(87)

Again as well as in (70) without any assumption onthe nature of the initial condition function 119891(119909) of theSchrodinger equation its solution

Ψ (119909 120591) = exp (minus119894Φ (119909 120591 119887)) 119891 (119909 + 1198871205912 119894120591) (88)

where 119891(119909 119905) is given by (85) and is expressed in terms ofthe function of two variables obtained by the consequentapplication of the heat propagation and translation operators(85) and (84) to119891(119909) So far we have demonstrated asGauss-Weierstrass transform (A1) describe the action of 119878on the initial probability amplitude and how the shift (84)finally yields the explicit form of the solution of Schrodingerequation It means that the result of the action of evolutionoperator (83) on 119891(119909) is the product of combined actionof translation operator Θ and heat propagation operator 119878representing the evolution operator of the free particle

Now let us consider another Fokker-Plank type equationthat is the following example

120597119905

119865 (119909 119905) = 1205721205972

119909

119865 (119909 119905) + 120573119909120597119909

119865 (119909 119905) (89)

with initial condition (55) Proceeding along the aboveoutlined scheme of the solution of (54) we write the solutionof (89) in general form (56)

119865 (119909 119905) = 119891 (119909) = exp (119860 + 119861) (90)

where operators 119860 and 119861 are defined as follows

119860 = 1199051205721205972

119909

119861 = 119905120573119909120597119909

(91)

Quantities 119860 and 119861 evidently do not commute

[119860 119861] = 21205721205731199052

1205972

119909

= 119898119860 119898 = 2119905120573 (92)

It allows disentanglement of the operators in the exponentialaccording to the following rule

exp (119860 + 119861) = exp(1 minus exp (minus119898)119898

119860) exp (119861) (93)

The Scientific World Journal 7

Thus the evolution operator action on 119891(119909) given below

119891 (119909) = exp (1205901205972119909

) exp (120573119905119909120597119909

) 119891 (119909)

= exp (1205901205972119909

) 119891 (119890120573119905

119909)

(94)

simply reduces to a Gauss transforms where the parameter120590 = 120590(119905) reads as follows

120590 (119905) =(1 minus exp (minus2120573119905)) 120572

120573 (95)

Upon the trivial change of variables we obtain

119891 (119909) = 119878119891 (119910) 119910 = 119909 exp (119887) 119878 = exp(120588

21205972

119910

)

(96)

where

120588 (119905) =120572

120573(1198902120573119905

minus 1) (97)

and eventually we end up with the following simple solutionof (89)

119865 (119909 119905) =1

radic2120587120588int

infin

minusinfin

119890minus(exp(120573119905)119909minus120585)22120588

119891 (120585) 119889120585 (98)

The same example of the initial Gaussian function 119891(119909) =exp(minus1199092) as in the case of (54) yields (compare with (71))

119865(119909 119905)|119891(119909)=exp(minus1199092) =

1

radic1 + 2120588 (119905)exp(minus 119890

2120573119905

1199092

1 + 2120588 (119905))

(99)

where 120588 is defined in (97) Note that we can meet thefollowing modified form of Fokker-Plank type equation (89)

120597119905

119865 (119909 119905) = 1205721205972

119909

119865 (119909 119905) + 120573120597119909

119909119865 (119909 119905) (100)

in problems related to propagation of electron beams inaccelerators Its solution arises from (99) immediately anddiffers from it just by a factor exp(120573119905) as written below fora Gaussian 119891(119909)

119865(119909 119905)|119891(119909)=exp(minus1199092) =

119890120573119905

radic1 + 2120588 (119905)exp(minus 119890

2120573119905

1199092

1 + 2120588 (119905))

=1

radic120578 (119905)exp(minus 119909

2

120578 (119905))

120578 (119905) = 2120572

120573(1 minus 119890

minus2120573119905

+120573

2120572119890minus2120573119905

)

(101)

However differently from the solution of (54) where we hadconsequent transforms of the initial condition function 119891(119909)by operators of translation and heat diffusion 119878 (63) (see also[15]) here we have just the action of 119878 alone with much morecomplicated dependence of the solution on time

5 Conclusions

Operational method is fast and universal mathematical toolfor obtaining solutions of differential equations Combina-tion of operational method integral transforms and theoryof special functions together with orthogonal polynomi-als closely related to them provides a powerful analyti-cal instrument for solving a wide spectrum of differentialequations and relevant physical problems The technique ofinverse operator applied for derivatives of various ordersand combined with integral transforms allows for easyand straightforward solutions of various types of differentialequations With operational approach we developed themethodology of inverse differential operators and deriveda number of operational identities with them We havedemonstrated that using the technique of inverse derivativesand inverse differential operators combinedwith exponentialoperator integral transforms and special functions we canmake significant progress in solution of variousmathematicalproblems and relevant physical applications described bydifferential equations

Appendix

In complete analogy with the heat equation solution byGauss-Weierstrass transform [16]

exp (1198871205972119909

) 119892 (119909) =1

2radic120587119887

int

infin

minusinfin

exp(minus(119909 minus 120585)2

4119887)119892 (120585) 119889120585

(A1)

accounting for noncommutative relation for operators ofinverse derivative 119863minus1

119909

and119871

119863119909

defined through the opera-tional relation (14) and accounting for (31) and (32) we writethe solution of (40) in the following form

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

exp(minus(119863minus1

119909

minus 120585)2

4119905) 1120593 (120585) 119889120585 (A2)

where 120593 is the image (32) of the initial condition function119892 The kernel of the integral in the above formula can beexpanded into series of two-variable Hermite polynomials119867119899

(119909 119910)

exp(minus(119863minus1

119909

minus 120585)2

4119905) 1

= 119890minus120585

24119905

infin

sum

119899=0

119863minus119899

119909

119899119867119899

(120585

2119905 minus1

4119905)

= 119890minus120585

24119905

infin

sum

119899=0

119909119899

(119899)2

119867119899

(120585

2119905 minus1

4119905)

(A3)

Taking into account formula (12) for 119909119899119867119899

(1205852119905 minus14119905) inthe operational identity above which can be viewed asa generating function in terms of inverse derivative for

8 The Scientific World Journal

119867119899

(119909 119910) we obtain the following expansion for the kernel ofthe integral

exp(minus(119863minus1

119909

minus 120585)

4119905) 1 = 119890minus120585

24119905

infin

sum

119899=0

1

(119899)2

119867119899

(119909120585

2119905 minus1199092

4119905)

(A4)

where the series of Hermite polynomials of two arguments119867119899

(119909 119910) can be expressed in terms of Hermite-Bessel-Tricomi functions

119867

119862119899

(119909 119910)mdashgeneralization of Bessel-Tricomi functionsmdashand related to Bessel-Wright functionsand to common Bessel functions [17]

119867

119862119899

(119909 119910) =

infin

sum

119898=0

119867119898

(minus119909 119910)

119903 (119899 + 119898) 119899 isin 119873

0

(A5)

In particular for 119899 = 0 we immediately find our series

119867

1198620

(119909 119910) =

infin

sum

119898=0

119867119898

(minus119909 119910)

(119898)2

(A6)

which obviously leads to the solution of the heat-typepropagation problem (40) by the following appropriate Gausstransform

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

119890minus120585

24119905

119867

1198620

(minus120585119909

2119905 minus1199092

4119905)120593 (120585) 119889120585

(A7)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] A Appel and J Kampe de Feriet Fonctions Hypergeometriqueset Hyperspheriques Polynomes drsquoHermite Gauthier-VillarsParis France 1926

[2] D T Haimo and C Markett ldquoA representation theory for solu-tions of a higher order heat equation Irdquo Journal ofMathematicalAnalysis and Applications vol 168 no 1 pp 89ndash107 1992

[3] D T Haimo and C Markett ldquoA representation theory forsolutions of a higher order heat equation IIrdquo Journal ofMathematical Analysis and Applications vol 168 no 2 pp 289ndash305 1992

[4] G Dattoli ldquoGeneralized polynomials operational identitiesand their applicationsrdquo Journal of Computational and AppliedMathematics vol 118 no 1-2 pp 111ndash123 2000

[5] G Dattoli H M Srivastava and K Zhukovsky ldquoOrthogonalityproperties of the Hermite and related polynomialsrdquo Journal ofComputational andAppliedMathematics vol 182 no 1 pp 165ndash172 2005

[6] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol 2 McGraw-Hill BookCompany New York NY USA 1953

[7] G Dattoli B GermanoM RMartinelli and P E Ricci ldquoNega-tive derivatives and special functionsrdquoAppliedMathematics andComputation vol 217 no 8 pp 3924ndash3928 2010

[8] G Dattoli M Migliorati and S Khan ldquoSolutions of integro-differential equations and operational methodsrdquo Applied Math-ematics and Computation vol 186 no 1 pp 302ndash308 2007

[9] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Mathematics and its Applications (Ellis HorwoodLtd) Halsted Press JohnWiley and Sons Chichester UK 1984

[10] GDattoli HM Srivastava andK Zhukovsky ldquoA new family ofintegral transforms and their applicationsrdquo Integral Transformsand Special Functions vol 17 no 1 pp 31ndash37 2006

[11] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 no 1 pp 1ndash174 1962

[12] G Dattoli H M Srivastava and K Zhukovsky ldquoOperationalmethods and differential equations with applications to initial-value problemsrdquo Applied Mathematics and Computation vol184 no 2 pp 979ndash1001 2007

[13] G Dattoli and K V Zhukovsky ldquoAppel polynomial seriesexpansionrdquo InternationalMathematical Forum vol 5 no 14 pp649ndash6662 2010

[14] G Dattoli M Migliorati and K Zhukovsky ldquoSummationformulae and stirling numbersrdquo International MathematicalForum vol 4 no 41 pp 2017ndash22040 2009

[15] J E Avron and I W Herbst ldquoSpectral and scattering theory ofSchrodinger operators related to the stark effectrdquo Communica-tions in Mathematical Physics vol 52 no 3 pp 239ndash254 1977

[16] K B Wolf Integral Transforms in Science and EngineeringPlenum Press New York NY USA 1979

[17] G N Watson A Treatise on the Theory of Bessel FunctionsCambridgeUniversity Press CambridgeUK 2nd edition 1944

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4 The Scientific World Journal

In conclusion of the present chapter we consider theexample of the solution of a heat propagation type equationby operational method involving the inverse derivativeoperator and exponential operator technique We recall thatthe common heat equation with initial condition problem

120597

120597119905119891 (119909 119905) = 120597

2

119909

119891 (119909 119905) 119891 (119909 0) = 119892 (119909) (38)

can be solved by Gauss transforms

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

exp(minus(119909 minus 120585)2

4119905) 119892 (120585) 119889120585 (39)

In complete analogy with the above statements we can solvethe heat-type equation with differential operator

119871

119863119909

(14)with for example the following initial condition

120597

120597119905119891 (119909 119905) =

119871

1198632

119909

119891 (119909 119905) 119891 (119909 0) = 119892 (119909) (40)

The Laguerre heat-type propagation problem (40) possessesthe following solution (see the Appendix)

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

119890minus120585

24119905

119867

1198620

(minus120585119909

2119905 minus1199092

4119905)120593 (120585) 119889120585

(41)

4 Operational Approach and Other Types ofDifferential Equations

Operational approach to solution of partial differential equa-tions demonstrated on the examples of diffusion-like andheat-like equations with 120597

119909

119909120597119909

derivatives can be furtherextended to other equation types Consider the followingexample of a rather complicated differential equation

1

120588

120597

120597119905119860 (119909 119905) = 119909

2

1205972

1205971199092119860 (119909 119905) + 120582119909

120597

120597119909119860 (119909 119905)

minus 120583119860 (119909 119905) 119892 (119909) = 119860 (119909 0)

(42)

where 120588 120582 and 120583 are some arbitrary constant coefficientsand function 119892(119909) = 119860(119909 119905 = 0) is the initial condition Byintroducing operator119863 = 119909120597

119909

and distinguishing the perfectsquare we rewrite (42)

1

120588

120597

120597 119905119860 (119909 119905) = ((119863 +

120582

2)

2

minus 120576)119860 (119909 119905) (43)

where

120576 = 120583 + (120582

2)

2

(44)

Thus the following exponential solution for (42) appears

119860 (119909 119905) = exp120588119905((119863 + 1205822)

2

minus 120576)119892 (119909) (45)

Now making use of the operational identity

exp (1199012) = 1

radic120587int

infin

minusinfin

exp (minus1205852 + 2120585119901) 119889120585 (46)

and applying exp(119886119863) according to

exp (119886119909120597119909

) 119891 (119909) = 119891 (119890119886

119909) (47)

we obtain the following compact expression for 119860(119909 119905)

119860 (119909 119905) =exp (minus120588120576119905)radic120587

times int

infin

minusinfin

exp [minus1205902 + 120590120572 1205822120588] 119892 (119909 exp (120590120572)) 119889120590

(48)

where

120572 = 120572 (119905) = 2radic120588119905 (49)

Consider two simple examples of initial condition functionsThe first one is

119892 (119909) = 119909119899

(50)

Then we immediately obtain the solution of (42) as follows

119860 (119909 119905) = 119909119899 exp 120588119905 (1198992 + 120582119899 minus 120583) (51)

The second example is given by the following initial conditionfunction

119892 (119909) = ln119909 (52)

Trivial computations yield the following solution

119860 (119909 119905) = (ln119909 + 120588119905120582) exp (minus120588119905120583) (53)

Thus operational technique combined with integraltransforms operational identities and extended forms oforthogonal polynomials represents powerful tool for findingsolutions of various classes of differential equations and initialvalue problems Note that within the framework of inversedifferential operators developed and described above theusage of the evolution operator method opens new possibili-ties which we will elucidate in what follows

Let us consider the following generalization of the heatequation

120597119905

119865 (119909 119905) = 1205721205972

119909

119865 (119909 119905) + 120573119909119865 (119909 119905) (54)

with the initial condition

119865 (119909 0) = 119891 (119909) (55)

The evolution type equation (54) contains linear coordinateterm in addition to the second order derivative Its formalsolution can be written via the evolution operator

119865 (119909 119905) = 119891 (119909) (56)

The Scientific World Journal 5

where

= exp (119860 + 119861) 119860 = 1205721199051205972

119909

119861 = 120573119905119909 (57)

The exponential of the evolution operator in (56) is thesum of two noncommuting operators and it can be writtenas the ordered product of two exponential operators Indeedthe commutator of 119860 and 119861 has the following nonzero value

[119860 119861] = 11989811986012

= 21205721205731199052

120597119909

119898 = 212057311990532

12057212

(58)

Then we can apply the disentanglement operational identity

119890

119860+

119861

= 119890(119898

212)minus(1198982)

119860

12+

119860

119890

119861

(59)

and the following chain rule

119890119901120597

2

119909119890119902119909

119892 (119909) = 119890119901119902

2

119890119902119909

1198902119901119902120597

119909119890119901120597

2

119909119892 (119909) (60)

where119901 119902 are constant parametersWith their help we obtainthe evolution operator for (54)

= 119890Φ(119909119905120573)

Θ119878 (61)

Note that we have factorized two commuting operators theoperator of translation in space

Θ = 119890120572120573119905

2120597

119909 (62)

and operator

119878 = 119890120572119905120597

2

119909 (63)

which is in fact operator 119878 = exp(1199051198632119909

) The phase in iswritten as follows

Φ(119909 119905 120572 120573) =1

31205721205732

1199053

+ 120573119905119909 (64)

The action of on the initial condition function 119891(119909) yieldsthe following solution for our problem

119865 (119909 119905) = 119890Φ(119909119905120572120573)

Θ119878119891 (119909) (65)

Thus we conclude from the form of (65) that the problem(54) with the initial condition (55) can be solved by theconsequent application of commuting operators Θ (62) and 119878(63) to 119891(119909) apart from the factor 119890Φ(119909119905120572120573) Now the explicitform of the solution (65) can be obtained by recalling that Θacts as a translation operator

119890119904120597

119909119892 (119909) = 119892 (119909 + 119904) (66)

and that the action of 119878 on the function 119892(119909) yields thesolution of the ordinary heat equation through Gauss-Weierstrass transform (A1) Accordingly we denote

119891 (119909 119905) equiv 119878119891 (119909) equiv 119890120572119905120597

2

119909119891 (119909) (67)

and we write

Θ119891 (119909 119905) = 119891 (119909 + 1205721205731199052

119905) (68)

Thus (54)with initial condition (55) has the following explicitsolution119865 (119909 119905)

= 119890Φ(119909119905120572120573)

1

2radic120587120572119905int

infin

minusinfin

119890minus(119909+120572120573119905

2minus120585)

2

4119905120572

119891 (120585) 119889120585

(69)

provided that the integral converges Summarizing the aboveoutlined procedure we conclude that a solution for (54)consists in finding a Gauss transformed function 119891 with ashifted argument

119865 (119909 119905) = 119890Φ(119909119905120572120573)

119891 (119909 + 1205721205731199052

119905) (70)

Moreover this is a general observation for this type ofequation valid for any function 119891(119909) (provided the integralconverges) In other words we have obtained the solution ofFokker plank equation as a consequent action of operator 119878of heat diffusion and operator Θ of translation on the initialcondition functionNote that119891(119909 119905) is the solution of the heatequation representing a natural propagation phenomenon

The effect produced by the translation operator Θ and theoperator 119878 is best illustrated with the example of Gaussian119891(119909) = exp(minus1199092) evolution when (69) becomes

119865(119909 119905)|119891(119909)=exp(minus1199092) =

exp (Φ (119909 119905))radic1 + 4119905

exp(minus(119909 + 120573119905

2

)2

1 + 4119905)

119905 ge 0

(71)

The above result is exactly the generalization of Gleisherrule (8) considered earlier in the context of the heat equationThus (71) is the solution of the ordinary heat equation with120573 = 0 when the initial function is Gaussian

Another interesting example of solving (54) appearswhenthe initial function119891(119909) allows the expansion in the followingseries

119891 (119909) = sum

119896

119888119896

119909119896

119890120574119909

(72)

In this case we refer to the identity

exp (1199101198632119909

) 119909119896

119890120574 119909

= 119890(120574 119909+120574

2119910)

119867119896

(119909 + 2120574119910 119910) (73)

which arises from

exp(119910 120597119898

120597119909119898)119891 (119909) = 119891(119909 + 119898119910

120597119898minus1

120597119909119898minus1) 1 (74)

with account for the generating function of Hermite polyno-mials (11) Then the action (67) of operator 119878 on the initialfunction (72) produces

119891 (119909 119905) = sum

119896

119888119896

119891119896

(119909 119905)

119891119896

(119909 119905) = 119890120574(119909+120574119886)

119867119896

(119909 + 2120574119886 119886) 119886 = 120572119905

(75)

6 The Scientific World Journal

In addition the translation operated by Θ shifts the argu-ment 119891(119909 119905) rarr 119891(119909 + 119886119887 119905) 119887 = 120573119905 Thus we obtain thesolution in a form of Hermite polynomial as follows

119865 (119909 119905) = sum

119896

119888119896

119890Φ+Φ

1119867119896

(119909 +1205721205742

120573(2119905120573

120574+ 1199052

1205732

1205742) 120572119905)

(76)

where Φ is defined by (64) and

Φ1

= 120574(119909 +1205721205742

120573(119905120573

120574+ (119905

120573

120574)

2

)) (77)

It is now evident that for a short time when 119905 ≪ 120574120573 thesolution will be spanning in space off the initial functionmodulated by 119867

119896

(119909 120572119905) and exponentially depending ontime

119865(119909 119905)|119905≪120574120573

asymp sum

119896

119888119896

exp (1205721205742119905 + 120574119909)119867119896

(119909 120572119905)

119865(119909 119905)|119905≪120574120573119905≪1

asymp sum

119896

119888119896

119890120572120574

2119905

119890120574119909

119909119896

(78)

Note that for extended times 119905 ≫ 120574120573 we have dominanceof time variable Φ ≫ Φ

1

and the solution asymptoticallybehaves as exp(120573119905119909) while119909 playsminor role in119867

119896

(119909+2120574120572119905+

1205721205731199052

120572119905)The same operational technique as employed for the

treatise of (54) can be easily adopted for the solution of theSchrodinger equation

119894ℏ120597119905

Ψ (119909 119905) = minusℏ2

21198981205972

119909

Ψ (119909 119905) + 119865119909Ψ (119909 119905)

Ψ (119909 0) = 119891 (119909)

(79)

where 119865 is constant (119865 has the dimension of force) Indeedrescaling variables in (79) we obtain the form of equationsimilar to (54)

119894120597120591

Ψ (119909 120591) = minus1205972

119909

Ψ (119909 120591) + 119887119909Ψ (119909 120591) (80)

where

120591 =ℏ119905

2119898 119887 =

2119865119898

ℏ2 (81)

Following the operational methodology developed for (54)we write the following solution of (80) in the form (56)

Ψ (119909 120591) =119880119891 (119909) (82)

where119880 = exp (119894120591) = 120597

2

119909

minus 119887119909 (83)

Then on account of the substitution 119905 rarr 119894120591 120573 rarr minus119887operators

Θ = exp (1198871205912120597

119909

) Θ119891 (119909 120591) = 119891 (119909 + 119887120591

2

120591) (84)

119878 = exp (1198941205911205972

119909

) 119878119891 (119909) = 119891 (119909 119894120591) (85)

ariseThus the solution of the Schrodinger equation is a resultof consequent action of the operator 119878 and further action ofΘ on the initial condition function

Ψ (119909 119905) = exp (minus119894Φ (119909 120591 119887)) Θ 119878119891 (119909) (86)

where Φ is defined by (64) The integral form of the solutionthen is written as follows

Ψ (119909 119905) = exp (minus119894Φ (119909 120591 119887)) 1

2radic119894120587120591

times int

infin

minusinfin

exp(minus(119909 + 119887120591

2

minus 120585)2

4119894120591)119891 (120585) 119889120585

(87)

Again as well as in (70) without any assumption onthe nature of the initial condition function 119891(119909) of theSchrodinger equation its solution

Ψ (119909 120591) = exp (minus119894Φ (119909 120591 119887)) 119891 (119909 + 1198871205912 119894120591) (88)

where 119891(119909 119905) is given by (85) and is expressed in terms ofthe function of two variables obtained by the consequentapplication of the heat propagation and translation operators(85) and (84) to119891(119909) So far we have demonstrated asGauss-Weierstrass transform (A1) describe the action of 119878on the initial probability amplitude and how the shift (84)finally yields the explicit form of the solution of Schrodingerequation It means that the result of the action of evolutionoperator (83) on 119891(119909) is the product of combined actionof translation operator Θ and heat propagation operator 119878representing the evolution operator of the free particle

Now let us consider another Fokker-Plank type equationthat is the following example

120597119905

119865 (119909 119905) = 1205721205972

119909

119865 (119909 119905) + 120573119909120597119909

119865 (119909 119905) (89)

with initial condition (55) Proceeding along the aboveoutlined scheme of the solution of (54) we write the solutionof (89) in general form (56)

119865 (119909 119905) = 119891 (119909) = exp (119860 + 119861) (90)

where operators 119860 and 119861 are defined as follows

119860 = 1199051205721205972

119909

119861 = 119905120573119909120597119909

(91)

Quantities 119860 and 119861 evidently do not commute

[119860 119861] = 21205721205731199052

1205972

119909

= 119898119860 119898 = 2119905120573 (92)

It allows disentanglement of the operators in the exponentialaccording to the following rule

exp (119860 + 119861) = exp(1 minus exp (minus119898)119898

119860) exp (119861) (93)

The Scientific World Journal 7

Thus the evolution operator action on 119891(119909) given below

119891 (119909) = exp (1205901205972119909

) exp (120573119905119909120597119909

) 119891 (119909)

= exp (1205901205972119909

) 119891 (119890120573119905

119909)

(94)

simply reduces to a Gauss transforms where the parameter120590 = 120590(119905) reads as follows

120590 (119905) =(1 minus exp (minus2120573119905)) 120572

120573 (95)

Upon the trivial change of variables we obtain

119891 (119909) = 119878119891 (119910) 119910 = 119909 exp (119887) 119878 = exp(120588

21205972

119910

)

(96)

where

120588 (119905) =120572

120573(1198902120573119905

minus 1) (97)

and eventually we end up with the following simple solutionof (89)

119865 (119909 119905) =1

radic2120587120588int

infin

minusinfin

119890minus(exp(120573119905)119909minus120585)22120588

119891 (120585) 119889120585 (98)

The same example of the initial Gaussian function 119891(119909) =exp(minus1199092) as in the case of (54) yields (compare with (71))

119865(119909 119905)|119891(119909)=exp(minus1199092) =

1

radic1 + 2120588 (119905)exp(minus 119890

2120573119905

1199092

1 + 2120588 (119905))

(99)

where 120588 is defined in (97) Note that we can meet thefollowing modified form of Fokker-Plank type equation (89)

120597119905

119865 (119909 119905) = 1205721205972

119909

119865 (119909 119905) + 120573120597119909

119909119865 (119909 119905) (100)

in problems related to propagation of electron beams inaccelerators Its solution arises from (99) immediately anddiffers from it just by a factor exp(120573119905) as written below fora Gaussian 119891(119909)

119865(119909 119905)|119891(119909)=exp(minus1199092) =

119890120573119905

radic1 + 2120588 (119905)exp(minus 119890

2120573119905

1199092

1 + 2120588 (119905))

=1

radic120578 (119905)exp(minus 119909

2

120578 (119905))

120578 (119905) = 2120572

120573(1 minus 119890

minus2120573119905

+120573

2120572119890minus2120573119905

)

(101)

However differently from the solution of (54) where we hadconsequent transforms of the initial condition function 119891(119909)by operators of translation and heat diffusion 119878 (63) (see also[15]) here we have just the action of 119878 alone with much morecomplicated dependence of the solution on time

5 Conclusions

Operational method is fast and universal mathematical toolfor obtaining solutions of differential equations Combina-tion of operational method integral transforms and theoryof special functions together with orthogonal polynomi-als closely related to them provides a powerful analyti-cal instrument for solving a wide spectrum of differentialequations and relevant physical problems The technique ofinverse operator applied for derivatives of various ordersand combined with integral transforms allows for easyand straightforward solutions of various types of differentialequations With operational approach we developed themethodology of inverse differential operators and deriveda number of operational identities with them We havedemonstrated that using the technique of inverse derivativesand inverse differential operators combinedwith exponentialoperator integral transforms and special functions we canmake significant progress in solution of variousmathematicalproblems and relevant physical applications described bydifferential equations

Appendix

In complete analogy with the heat equation solution byGauss-Weierstrass transform [16]

exp (1198871205972119909

) 119892 (119909) =1

2radic120587119887

int

infin

minusinfin

exp(minus(119909 minus 120585)2

4119887)119892 (120585) 119889120585

(A1)

accounting for noncommutative relation for operators ofinverse derivative 119863minus1

119909

and119871

119863119909

defined through the opera-tional relation (14) and accounting for (31) and (32) we writethe solution of (40) in the following form

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

exp(minus(119863minus1

119909

minus 120585)2

4119905) 1120593 (120585) 119889120585 (A2)

where 120593 is the image (32) of the initial condition function119892 The kernel of the integral in the above formula can beexpanded into series of two-variable Hermite polynomials119867119899

(119909 119910)

exp(minus(119863minus1

119909

minus 120585)2

4119905) 1

= 119890minus120585

24119905

infin

sum

119899=0

119863minus119899

119909

119899119867119899

(120585

2119905 minus1

4119905)

= 119890minus120585

24119905

infin

sum

119899=0

119909119899

(119899)2

119867119899

(120585

2119905 minus1

4119905)

(A3)

Taking into account formula (12) for 119909119899119867119899

(1205852119905 minus14119905) inthe operational identity above which can be viewed asa generating function in terms of inverse derivative for

8 The Scientific World Journal

119867119899

(119909 119910) we obtain the following expansion for the kernel ofthe integral

exp(minus(119863minus1

119909

minus 120585)

4119905) 1 = 119890minus120585

24119905

infin

sum

119899=0

1

(119899)2

119867119899

(119909120585

2119905 minus1199092

4119905)

(A4)

where the series of Hermite polynomials of two arguments119867119899

(119909 119910) can be expressed in terms of Hermite-Bessel-Tricomi functions

119867

119862119899

(119909 119910)mdashgeneralization of Bessel-Tricomi functionsmdashand related to Bessel-Wright functionsand to common Bessel functions [17]

119867

119862119899

(119909 119910) =

infin

sum

119898=0

119867119898

(minus119909 119910)

119903 (119899 + 119898) 119899 isin 119873

0

(A5)

In particular for 119899 = 0 we immediately find our series

119867

1198620

(119909 119910) =

infin

sum

119898=0

119867119898

(minus119909 119910)

(119898)2

(A6)

which obviously leads to the solution of the heat-typepropagation problem (40) by the following appropriate Gausstransform

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

119890minus120585

24119905

119867

1198620

(minus120585119909

2119905 minus1199092

4119905)120593 (120585) 119889120585

(A7)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] A Appel and J Kampe de Feriet Fonctions Hypergeometriqueset Hyperspheriques Polynomes drsquoHermite Gauthier-VillarsParis France 1926

[2] D T Haimo and C Markett ldquoA representation theory for solu-tions of a higher order heat equation Irdquo Journal ofMathematicalAnalysis and Applications vol 168 no 1 pp 89ndash107 1992

[3] D T Haimo and C Markett ldquoA representation theory forsolutions of a higher order heat equation IIrdquo Journal ofMathematical Analysis and Applications vol 168 no 2 pp 289ndash305 1992

[4] G Dattoli ldquoGeneralized polynomials operational identitiesand their applicationsrdquo Journal of Computational and AppliedMathematics vol 118 no 1-2 pp 111ndash123 2000

[5] G Dattoli H M Srivastava and K Zhukovsky ldquoOrthogonalityproperties of the Hermite and related polynomialsrdquo Journal ofComputational andAppliedMathematics vol 182 no 1 pp 165ndash172 2005

[6] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol 2 McGraw-Hill BookCompany New York NY USA 1953

[7] G Dattoli B GermanoM RMartinelli and P E Ricci ldquoNega-tive derivatives and special functionsrdquoAppliedMathematics andComputation vol 217 no 8 pp 3924ndash3928 2010

[8] G Dattoli M Migliorati and S Khan ldquoSolutions of integro-differential equations and operational methodsrdquo Applied Math-ematics and Computation vol 186 no 1 pp 302ndash308 2007

[9] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Mathematics and its Applications (Ellis HorwoodLtd) Halsted Press JohnWiley and Sons Chichester UK 1984

[10] GDattoli HM Srivastava andK Zhukovsky ldquoA new family ofintegral transforms and their applicationsrdquo Integral Transformsand Special Functions vol 17 no 1 pp 31ndash37 2006

[11] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 no 1 pp 1ndash174 1962

[12] G Dattoli H M Srivastava and K Zhukovsky ldquoOperationalmethods and differential equations with applications to initial-value problemsrdquo Applied Mathematics and Computation vol184 no 2 pp 979ndash1001 2007

[13] G Dattoli and K V Zhukovsky ldquoAppel polynomial seriesexpansionrdquo InternationalMathematical Forum vol 5 no 14 pp649ndash6662 2010

[14] G Dattoli M Migliorati and K Zhukovsky ldquoSummationformulae and stirling numbersrdquo International MathematicalForum vol 4 no 41 pp 2017ndash22040 2009

[15] J E Avron and I W Herbst ldquoSpectral and scattering theory ofSchrodinger operators related to the stark effectrdquo Communica-tions in Mathematical Physics vol 52 no 3 pp 239ndash254 1977

[16] K B Wolf Integral Transforms in Science and EngineeringPlenum Press New York NY USA 1979

[17] G N Watson A Treatise on the Theory of Bessel FunctionsCambridgeUniversity Press CambridgeUK 2nd edition 1944

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 5

where

= exp (119860 + 119861) 119860 = 1205721199051205972

119909

119861 = 120573119905119909 (57)

The exponential of the evolution operator in (56) is thesum of two noncommuting operators and it can be writtenas the ordered product of two exponential operators Indeedthe commutator of 119860 and 119861 has the following nonzero value

[119860 119861] = 11989811986012

= 21205721205731199052

120597119909

119898 = 212057311990532

12057212

(58)

Then we can apply the disentanglement operational identity

119890

119860+

119861

= 119890(119898

212)minus(1198982)

119860

12+

119860

119890

119861

(59)

and the following chain rule

119890119901120597

2

119909119890119902119909

119892 (119909) = 119890119901119902

2

119890119902119909

1198902119901119902120597

119909119890119901120597

2

119909119892 (119909) (60)

where119901 119902 are constant parametersWith their help we obtainthe evolution operator for (54)

= 119890Φ(119909119905120573)

Θ119878 (61)

Note that we have factorized two commuting operators theoperator of translation in space

Θ = 119890120572120573119905

2120597

119909 (62)

and operator

119878 = 119890120572119905120597

2

119909 (63)

which is in fact operator 119878 = exp(1199051198632119909

) The phase in iswritten as follows

Φ(119909 119905 120572 120573) =1

31205721205732

1199053

+ 120573119905119909 (64)

The action of on the initial condition function 119891(119909) yieldsthe following solution for our problem

119865 (119909 119905) = 119890Φ(119909119905120572120573)

Θ119878119891 (119909) (65)

Thus we conclude from the form of (65) that the problem(54) with the initial condition (55) can be solved by theconsequent application of commuting operators Θ (62) and 119878(63) to 119891(119909) apart from the factor 119890Φ(119909119905120572120573) Now the explicitform of the solution (65) can be obtained by recalling that Θacts as a translation operator

119890119904120597

119909119892 (119909) = 119892 (119909 + 119904) (66)

and that the action of 119878 on the function 119892(119909) yields thesolution of the ordinary heat equation through Gauss-Weierstrass transform (A1) Accordingly we denote

119891 (119909 119905) equiv 119878119891 (119909) equiv 119890120572119905120597

2

119909119891 (119909) (67)

and we write

Θ119891 (119909 119905) = 119891 (119909 + 1205721205731199052

119905) (68)

Thus (54)with initial condition (55) has the following explicitsolution119865 (119909 119905)

= 119890Φ(119909119905120572120573)

1

2radic120587120572119905int

infin

minusinfin

119890minus(119909+120572120573119905

2minus120585)

2

4119905120572

119891 (120585) 119889120585

(69)

provided that the integral converges Summarizing the aboveoutlined procedure we conclude that a solution for (54)consists in finding a Gauss transformed function 119891 with ashifted argument

119865 (119909 119905) = 119890Φ(119909119905120572120573)

119891 (119909 + 1205721205731199052

119905) (70)

Moreover this is a general observation for this type ofequation valid for any function 119891(119909) (provided the integralconverges) In other words we have obtained the solution ofFokker plank equation as a consequent action of operator 119878of heat diffusion and operator Θ of translation on the initialcondition functionNote that119891(119909 119905) is the solution of the heatequation representing a natural propagation phenomenon

The effect produced by the translation operator Θ and theoperator 119878 is best illustrated with the example of Gaussian119891(119909) = exp(minus1199092) evolution when (69) becomes

119865(119909 119905)|119891(119909)=exp(minus1199092) =

exp (Φ (119909 119905))radic1 + 4119905

exp(minus(119909 + 120573119905

2

)2

1 + 4119905)

119905 ge 0

(71)

The above result is exactly the generalization of Gleisherrule (8) considered earlier in the context of the heat equationThus (71) is the solution of the ordinary heat equation with120573 = 0 when the initial function is Gaussian

Another interesting example of solving (54) appearswhenthe initial function119891(119909) allows the expansion in the followingseries

119891 (119909) = sum

119896

119888119896

119909119896

119890120574119909

(72)

In this case we refer to the identity

exp (1199101198632119909

) 119909119896

119890120574 119909

= 119890(120574 119909+120574

2119910)

119867119896

(119909 + 2120574119910 119910) (73)

which arises from

exp(119910 120597119898

120597119909119898)119891 (119909) = 119891(119909 + 119898119910

120597119898minus1

120597119909119898minus1) 1 (74)

with account for the generating function of Hermite polyno-mials (11) Then the action (67) of operator 119878 on the initialfunction (72) produces

119891 (119909 119905) = sum

119896

119888119896

119891119896

(119909 119905)

119891119896

(119909 119905) = 119890120574(119909+120574119886)

119867119896

(119909 + 2120574119886 119886) 119886 = 120572119905

(75)

6 The Scientific World Journal

In addition the translation operated by Θ shifts the argu-ment 119891(119909 119905) rarr 119891(119909 + 119886119887 119905) 119887 = 120573119905 Thus we obtain thesolution in a form of Hermite polynomial as follows

119865 (119909 119905) = sum

119896

119888119896

119890Φ+Φ

1119867119896

(119909 +1205721205742

120573(2119905120573

120574+ 1199052

1205732

1205742) 120572119905)

(76)

where Φ is defined by (64) and

Φ1

= 120574(119909 +1205721205742

120573(119905120573

120574+ (119905

120573

120574)

2

)) (77)

It is now evident that for a short time when 119905 ≪ 120574120573 thesolution will be spanning in space off the initial functionmodulated by 119867

119896

(119909 120572119905) and exponentially depending ontime

119865(119909 119905)|119905≪120574120573

asymp sum

119896

119888119896

exp (1205721205742119905 + 120574119909)119867119896

(119909 120572119905)

119865(119909 119905)|119905≪120574120573119905≪1

asymp sum

119896

119888119896

119890120572120574

2119905

119890120574119909

119909119896

(78)

Note that for extended times 119905 ≫ 120574120573 we have dominanceof time variable Φ ≫ Φ

1

and the solution asymptoticallybehaves as exp(120573119905119909) while119909 playsminor role in119867

119896

(119909+2120574120572119905+

1205721205731199052

120572119905)The same operational technique as employed for the

treatise of (54) can be easily adopted for the solution of theSchrodinger equation

119894ℏ120597119905

Ψ (119909 119905) = minusℏ2

21198981205972

119909

Ψ (119909 119905) + 119865119909Ψ (119909 119905)

Ψ (119909 0) = 119891 (119909)

(79)

where 119865 is constant (119865 has the dimension of force) Indeedrescaling variables in (79) we obtain the form of equationsimilar to (54)

119894120597120591

Ψ (119909 120591) = minus1205972

119909

Ψ (119909 120591) + 119887119909Ψ (119909 120591) (80)

where

120591 =ℏ119905

2119898 119887 =

2119865119898

ℏ2 (81)

Following the operational methodology developed for (54)we write the following solution of (80) in the form (56)

Ψ (119909 120591) =119880119891 (119909) (82)

where119880 = exp (119894120591) = 120597

2

119909

minus 119887119909 (83)

Then on account of the substitution 119905 rarr 119894120591 120573 rarr minus119887operators

Θ = exp (1198871205912120597

119909

) Θ119891 (119909 120591) = 119891 (119909 + 119887120591

2

120591) (84)

119878 = exp (1198941205911205972

119909

) 119878119891 (119909) = 119891 (119909 119894120591) (85)

ariseThus the solution of the Schrodinger equation is a resultof consequent action of the operator 119878 and further action ofΘ on the initial condition function

Ψ (119909 119905) = exp (minus119894Φ (119909 120591 119887)) Θ 119878119891 (119909) (86)

where Φ is defined by (64) The integral form of the solutionthen is written as follows

Ψ (119909 119905) = exp (minus119894Φ (119909 120591 119887)) 1

2radic119894120587120591

times int

infin

minusinfin

exp(minus(119909 + 119887120591

2

minus 120585)2

4119894120591)119891 (120585) 119889120585

(87)

Again as well as in (70) without any assumption onthe nature of the initial condition function 119891(119909) of theSchrodinger equation its solution

Ψ (119909 120591) = exp (minus119894Φ (119909 120591 119887)) 119891 (119909 + 1198871205912 119894120591) (88)

where 119891(119909 119905) is given by (85) and is expressed in terms ofthe function of two variables obtained by the consequentapplication of the heat propagation and translation operators(85) and (84) to119891(119909) So far we have demonstrated asGauss-Weierstrass transform (A1) describe the action of 119878on the initial probability amplitude and how the shift (84)finally yields the explicit form of the solution of Schrodingerequation It means that the result of the action of evolutionoperator (83) on 119891(119909) is the product of combined actionof translation operator Θ and heat propagation operator 119878representing the evolution operator of the free particle

Now let us consider another Fokker-Plank type equationthat is the following example

120597119905

119865 (119909 119905) = 1205721205972

119909

119865 (119909 119905) + 120573119909120597119909

119865 (119909 119905) (89)

with initial condition (55) Proceeding along the aboveoutlined scheme of the solution of (54) we write the solutionof (89) in general form (56)

119865 (119909 119905) = 119891 (119909) = exp (119860 + 119861) (90)

where operators 119860 and 119861 are defined as follows

119860 = 1199051205721205972

119909

119861 = 119905120573119909120597119909

(91)

Quantities 119860 and 119861 evidently do not commute

[119860 119861] = 21205721205731199052

1205972

119909

= 119898119860 119898 = 2119905120573 (92)

It allows disentanglement of the operators in the exponentialaccording to the following rule

exp (119860 + 119861) = exp(1 minus exp (minus119898)119898

119860) exp (119861) (93)

The Scientific World Journal 7

Thus the evolution operator action on 119891(119909) given below

119891 (119909) = exp (1205901205972119909

) exp (120573119905119909120597119909

) 119891 (119909)

= exp (1205901205972119909

) 119891 (119890120573119905

119909)

(94)

simply reduces to a Gauss transforms where the parameter120590 = 120590(119905) reads as follows

120590 (119905) =(1 minus exp (minus2120573119905)) 120572

120573 (95)

Upon the trivial change of variables we obtain

119891 (119909) = 119878119891 (119910) 119910 = 119909 exp (119887) 119878 = exp(120588

21205972

119910

)

(96)

where

120588 (119905) =120572

120573(1198902120573119905

minus 1) (97)

and eventually we end up with the following simple solutionof (89)

119865 (119909 119905) =1

radic2120587120588int

infin

minusinfin

119890minus(exp(120573119905)119909minus120585)22120588

119891 (120585) 119889120585 (98)

The same example of the initial Gaussian function 119891(119909) =exp(minus1199092) as in the case of (54) yields (compare with (71))

119865(119909 119905)|119891(119909)=exp(minus1199092) =

1

radic1 + 2120588 (119905)exp(minus 119890

2120573119905

1199092

1 + 2120588 (119905))

(99)

where 120588 is defined in (97) Note that we can meet thefollowing modified form of Fokker-Plank type equation (89)

120597119905

119865 (119909 119905) = 1205721205972

119909

119865 (119909 119905) + 120573120597119909

119909119865 (119909 119905) (100)

in problems related to propagation of electron beams inaccelerators Its solution arises from (99) immediately anddiffers from it just by a factor exp(120573119905) as written below fora Gaussian 119891(119909)

119865(119909 119905)|119891(119909)=exp(minus1199092) =

119890120573119905

radic1 + 2120588 (119905)exp(minus 119890

2120573119905

1199092

1 + 2120588 (119905))

=1

radic120578 (119905)exp(minus 119909

2

120578 (119905))

120578 (119905) = 2120572

120573(1 minus 119890

minus2120573119905

+120573

2120572119890minus2120573119905

)

(101)

However differently from the solution of (54) where we hadconsequent transforms of the initial condition function 119891(119909)by operators of translation and heat diffusion 119878 (63) (see also[15]) here we have just the action of 119878 alone with much morecomplicated dependence of the solution on time

5 Conclusions

Operational method is fast and universal mathematical toolfor obtaining solutions of differential equations Combina-tion of operational method integral transforms and theoryof special functions together with orthogonal polynomi-als closely related to them provides a powerful analyti-cal instrument for solving a wide spectrum of differentialequations and relevant physical problems The technique ofinverse operator applied for derivatives of various ordersand combined with integral transforms allows for easyand straightforward solutions of various types of differentialequations With operational approach we developed themethodology of inverse differential operators and deriveda number of operational identities with them We havedemonstrated that using the technique of inverse derivativesand inverse differential operators combinedwith exponentialoperator integral transforms and special functions we canmake significant progress in solution of variousmathematicalproblems and relevant physical applications described bydifferential equations

Appendix

In complete analogy with the heat equation solution byGauss-Weierstrass transform [16]

exp (1198871205972119909

) 119892 (119909) =1

2radic120587119887

int

infin

minusinfin

exp(minus(119909 minus 120585)2

4119887)119892 (120585) 119889120585

(A1)

accounting for noncommutative relation for operators ofinverse derivative 119863minus1

119909

and119871

119863119909

defined through the opera-tional relation (14) and accounting for (31) and (32) we writethe solution of (40) in the following form

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

exp(minus(119863minus1

119909

minus 120585)2

4119905) 1120593 (120585) 119889120585 (A2)

where 120593 is the image (32) of the initial condition function119892 The kernel of the integral in the above formula can beexpanded into series of two-variable Hermite polynomials119867119899

(119909 119910)

exp(minus(119863minus1

119909

minus 120585)2

4119905) 1

= 119890minus120585

24119905

infin

sum

119899=0

119863minus119899

119909

119899119867119899

(120585

2119905 minus1

4119905)

= 119890minus120585

24119905

infin

sum

119899=0

119909119899

(119899)2

119867119899

(120585

2119905 minus1

4119905)

(A3)

Taking into account formula (12) for 119909119899119867119899

(1205852119905 minus14119905) inthe operational identity above which can be viewed asa generating function in terms of inverse derivative for

8 The Scientific World Journal

119867119899

(119909 119910) we obtain the following expansion for the kernel ofthe integral

exp(minus(119863minus1

119909

minus 120585)

4119905) 1 = 119890minus120585

24119905

infin

sum

119899=0

1

(119899)2

119867119899

(119909120585

2119905 minus1199092

4119905)

(A4)

where the series of Hermite polynomials of two arguments119867119899

(119909 119910) can be expressed in terms of Hermite-Bessel-Tricomi functions

119867

119862119899

(119909 119910)mdashgeneralization of Bessel-Tricomi functionsmdashand related to Bessel-Wright functionsand to common Bessel functions [17]

119867

119862119899

(119909 119910) =

infin

sum

119898=0

119867119898

(minus119909 119910)

119903 (119899 + 119898) 119899 isin 119873

0

(A5)

In particular for 119899 = 0 we immediately find our series

119867

1198620

(119909 119910) =

infin

sum

119898=0

119867119898

(minus119909 119910)

(119898)2

(A6)

which obviously leads to the solution of the heat-typepropagation problem (40) by the following appropriate Gausstransform

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

119890minus120585

24119905

119867

1198620

(minus120585119909

2119905 minus1199092

4119905)120593 (120585) 119889120585

(A7)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] A Appel and J Kampe de Feriet Fonctions Hypergeometriqueset Hyperspheriques Polynomes drsquoHermite Gauthier-VillarsParis France 1926

[2] D T Haimo and C Markett ldquoA representation theory for solu-tions of a higher order heat equation Irdquo Journal ofMathematicalAnalysis and Applications vol 168 no 1 pp 89ndash107 1992

[3] D T Haimo and C Markett ldquoA representation theory forsolutions of a higher order heat equation IIrdquo Journal ofMathematical Analysis and Applications vol 168 no 2 pp 289ndash305 1992

[4] G Dattoli ldquoGeneralized polynomials operational identitiesand their applicationsrdquo Journal of Computational and AppliedMathematics vol 118 no 1-2 pp 111ndash123 2000

[5] G Dattoli H M Srivastava and K Zhukovsky ldquoOrthogonalityproperties of the Hermite and related polynomialsrdquo Journal ofComputational andAppliedMathematics vol 182 no 1 pp 165ndash172 2005

[6] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol 2 McGraw-Hill BookCompany New York NY USA 1953

[7] G Dattoli B GermanoM RMartinelli and P E Ricci ldquoNega-tive derivatives and special functionsrdquoAppliedMathematics andComputation vol 217 no 8 pp 3924ndash3928 2010

[8] G Dattoli M Migliorati and S Khan ldquoSolutions of integro-differential equations and operational methodsrdquo Applied Math-ematics and Computation vol 186 no 1 pp 302ndash308 2007

[9] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Mathematics and its Applications (Ellis HorwoodLtd) Halsted Press JohnWiley and Sons Chichester UK 1984

[10] GDattoli HM Srivastava andK Zhukovsky ldquoA new family ofintegral transforms and their applicationsrdquo Integral Transformsand Special Functions vol 17 no 1 pp 31ndash37 2006

[11] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 no 1 pp 1ndash174 1962

[12] G Dattoli H M Srivastava and K Zhukovsky ldquoOperationalmethods and differential equations with applications to initial-value problemsrdquo Applied Mathematics and Computation vol184 no 2 pp 979ndash1001 2007

[13] G Dattoli and K V Zhukovsky ldquoAppel polynomial seriesexpansionrdquo InternationalMathematical Forum vol 5 no 14 pp649ndash6662 2010

[14] G Dattoli M Migliorati and K Zhukovsky ldquoSummationformulae and stirling numbersrdquo International MathematicalForum vol 4 no 41 pp 2017ndash22040 2009

[15] J E Avron and I W Herbst ldquoSpectral and scattering theory ofSchrodinger operators related to the stark effectrdquo Communica-tions in Mathematical Physics vol 52 no 3 pp 239ndash254 1977

[16] K B Wolf Integral Transforms in Science and EngineeringPlenum Press New York NY USA 1979

[17] G N Watson A Treatise on the Theory of Bessel FunctionsCambridgeUniversity Press CambridgeUK 2nd edition 1944

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 The Scientific World Journal

In addition the translation operated by Θ shifts the argu-ment 119891(119909 119905) rarr 119891(119909 + 119886119887 119905) 119887 = 120573119905 Thus we obtain thesolution in a form of Hermite polynomial as follows

119865 (119909 119905) = sum

119896

119888119896

119890Φ+Φ

1119867119896

(119909 +1205721205742

120573(2119905120573

120574+ 1199052

1205732

1205742) 120572119905)

(76)

where Φ is defined by (64) and

Φ1

= 120574(119909 +1205721205742

120573(119905120573

120574+ (119905

120573

120574)

2

)) (77)

It is now evident that for a short time when 119905 ≪ 120574120573 thesolution will be spanning in space off the initial functionmodulated by 119867

119896

(119909 120572119905) and exponentially depending ontime

119865(119909 119905)|119905≪120574120573

asymp sum

119896

119888119896

exp (1205721205742119905 + 120574119909)119867119896

(119909 120572119905)

119865(119909 119905)|119905≪120574120573119905≪1

asymp sum

119896

119888119896

119890120572120574

2119905

119890120574119909

119909119896

(78)

Note that for extended times 119905 ≫ 120574120573 we have dominanceof time variable Φ ≫ Φ

1

and the solution asymptoticallybehaves as exp(120573119905119909) while119909 playsminor role in119867

119896

(119909+2120574120572119905+

1205721205731199052

120572119905)The same operational technique as employed for the

treatise of (54) can be easily adopted for the solution of theSchrodinger equation

119894ℏ120597119905

Ψ (119909 119905) = minusℏ2

21198981205972

119909

Ψ (119909 119905) + 119865119909Ψ (119909 119905)

Ψ (119909 0) = 119891 (119909)

(79)

where 119865 is constant (119865 has the dimension of force) Indeedrescaling variables in (79) we obtain the form of equationsimilar to (54)

119894120597120591

Ψ (119909 120591) = minus1205972

119909

Ψ (119909 120591) + 119887119909Ψ (119909 120591) (80)

where

120591 =ℏ119905

2119898 119887 =

2119865119898

ℏ2 (81)

Following the operational methodology developed for (54)we write the following solution of (80) in the form (56)

Ψ (119909 120591) =119880119891 (119909) (82)

where119880 = exp (119894120591) = 120597

2

119909

minus 119887119909 (83)

Then on account of the substitution 119905 rarr 119894120591 120573 rarr minus119887operators

Θ = exp (1198871205912120597

119909

) Θ119891 (119909 120591) = 119891 (119909 + 119887120591

2

120591) (84)

119878 = exp (1198941205911205972

119909

) 119878119891 (119909) = 119891 (119909 119894120591) (85)

ariseThus the solution of the Schrodinger equation is a resultof consequent action of the operator 119878 and further action ofΘ on the initial condition function

Ψ (119909 119905) = exp (minus119894Φ (119909 120591 119887)) Θ 119878119891 (119909) (86)

where Φ is defined by (64) The integral form of the solutionthen is written as follows

Ψ (119909 119905) = exp (minus119894Φ (119909 120591 119887)) 1

2radic119894120587120591

times int

infin

minusinfin

exp(minus(119909 + 119887120591

2

minus 120585)2

4119894120591)119891 (120585) 119889120585

(87)

Again as well as in (70) without any assumption onthe nature of the initial condition function 119891(119909) of theSchrodinger equation its solution

Ψ (119909 120591) = exp (minus119894Φ (119909 120591 119887)) 119891 (119909 + 1198871205912 119894120591) (88)

where 119891(119909 119905) is given by (85) and is expressed in terms ofthe function of two variables obtained by the consequentapplication of the heat propagation and translation operators(85) and (84) to119891(119909) So far we have demonstrated asGauss-Weierstrass transform (A1) describe the action of 119878on the initial probability amplitude and how the shift (84)finally yields the explicit form of the solution of Schrodingerequation It means that the result of the action of evolutionoperator (83) on 119891(119909) is the product of combined actionof translation operator Θ and heat propagation operator 119878representing the evolution operator of the free particle

Now let us consider another Fokker-Plank type equationthat is the following example

120597119905

119865 (119909 119905) = 1205721205972

119909

119865 (119909 119905) + 120573119909120597119909

119865 (119909 119905) (89)

with initial condition (55) Proceeding along the aboveoutlined scheme of the solution of (54) we write the solutionof (89) in general form (56)

119865 (119909 119905) = 119891 (119909) = exp (119860 + 119861) (90)

where operators 119860 and 119861 are defined as follows

119860 = 1199051205721205972

119909

119861 = 119905120573119909120597119909

(91)

Quantities 119860 and 119861 evidently do not commute

[119860 119861] = 21205721205731199052

1205972

119909

= 119898119860 119898 = 2119905120573 (92)

It allows disentanglement of the operators in the exponentialaccording to the following rule

exp (119860 + 119861) = exp(1 minus exp (minus119898)119898

119860) exp (119861) (93)

The Scientific World Journal 7

Thus the evolution operator action on 119891(119909) given below

119891 (119909) = exp (1205901205972119909

) exp (120573119905119909120597119909

) 119891 (119909)

= exp (1205901205972119909

) 119891 (119890120573119905

119909)

(94)

simply reduces to a Gauss transforms where the parameter120590 = 120590(119905) reads as follows

120590 (119905) =(1 minus exp (minus2120573119905)) 120572

120573 (95)

Upon the trivial change of variables we obtain

119891 (119909) = 119878119891 (119910) 119910 = 119909 exp (119887) 119878 = exp(120588

21205972

119910

)

(96)

where

120588 (119905) =120572

120573(1198902120573119905

minus 1) (97)

and eventually we end up with the following simple solutionof (89)

119865 (119909 119905) =1

radic2120587120588int

infin

minusinfin

119890minus(exp(120573119905)119909minus120585)22120588

119891 (120585) 119889120585 (98)

The same example of the initial Gaussian function 119891(119909) =exp(minus1199092) as in the case of (54) yields (compare with (71))

119865(119909 119905)|119891(119909)=exp(minus1199092) =

1

radic1 + 2120588 (119905)exp(minus 119890

2120573119905

1199092

1 + 2120588 (119905))

(99)

where 120588 is defined in (97) Note that we can meet thefollowing modified form of Fokker-Plank type equation (89)

120597119905

119865 (119909 119905) = 1205721205972

119909

119865 (119909 119905) + 120573120597119909

119909119865 (119909 119905) (100)

in problems related to propagation of electron beams inaccelerators Its solution arises from (99) immediately anddiffers from it just by a factor exp(120573119905) as written below fora Gaussian 119891(119909)

119865(119909 119905)|119891(119909)=exp(minus1199092) =

119890120573119905

radic1 + 2120588 (119905)exp(minus 119890

2120573119905

1199092

1 + 2120588 (119905))

=1

radic120578 (119905)exp(minus 119909

2

120578 (119905))

120578 (119905) = 2120572

120573(1 minus 119890

minus2120573119905

+120573

2120572119890minus2120573119905

)

(101)

However differently from the solution of (54) where we hadconsequent transforms of the initial condition function 119891(119909)by operators of translation and heat diffusion 119878 (63) (see also[15]) here we have just the action of 119878 alone with much morecomplicated dependence of the solution on time

5 Conclusions

Operational method is fast and universal mathematical toolfor obtaining solutions of differential equations Combina-tion of operational method integral transforms and theoryof special functions together with orthogonal polynomi-als closely related to them provides a powerful analyti-cal instrument for solving a wide spectrum of differentialequations and relevant physical problems The technique ofinverse operator applied for derivatives of various ordersand combined with integral transforms allows for easyand straightforward solutions of various types of differentialequations With operational approach we developed themethodology of inverse differential operators and deriveda number of operational identities with them We havedemonstrated that using the technique of inverse derivativesand inverse differential operators combinedwith exponentialoperator integral transforms and special functions we canmake significant progress in solution of variousmathematicalproblems and relevant physical applications described bydifferential equations

Appendix

In complete analogy with the heat equation solution byGauss-Weierstrass transform [16]

exp (1198871205972119909

) 119892 (119909) =1

2radic120587119887

int

infin

minusinfin

exp(minus(119909 minus 120585)2

4119887)119892 (120585) 119889120585

(A1)

accounting for noncommutative relation for operators ofinverse derivative 119863minus1

119909

and119871

119863119909

defined through the opera-tional relation (14) and accounting for (31) and (32) we writethe solution of (40) in the following form

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

exp(minus(119863minus1

119909

minus 120585)2

4119905) 1120593 (120585) 119889120585 (A2)

where 120593 is the image (32) of the initial condition function119892 The kernel of the integral in the above formula can beexpanded into series of two-variable Hermite polynomials119867119899

(119909 119910)

exp(minus(119863minus1

119909

minus 120585)2

4119905) 1

= 119890minus120585

24119905

infin

sum

119899=0

119863minus119899

119909

119899119867119899

(120585

2119905 minus1

4119905)

= 119890minus120585

24119905

infin

sum

119899=0

119909119899

(119899)2

119867119899

(120585

2119905 minus1

4119905)

(A3)

Taking into account formula (12) for 119909119899119867119899

(1205852119905 minus14119905) inthe operational identity above which can be viewed asa generating function in terms of inverse derivative for

8 The Scientific World Journal

119867119899

(119909 119910) we obtain the following expansion for the kernel ofthe integral

exp(minus(119863minus1

119909

minus 120585)

4119905) 1 = 119890minus120585

24119905

infin

sum

119899=0

1

(119899)2

119867119899

(119909120585

2119905 minus1199092

4119905)

(A4)

where the series of Hermite polynomials of two arguments119867119899

(119909 119910) can be expressed in terms of Hermite-Bessel-Tricomi functions

119867

119862119899

(119909 119910)mdashgeneralization of Bessel-Tricomi functionsmdashand related to Bessel-Wright functionsand to common Bessel functions [17]

119867

119862119899

(119909 119910) =

infin

sum

119898=0

119867119898

(minus119909 119910)

119903 (119899 + 119898) 119899 isin 119873

0

(A5)

In particular for 119899 = 0 we immediately find our series

119867

1198620

(119909 119910) =

infin

sum

119898=0

119867119898

(minus119909 119910)

(119898)2

(A6)

which obviously leads to the solution of the heat-typepropagation problem (40) by the following appropriate Gausstransform

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

119890minus120585

24119905

119867

1198620

(minus120585119909

2119905 minus1199092

4119905)120593 (120585) 119889120585

(A7)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] A Appel and J Kampe de Feriet Fonctions Hypergeometriqueset Hyperspheriques Polynomes drsquoHermite Gauthier-VillarsParis France 1926

[2] D T Haimo and C Markett ldquoA representation theory for solu-tions of a higher order heat equation Irdquo Journal ofMathematicalAnalysis and Applications vol 168 no 1 pp 89ndash107 1992

[3] D T Haimo and C Markett ldquoA representation theory forsolutions of a higher order heat equation IIrdquo Journal ofMathematical Analysis and Applications vol 168 no 2 pp 289ndash305 1992

[4] G Dattoli ldquoGeneralized polynomials operational identitiesand their applicationsrdquo Journal of Computational and AppliedMathematics vol 118 no 1-2 pp 111ndash123 2000

[5] G Dattoli H M Srivastava and K Zhukovsky ldquoOrthogonalityproperties of the Hermite and related polynomialsrdquo Journal ofComputational andAppliedMathematics vol 182 no 1 pp 165ndash172 2005

[6] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol 2 McGraw-Hill BookCompany New York NY USA 1953

[7] G Dattoli B GermanoM RMartinelli and P E Ricci ldquoNega-tive derivatives and special functionsrdquoAppliedMathematics andComputation vol 217 no 8 pp 3924ndash3928 2010

[8] G Dattoli M Migliorati and S Khan ldquoSolutions of integro-differential equations and operational methodsrdquo Applied Math-ematics and Computation vol 186 no 1 pp 302ndash308 2007

[9] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Mathematics and its Applications (Ellis HorwoodLtd) Halsted Press JohnWiley and Sons Chichester UK 1984

[10] GDattoli HM Srivastava andK Zhukovsky ldquoA new family ofintegral transforms and their applicationsrdquo Integral Transformsand Special Functions vol 17 no 1 pp 31ndash37 2006

[11] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 no 1 pp 1ndash174 1962

[12] G Dattoli H M Srivastava and K Zhukovsky ldquoOperationalmethods and differential equations with applications to initial-value problemsrdquo Applied Mathematics and Computation vol184 no 2 pp 979ndash1001 2007

[13] G Dattoli and K V Zhukovsky ldquoAppel polynomial seriesexpansionrdquo InternationalMathematical Forum vol 5 no 14 pp649ndash6662 2010

[14] G Dattoli M Migliorati and K Zhukovsky ldquoSummationformulae and stirling numbersrdquo International MathematicalForum vol 4 no 41 pp 2017ndash22040 2009

[15] J E Avron and I W Herbst ldquoSpectral and scattering theory ofSchrodinger operators related to the stark effectrdquo Communica-tions in Mathematical Physics vol 52 no 3 pp 239ndash254 1977

[16] K B Wolf Integral Transforms in Science and EngineeringPlenum Press New York NY USA 1979

[17] G N Watson A Treatise on the Theory of Bessel FunctionsCambridgeUniversity Press CambridgeUK 2nd edition 1944

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 7

Thus the evolution operator action on 119891(119909) given below

119891 (119909) = exp (1205901205972119909

) exp (120573119905119909120597119909

) 119891 (119909)

= exp (1205901205972119909

) 119891 (119890120573119905

119909)

(94)

simply reduces to a Gauss transforms where the parameter120590 = 120590(119905) reads as follows

120590 (119905) =(1 minus exp (minus2120573119905)) 120572

120573 (95)

Upon the trivial change of variables we obtain

119891 (119909) = 119878119891 (119910) 119910 = 119909 exp (119887) 119878 = exp(120588

21205972

119910

)

(96)

where

120588 (119905) =120572

120573(1198902120573119905

minus 1) (97)

and eventually we end up with the following simple solutionof (89)

119865 (119909 119905) =1

radic2120587120588int

infin

minusinfin

119890minus(exp(120573119905)119909minus120585)22120588

119891 (120585) 119889120585 (98)

The same example of the initial Gaussian function 119891(119909) =exp(minus1199092) as in the case of (54) yields (compare with (71))

119865(119909 119905)|119891(119909)=exp(minus1199092) =

1

radic1 + 2120588 (119905)exp(minus 119890

2120573119905

1199092

1 + 2120588 (119905))

(99)

where 120588 is defined in (97) Note that we can meet thefollowing modified form of Fokker-Plank type equation (89)

120597119905

119865 (119909 119905) = 1205721205972

119909

119865 (119909 119905) + 120573120597119909

119909119865 (119909 119905) (100)

in problems related to propagation of electron beams inaccelerators Its solution arises from (99) immediately anddiffers from it just by a factor exp(120573119905) as written below fora Gaussian 119891(119909)

119865(119909 119905)|119891(119909)=exp(minus1199092) =

119890120573119905

radic1 + 2120588 (119905)exp(minus 119890

2120573119905

1199092

1 + 2120588 (119905))

=1

radic120578 (119905)exp(minus 119909

2

120578 (119905))

120578 (119905) = 2120572

120573(1 minus 119890

minus2120573119905

+120573

2120572119890minus2120573119905

)

(101)

However differently from the solution of (54) where we hadconsequent transforms of the initial condition function 119891(119909)by operators of translation and heat diffusion 119878 (63) (see also[15]) here we have just the action of 119878 alone with much morecomplicated dependence of the solution on time

5 Conclusions

Operational method is fast and universal mathematical toolfor obtaining solutions of differential equations Combina-tion of operational method integral transforms and theoryof special functions together with orthogonal polynomi-als closely related to them provides a powerful analyti-cal instrument for solving a wide spectrum of differentialequations and relevant physical problems The technique ofinverse operator applied for derivatives of various ordersand combined with integral transforms allows for easyand straightforward solutions of various types of differentialequations With operational approach we developed themethodology of inverse differential operators and deriveda number of operational identities with them We havedemonstrated that using the technique of inverse derivativesand inverse differential operators combinedwith exponentialoperator integral transforms and special functions we canmake significant progress in solution of variousmathematicalproblems and relevant physical applications described bydifferential equations

Appendix

In complete analogy with the heat equation solution byGauss-Weierstrass transform [16]

exp (1198871205972119909

) 119892 (119909) =1

2radic120587119887

int

infin

minusinfin

exp(minus(119909 minus 120585)2

4119887)119892 (120585) 119889120585

(A1)

accounting for noncommutative relation for operators ofinverse derivative 119863minus1

119909

and119871

119863119909

defined through the opera-tional relation (14) and accounting for (31) and (32) we writethe solution of (40) in the following form

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

exp(minus(119863minus1

119909

minus 120585)2

4119905) 1120593 (120585) 119889120585 (A2)

where 120593 is the image (32) of the initial condition function119892 The kernel of the integral in the above formula can beexpanded into series of two-variable Hermite polynomials119867119899

(119909 119910)

exp(minus(119863minus1

119909

minus 120585)2

4119905) 1

= 119890minus120585

24119905

infin

sum

119899=0

119863minus119899

119909

119899119867119899

(120585

2119905 minus1

4119905)

= 119890minus120585

24119905

infin

sum

119899=0

119909119899

(119899)2

119867119899

(120585

2119905 minus1

4119905)

(A3)

Taking into account formula (12) for 119909119899119867119899

(1205852119905 minus14119905) inthe operational identity above which can be viewed asa generating function in terms of inverse derivative for

8 The Scientific World Journal

119867119899

(119909 119910) we obtain the following expansion for the kernel ofthe integral

exp(minus(119863minus1

119909

minus 120585)

4119905) 1 = 119890minus120585

24119905

infin

sum

119899=0

1

(119899)2

119867119899

(119909120585

2119905 minus1199092

4119905)

(A4)

where the series of Hermite polynomials of two arguments119867119899

(119909 119910) can be expressed in terms of Hermite-Bessel-Tricomi functions

119867

119862119899

(119909 119910)mdashgeneralization of Bessel-Tricomi functionsmdashand related to Bessel-Wright functionsand to common Bessel functions [17]

119867

119862119899

(119909 119910) =

infin

sum

119898=0

119867119898

(minus119909 119910)

119903 (119899 + 119898) 119899 isin 119873

0

(A5)

In particular for 119899 = 0 we immediately find our series

119867

1198620

(119909 119910) =

infin

sum

119898=0

119867119898

(minus119909 119910)

(119898)2

(A6)

which obviously leads to the solution of the heat-typepropagation problem (40) by the following appropriate Gausstransform

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

119890minus120585

24119905

119867

1198620

(minus120585119909

2119905 minus1199092

4119905)120593 (120585) 119889120585

(A7)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] A Appel and J Kampe de Feriet Fonctions Hypergeometriqueset Hyperspheriques Polynomes drsquoHermite Gauthier-VillarsParis France 1926

[2] D T Haimo and C Markett ldquoA representation theory for solu-tions of a higher order heat equation Irdquo Journal ofMathematicalAnalysis and Applications vol 168 no 1 pp 89ndash107 1992

[3] D T Haimo and C Markett ldquoA representation theory forsolutions of a higher order heat equation IIrdquo Journal ofMathematical Analysis and Applications vol 168 no 2 pp 289ndash305 1992

[4] G Dattoli ldquoGeneralized polynomials operational identitiesand their applicationsrdquo Journal of Computational and AppliedMathematics vol 118 no 1-2 pp 111ndash123 2000

[5] G Dattoli H M Srivastava and K Zhukovsky ldquoOrthogonalityproperties of the Hermite and related polynomialsrdquo Journal ofComputational andAppliedMathematics vol 182 no 1 pp 165ndash172 2005

[6] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol 2 McGraw-Hill BookCompany New York NY USA 1953

[7] G Dattoli B GermanoM RMartinelli and P E Ricci ldquoNega-tive derivatives and special functionsrdquoAppliedMathematics andComputation vol 217 no 8 pp 3924ndash3928 2010

[8] G Dattoli M Migliorati and S Khan ldquoSolutions of integro-differential equations and operational methodsrdquo Applied Math-ematics and Computation vol 186 no 1 pp 302ndash308 2007

[9] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Mathematics and its Applications (Ellis HorwoodLtd) Halsted Press JohnWiley and Sons Chichester UK 1984

[10] GDattoli HM Srivastava andK Zhukovsky ldquoA new family ofintegral transforms and their applicationsrdquo Integral Transformsand Special Functions vol 17 no 1 pp 31ndash37 2006

[11] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 no 1 pp 1ndash174 1962

[12] G Dattoli H M Srivastava and K Zhukovsky ldquoOperationalmethods and differential equations with applications to initial-value problemsrdquo Applied Mathematics and Computation vol184 no 2 pp 979ndash1001 2007

[13] G Dattoli and K V Zhukovsky ldquoAppel polynomial seriesexpansionrdquo InternationalMathematical Forum vol 5 no 14 pp649ndash6662 2010

[14] G Dattoli M Migliorati and K Zhukovsky ldquoSummationformulae and stirling numbersrdquo International MathematicalForum vol 4 no 41 pp 2017ndash22040 2009

[15] J E Avron and I W Herbst ldquoSpectral and scattering theory ofSchrodinger operators related to the stark effectrdquo Communica-tions in Mathematical Physics vol 52 no 3 pp 239ndash254 1977

[16] K B Wolf Integral Transforms in Science and EngineeringPlenum Press New York NY USA 1979

[17] G N Watson A Treatise on the Theory of Bessel FunctionsCambridgeUniversity Press CambridgeUK 2nd edition 1944

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 The Scientific World Journal

119867119899

(119909 119910) we obtain the following expansion for the kernel ofthe integral

exp(minus(119863minus1

119909

minus 120585)

4119905) 1 = 119890minus120585

24119905

infin

sum

119899=0

1

(119899)2

119867119899

(119909120585

2119905 minus1199092

4119905)

(A4)

where the series of Hermite polynomials of two arguments119867119899

(119909 119910) can be expressed in terms of Hermite-Bessel-Tricomi functions

119867

119862119899

(119909 119910)mdashgeneralization of Bessel-Tricomi functionsmdashand related to Bessel-Wright functionsand to common Bessel functions [17]

119867

119862119899

(119909 119910) =

infin

sum

119898=0

119867119898

(minus119909 119910)

119903 (119899 + 119898) 119899 isin 119873

0

(A5)

In particular for 119899 = 0 we immediately find our series

119867

1198620

(119909 119910) =

infin

sum

119898=0

119867119898

(minus119909 119910)

(119898)2

(A6)

which obviously leads to the solution of the heat-typepropagation problem (40) by the following appropriate Gausstransform

119891 (119909 119905) =1

2radic120587119905int

infin

minusinfin

119890minus120585

24119905

119867

1198620

(minus120585119909

2119905 minus1199092

4119905)120593 (120585) 119889120585

(A7)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] A Appel and J Kampe de Feriet Fonctions Hypergeometriqueset Hyperspheriques Polynomes drsquoHermite Gauthier-VillarsParis France 1926

[2] D T Haimo and C Markett ldquoA representation theory for solu-tions of a higher order heat equation Irdquo Journal ofMathematicalAnalysis and Applications vol 168 no 1 pp 89ndash107 1992

[3] D T Haimo and C Markett ldquoA representation theory forsolutions of a higher order heat equation IIrdquo Journal ofMathematical Analysis and Applications vol 168 no 2 pp 289ndash305 1992

[4] G Dattoli ldquoGeneralized polynomials operational identitiesand their applicationsrdquo Journal of Computational and AppliedMathematics vol 118 no 1-2 pp 111ndash123 2000

[5] G Dattoli H M Srivastava and K Zhukovsky ldquoOrthogonalityproperties of the Hermite and related polynomialsrdquo Journal ofComputational andAppliedMathematics vol 182 no 1 pp 165ndash172 2005

[6] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol 2 McGraw-Hill BookCompany New York NY USA 1953

[7] G Dattoli B GermanoM RMartinelli and P E Ricci ldquoNega-tive derivatives and special functionsrdquoAppliedMathematics andComputation vol 217 no 8 pp 3924ndash3928 2010

[8] G Dattoli M Migliorati and S Khan ldquoSolutions of integro-differential equations and operational methodsrdquo Applied Math-ematics and Computation vol 186 no 1 pp 302ndash308 2007

[9] H M Srivastava and H L Manocha A Treatise on GeneratingFunctions Mathematics and its Applications (Ellis HorwoodLtd) Halsted Press JohnWiley and Sons Chichester UK 1984

[10] GDattoli HM Srivastava andK Zhukovsky ldquoA new family ofintegral transforms and their applicationsrdquo Integral Transformsand Special Functions vol 17 no 1 pp 31ndash37 2006

[11] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 no 1 pp 1ndash174 1962

[12] G Dattoli H M Srivastava and K Zhukovsky ldquoOperationalmethods and differential equations with applications to initial-value problemsrdquo Applied Mathematics and Computation vol184 no 2 pp 979ndash1001 2007

[13] G Dattoli and K V Zhukovsky ldquoAppel polynomial seriesexpansionrdquo InternationalMathematical Forum vol 5 no 14 pp649ndash6662 2010

[14] G Dattoli M Migliorati and K Zhukovsky ldquoSummationformulae and stirling numbersrdquo International MathematicalForum vol 4 no 41 pp 2017ndash22040 2009

[15] J E Avron and I W Herbst ldquoSpectral and scattering theory ofSchrodinger operators related to the stark effectrdquo Communica-tions in Mathematical Physics vol 52 no 3 pp 239ndash254 1977

[16] K B Wolf Integral Transforms in Science and EngineeringPlenum Press New York NY USA 1979

[17] G N Watson A Treatise on the Theory of Bessel FunctionsCambridgeUniversity Press CambridgeUK 2nd edition 1944

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of