Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2013 Article ID 421378 8 pageshttpdxdoiorg1011552013421378

Research ArticleUsing Homo-Separation of Variables for Solving Systems ofNonlinear Fractional Partial Differential Equations

Abdolamir Karbalaie Hamed Hamid Muhammed and Bjorn-Erik Erlandsson

Division of Informatics Logistics and Management School of Technology and Health (STH) Royal Institute of Technology (KTH)100 44 Stockholm Sweden

Correspondence should be addressed to Abdolamir Karbalaie abdolamirkarbalaiesthkthse

Received 18 March 2013 Accepted 24 May 2013

Academic Editor Irena Lasiecka

Copyright copy 2013 Abdolamir Karbalaie et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A new method proposed and coined by the authors as the homo-separation of variables method is utilized to solve systems oflinear and nonlinear fractional partial differential equations (FPDEs) The new method is a combination of two well-establishedmathematical methods namely the homotopy perturbation method (HPM) and the separation of variables method Whencompared to existing analytical and numerical methods the method resulting from our approach shows that it is capable ofsimplifying the target problem at hand and reducing the computational load that is required to solve it considerablyThe efficiencyand usefulness of this new general-purpose method is verified by several examples where different systems of linear and nonlinearFPDEs are solved

1 Introduction

In the recent years it has turned out that many phenomenain various technical and scientific fields can be describedvery successfully by using fractional calculus In particularfractional calculus can be employed to solve many problemswithin the biomedical research field and get better resultsSuch a practical application of fractional order models is touse these models to improve the behavior and efficiency ofbioelectrodes The importance of this application is based onthe fact that bioelectrodes are usually needed to be used forall types of biopotential recording and signal measurementpurposes such as electroencephalography (EEG) electro-cardiography (ECG) and electromyography (EMG) [1ndash3]Another promising biomedical application field is proposedby Arafa et al [4] where a fractional-order model of HIV-1 infection of CD4+ T cells is introduced Other examplesof applications of fractional calculus in life sciences andtechnology can be found in botanics [5] biology [6] rheology[7] and elastography [8]

Numerous analytical methods have been presented inthe literature to solve FPDEs such as the fractional Greensfunction method [9] the Fourier transform method [2]

the Sumudu transform method [10] the Laplace transformmethod and theMellin transformmethod [11] Some numer-ical methods have also widely been used to solve systemsof FPDEs such as the variational iteration method [12]the Adomian decomposition method [2] the homotopyperturbationmethod [13] and the homotopy analysis method[14] Some of these methods use specific transformations andothers give the solution as a series which converges to theexact solution

In addition some numerical methods use a combinationof utilizing specific transformations and obtaining serieswhich converge to the exact solutions An example of sucha method is the iterative Laplace transform method whichis a combination of the Laplace transform method and aniterative method [15] Another such a combination is thehomotopy perturbation transformation method which isconstructed by combining two powerful methods namelythe Laplace transform method and the homotopy perturba-tion method [16] A third example is the Sumudu decom-position method which is a combination of the Sumudutransform method and Adomian decomposition method[17] A fourth such an approach is combining the Sumudutransformation method with the homotopy perturbation

2 International Journal of Mathematics and Mathematical Sciences

method which gives a new method called the homotopyperturbation Sumudu transform method [18]

Recently the homotopy perturbation method and theAdomian decomposition method are frequently used forsolving nonlinear FPDE problems

According to the combinational methods mentionedabove it is possible to notice that HPM has a strong potentialto be combined with another method to produce a moreefficient approachThemain reason is thatHPM is an efficientmethod for solving PDEs and ODEs (ordinary differentialequations) with integer or fractional order Recently Kar-balaie et al [19] found the exact solution of one-dimensionalFPDEs by using truncated versions of modified HPMThere-fore by getting inspiration of the ideas methods and toolsof previous works the novel approach called the homo-separation of variables method is developed and utilized tofind the exact solutions of systems of FPDEs The methodspresented in Karbalaie et al [19] are extended and developedfurther to achieve methods that can solve n-dimensionalsystems of FPDEs This new approach is constructed by asmart combination of HPM and the separation of variablesmethod By using this method the system of FPDEs to besolved is changed into a system of FODEs Consequently theoriginal problem is simplified considerably which makes itmore straightforward and easier to solve

The structure of the current paper is as follows Foreasy reader-friendly reference some basic definitions andproperties of fractional differential equations are presentedin Section 2 In Section 3 the homo-separation of variablesmethod is described After that in Section 4 three examplesare presented to show the simplicity and efficiency (at thesame time) of the homo-separation of variables methodFinally in Section 5 relevant conclusions are drawn

2 Basic Definitions and Properties

In this section to make it easier to introduce the new ap-proach of this researchwork a number of good-to-know con-cepts and properties within the field of fractional differentialequations are defined and presented briefly

Definition 1 A real function 119891(119905) 119905 gt 0 is said to be in thespace 119862

120590 120590 isin R if there exists a real number 119901 gt 120590 such

that 119891(119905) = 1199051199011198911(119905) where 119891

1(119905) isin 119862[0infin) and it is said to be

in the space 119862119898120590if 119891119898 isin 119862

120590 119898 isin N

Definition 2 The left sided Riemann-Liouville fractionalintegral of order 120572 ge 0 of a function 119891 isin 119862

120590 120590 ge minus1 is

defined as

119869120572119905119891 (119905)

def=

1

Γ (120572)int119905

0

(119905 minus 120591)120572minus1119891 (120591) 119889120591 (1)

where 120572 gt 0 119905 gt 0 and Γ(120572) = intinfin0119903120572minus1119890minus119903119889119903 is the gamma

functionProperties of the operator 119869120572

119905(for119891 isin 119862

120583 120583 ge minus1 120572 120573 ge

0 120574 ge minus1) can be found in [20] and are defined as follows

(1) 1198690119905119891(119905) = 119891(119905)

(2) 119869120572119905119869120573

119905119891(119905) = 119869

120572+120573

119905119891(119905)

(3) 119869120572119905119905120574 = (Γ(120574 + 1)Γ(120572 + 120574 + 1))119905120572+120574

Definition 3 The left sided Caputo fractional derivative of119891 (119891 isin 119862119899

120583 119899 isin N cup 0) in Caputo sense is defined by [11]

as follows

119863120572119905119891 (119905)

def=

1

Γ (119899 minus 120572)int119905

0

(119905 minus 120591)119899minus120572minus1119891(119899) (120591) 119889120591 119899 minus 1 lt 120572 le 119899

119863119899119905119891 (119905) 120572 = 119899

(2)

Note that according to [21] (1) and (2) become

119869120572119905119891 (119909 119905) =

1

Γ (120572)int119905

0

(119905 minus 120591)120572minus1119891 (119909 120591) 119889120591 120572 gt 0 119905 gt 0

119863120572119905119891 (119909 119905) =

1

Γ (119899 minus 120572)int119905

0

(119905 minus 120591)119899minus120572minus1119891(119899) (119909 120591) 119889120591

119899 minus 1 lt 120572 le 119899

(3)

Definition 4 The single-parameter and the two-parametervariants of the Mittag-leffler function are denoted by 119864

120572(119905)

and 119864120572120573(119905) respectivelyThese two variants are relevant to be

considered here because of their connection with fractionalcalculus and are defined as

119864120572(119905) =

infin

sum119895=0

119905119895

Γ (120572119895 + 1) 120572 gt 0 119905 isin C

119864120572120573(119905) =

infin

sum119895=0

119905119895

Γ (120572119895 + 120573) 120572 120573 gt 0 119905 isin C

(4)

The 119896th derivatives of these two variants are

119864(119896)120572(119905) =

119889119896

119889119905119896119864120572(119905)

=infin

sum119895=0

(119896 + 119895)119905119895

119895Γ (120572119895 + 120572119896 + 1) 119896 = 0 1 2

119864(119896)120572120573(119905) =

119889119896

119889119905119896119864120572120573(119905)

=infin

sum119895=0

(119896 + 119895)119905119895

119895Γ (120572119895 + 120572119896 + 120573) 119896 = 0 1 2

(5)

Some special cases and properties of the Mittag-Lefflerfunctions are the following

(1) 1198641205721(119905) = 119864

120572(119905)

(2) 11986411(119905) = 119864

1(119905) = 119890119905

(3) 11986421(1199052) = cosh(119905)

(4) 11986422(1199052) = sinh(119905)119905

International Journal of Mathematics and Mathematical Sciences 3

(5) 119864120572120573(119905) + 119864

120572120573(minus119905) = 2119864

2120572120573(1199052)

(6) 119864120572120573(119905) minus 119864

120572120573(minus119905) = 2119905119864

2120572120572+120573(1199052)

(7) 119863120572119905(119905120572minus1119864

120572120572(119886119905120572)) = 119905120572minus1119864

120572120572(119886119905120572)

Further properties of the Mittag-Leffler functions canbe found in [21] These functions are generalizations ofthe exponential function therefore most linear differentialequations of fractional order (FPDEs) have solutions that areexpressed in terms of these functions

Important Note According to [22] 119864120572(119886119905120572) cannot satisfy the

following semigroup property

119864120572(119886(119905 + 119904)

120572) = 119864120572(119886119905120572) 119864

120572(119886119904120572) 119905 119904 ge 0 119886 isin R (6)

for any noninteger 120572 gt 0 The above property is not trueunless in the special cases where 120572 = 1 or = 0 Unfortunatelythis misunderstanding can be found in a number of publica-tions which can be misleading and confusing

Definition 5 Consider the following relaxation-oscillationequation which is discussed in [23]

119863120572119905119906 (119905) + 119860119906 (119905) = 119891 (119905) 119905 gt 0 (7)

subject to the initial condition

120597119894

120597119905119894119906(119905)

100381610038161003816100381610038161003816100381610038161003816119905=0= 119906119894 (0) = 119887

119894119894 = 1 2 119899 minus 1 (8)

where 119887119894(119894 = 1 2 119899minus1) are real constants and 119899minus1 lt 120572 le

119899 The solution of (7) is obtained according to [23] as follows

119906 (119905) = int119905

0

(119905 minus 120591)120572minus1119864120572120572(minus119860(119905 minus 120591)

120572) 119891 (120591) 119889120591

+119899minus1

sum119895=0

119887119895119863120572minus119895minus1

119905(119905120572minus1119864

120572120572(minus119860119905120572))

(9)

where 119864120572120572

is the Mittag-Leffler function It is easy to realizethat if 0 lt 120572 le 1 then the solution of (7) becomes as follows

119906 (119905) = int119905

0

(119905 minus 120591)120572minus1119864120572120572(minus119860(119905 minus 120591)

120572) 119891 (120591) 119889120591

+ 119863120572minus1119905

(119905120572minus1119864120572120572(minus119860119905120572))

(10)

which will be used to solve the systems of FPDEs in theexamples discussed in this paper

Definition 6 According to [24] consider the following linearsystem of fractional differential equations with its initialcondition

119863120572119905x (119905) = 119860x (119905) 0 lt 119905 le 119886

x (0) = x0

(11)

where x isin R119899 119886 gt 0 the matrix 119860 isin R119899 times R119899 and 119863120572119905is the

Caputo fractional derivative of order 120572 where 0 lt 120572 le 1Thegeneral analytical solution for (11) is given by

x (119905) = 1198881u(1)119864120572(1205821119905120572) + 119888

2u(2)119864120572(1205822t120572)

+ sdot sdot sdot + 119888119899u(119899)119864120572(120582119899119905120572)

(12)

where 1198881 1198882 119888

119899are arbitrary constants 119864

120572is the Mittag-

Leffler function and 1205821 1205822 120582

119899and u(1) u(2) u(119899) are

the eigenvalues and the corresponding eigenvectors of thecharacteristic equation (119860minus120582

119894I) u(119894) = 0 where 119894 = 1 2 119899

and I is an 119899 times 119899 identity matrix

3 The Homo-Separation of Variables Method

This approach is achieved by a novel combination of HPMand separation of variables In this section the algorithmof this new method is presented briefly by considering thefollowing system of FPDEs

119863120572119894

119905119906119894= 119891119894(119909 119905) +L

119894(1199061 1199062 119906

119899)

+N119894(1199061 1199062 119906

119899) 119894 = 1 2 119899

(13)

subject to the initial conditions

119906119894(119909 0) = 119892

119894(119909) (14)

where 119909 isin R119899minus1 and for 119894 = 1 2 119899 the termsL119894represent linear operators the terms N

119894are nonlinear

operators while the terms 119891119894represent known analytical

functions In addition119863120572119894119905= 120597120572119894120597119905120572119894 is the Caputo fractional

derivative of order 120572119894 where 0 lt 120572

119894le 1 for 119894 = 1 2 119899

According to the homotopy perturbation technique(HPM) we can construct the following homotopies

(119901 minus 1) (119863120572119894

119905119881119894minus 119863120572119894

1199051199061198940) + 119901 (119863

120572119894

119905119881119894minus L119894minus N119894minus 119891119894(119909 119905))

= 0 119894 = 1 2 119899

(15)

where L119894= L119894(1198811 1198812 119881

119899) N119894= N119894(1198811 1198812 119881

119899) and

the homotopy parameter denoted by119901 is considered as small(119901 isin [0 1]) In addition for 119894 = 1 2 119899 119906

1198940= 1199061198940(119909 119905) is

an initial approximation of the solution of (13) which satisfiesthe initial condition in (14)

We can assume that the solution of (15) can be expressedas a power series in 119901 as given below in (16)

1198811=infin

sum119895=0

1199011198951198811119895= 11988110+ 11990111988111+ 1199012119881

12+ 1199013119881

13+ sdot sdot sdot

1198812=infin

sum119895=0

1199011198951198812119895= 11988120+ 11990111988121+ 1199012119881

22+ 1199013119881

23+ sdot sdot sdot

119881119899=infin

sum119895=0

119901119895119881119899119895= 1198811198990+ 1199011198811198991+ 1199012119881

1198992+ 1199013119881

1198993+ sdot sdot sdot

(16)

4 International Journal of Mathematics and Mathematical Sciences

By substituting (16) into (15) and equating the terms withidentical powers of119901 a set of equations is obtained as follows

1199010 119863120572119894

1199051198811198940minus 119863120572119894

1199051199061198940= 0 (17)

1199011 119863120572119894

1199051198811198941+119863120572119894

11990511990610

minusL119894(11988110 11988120 119881

1198990) minusM

1198940minus 119891119894= 0

(18)

119901119895 119863120572119894

119905119881119894119895minusL119894(1198811119895minus1

1198812119895minus1

119881119899119895minus1

) minusM119894119895minus1

= 0

(19)

where the terms M119894119895(119894 = 1 2 119899 and 119895 = 0 1 ) are

obtained from (17)ndash(19) by equating the coefficients of thenonlinear operators N

119894119895(119894 = 1 2 119899 and 119895 = 0 1 )

with the identical powers 119895 of 119901 that is the terms containing119901119895

In case the 119901-parameter is considered as small the bestapproximate solution of (13) can be readily obtained asfollows

119906119894(119909 119905) = lim

119901rarr1

infin

sum119895=0

119901119895119881119894119895(119909 119905)

= 1198811198940(119909 119905) + 119881

1198941(119909 119905) + 119881

1198942(119909 119905) + sdot sdot sdot

119894 = 1 2 119899

(20)

where 119909 = (1199091 1199092 119909

119899minus1)

If in (20) there exists some 119881119894119873(119909 119905) = 0 where 119873 ge 1

and 119894 = 1 2 119899 then the exact solution can be written inthe following form

119906119894(119909 119905) = 119881

1198940(119909 119905) + 119901119881

1198941(119909 119905) + sdot sdot sdot + 119901

119873minus1119881119894119873minus1

(119909 119905)

119894 = 1 2 119899

(21)

For simplicity we assume that 1198811198941(119909 119905) equiv 0 in (21) which

means that the exact solution in (13) is simplified as follows119906119894(119909 119905) = 119881

1198940(119909 119905) 119894 = 1 2 119899 (22)

Now when solving (17) we obtain the following result1198811198940(119909 119905) = 119906

1198940(119909 119905) 119894 = 1 2 119899 (23)

By using (22) and (23) we have119906119894(119909 119905) = 119881

1198940(119909 119905) = 119906

1198940(119909 119905) 119894 = 1 2 119899 (24)

The core of the new method proposed in this researchwork is to formulate the initial approximation of (13) in theform of separation of variables as follows119906119894(119909 119905) = 119906

1198940(119909 119905) = 119906

119894(119909 0) 119888

1198941(119905)

+ (119899

sum119895=1

119863119909119895119906119894(119909 0)) 119888

1198942(119905)

=119892119894(119909) 1198881198941(119905)+(

119899

sum119895=1

119863119909119895119892119894(119909)) 119888

1198942(119905) 119894=1 2 119899

(25)

The task now is to find the terms 1198881198941(119905) and 119888

1198942(119905) to obtain

the exact solution of the system of FPDEs in (13) Since (25)satisfies the initial conditions in (14) we get

1198881198941(0) = 1 119888

1198942(0) = 0 119894 = 1 2 119899 (26)

By substituting (25) into (18) we get

119863120572119894

1199051198811198941= minus 119863

120572119894

119905(119892119894(119909) 1198881198941(119905) + (

119899

sum119895=1

119863119909119895119892119894(119909)) 119888

1198942(119905))

minus L119894minusM1198940+ 119891119894(119909 119905) equiv 0

(27)

where 119894 = 1 2 119899 and

L119894= L119894(1198921(119909) 11988811(119905) + (

119899

sum119895=1

1198631199091198951198921(119909)) 119888

12(119905)

(119892119899(119909) 1198881198991(119905) + (

119899

sum119895=1

119863119909119895119892119899(119909)) 119888

1198992(119905)))

M1198940= M1198940(1198921(119909) 11988811(119905) + (

119899

sum119895=1

1198631199091198951198921(119909)) 119888

12(119905)

(119892119899(119909) 1198881198991(119905) + (

119899

sum119895=1

119863119909119895119892119899(119909)) 119888

1198992(119905)))

(28)

Obviously our goal is finally achieved and the system ofFPDEs is changed into a system of FODEs Consequently theproblem at hand is simplified considerably as it is well knownthat solving a system of FODEs is generally more straightfor-ward andmuch easier than solving the corresponding systemof FPDEs

Finally the target unknowns 1198881198941(119905) and 119888

1198942(119905) can be

obtained by utilizing and solving the system of FODEs andthe initial conditions in (26)

4 Applications

In this section we illustrate the applicability simplicity andefficiency of the homo-separation of variables method forsolving systems of linear and nonlinear FPDEs

Example 7 Consider the following system of linear FPDEs

119863120572119905119906 = V119909minus V minus 119906

119863120572119905V = 119906119909minus V minus 119906

(29)

where 0 lt 120572 le 1 subject to the initial conditions119906 (119909 0) = sinh (119909) V (119909 0) = cosh (119909) (30)

To solve (29) and (30) by using the proposed homo-separation of variables method we choose the initial approx-imations for (29) as follows

1199060(119909 119905) = 119906 (119909 0) 119887

1(119905) + 119863

119909119906 (119909 0) 119887

2(119905)

V0(119909 119905) = V (119909 0) 119888

1(119905) + 119863

119909V (119909 0) 119888

2(119905)

(31)

International Journal of Mathematics and Mathematical Sciences 5

By substituting the two initial conditions given by (30)into these two initial approximations we get the followinginitial approximations

1199060(119909 119905) = 119887

1(119905) sinh (119909) + 119887

2(119905) cosh (119909)

V0(119909 119905) = 119888

1(119905) cosh (119909) + 119888

2(119905) sinh (119909)

(32)

By considering and following the approach whichresulted in (27) and substituting the results presented by (32)into the system of linear FPDEs in (29) we get

1198631199051199061= sinh (119909) (119863120572

1199051198871(119905) + 119887

1(119905) minus 119888

1(119905) + 119888

2(119905))

+ cosh (119909) (1198631205721199051198872(119905) + 119887

2(119905) + 119888

1(119905) minus 119888

2(119905)) equiv 0

119863119905V1= cosh (119909) (119863120572

1199051198881(119905) + 119888

1(119905) minus 119887

1(119905) + 119887

2(119905))

+ sinh (119909) (1198631205721199051198882(119905) + 119888

2(119905) + 119887

1(119905) minus 119887

2(119905)) equiv 0

(33)

By solving the latter system of linear FPDEs presented in(33) we obtain the following system of linear FODEs

1198631205721199051198871(119905) + 119887

1(119905) minus 119888

1(119905) + 119888

2(119905) = 0

1198631205721199051198872(119905) + 119887

2(119905) + 119888

1(119905) minus 119888

2(119905) = 0

1198631205721199051198881(119905) + 119888

1(119905) minus 119887

1(119905) + 119887

2(119905) = 0

1198631205721199051198882(119905) + 119888

2(119905) + 119887

1(119905) minus 119887

2(119905) = 0

(34)

together with the initial conditions

1198872(0) = 1 119887

2(0) = 0

1198881(0) = 1 119888

1(0) = 0

(35)

By solving (34) and (35) by considering and applyingDefinition 6 which is discussed in Section 2 we obtain thefollowing results

1198871(119905) =

1

2(119864120572(119905120572) + 119864

120572(minus119905120572))

1198872(119905) =

1

2(minus119864120572(119905120572) + 119864

120572(minus119905120572))

1198881(119905) =

1

2(119864120572(119905120572) + 119864

120572(minus119905120572))

1198882(119905) =

1

2(minus119864120572(119905120572) + 119864

120572(minus119905120572))

(36)

By substituting (36) into (32) and applying Definition 4in particular properties (2) (5) and (6) which are discussedin Section 2 we obtain the exact solution for the system oflinear FPDEs in (29) as follows

119906 (119909 119905) = 11986421205721

(1199052120572) sinh (119909) minus 1199051205721198642120572120572+1

(1199052120572) cosh (119909)

V (119909 119905) = 11986421205721

(1199052120572) cosh (119909) minus 1199051205721198642120572120572+1

(1199052120572) sinh (119909) (37)

If we now put 120572 rarr 1 in (37) or solve (29) and (30)for 120572 = 1 we obtain the following exact solution for thecorresponding system of linear PDEs

119906 (119909 119905) = sinh (119909 minus 119905)

V (119909 119905) = cosh (119909 minus 119905) (38)

Example 8 Consider the following system of inhomoge-neous FPDEs

119863120572119905119906 + 119906119909V + 119906 = 1

119863120573

119905V minus V119909119906 minus V = 1

(39)

where 0 lt 120572 and 120573 le 1 subject to the initial conditions

119906 (119909 0) = 119890119909 V (119909 0) = 119890minus119909 (40)

To solve (39) and (40) by using the new homo-separationof variables method we choose the initial approximations for(39) as follows

1199060(119909 119905) = 119906 (119909 0) 119888

1(119905) + 119863

119909119906 (119909 0) 119888

2(119905)

V0(119909 119905) = V (119909 0) 119887

1(119905) + 119863

119909V (119909 0) 119887

2(119905)

(41)

By substituting the two initial conditions given by (40)into these two initial approximations we get the followinginitial approximations

1199060(119909 119905) = 119890

1199091198881(119905) + 119890

1199091198882(119905)

V0(119909 119905) = 119890

minus1199091198871(119905) minus 119890

minus1199091198872(119905)

(42)

By considering and following the approach whichresulted in (27) and substituting the results presented by (42)into the system of inhomogeneous FPDEs that we would liketo solve which is presented by (39) we get

1198631205721199051199061= 119890119909 (119863120572

119905(1198881(119905) + 119888

2(119905)) + (119888

1(119905) + 119888

2(119905)))

+ (1198881(119905) + 119888

2(119905)) (119887

1(119905) minus 119887

2(119905)) minus 1 equiv 0

119863120573

119905V1= 119890119909 (119863

120573

119905(1198871(119905) minus 119887

2(119905)) minus (119887

1(119905) minus 119887

2(119905)))

+ (1198881(119905) + 119888

2(119905)) (119887

1(119905) minus 119887

2(119905)) minus 1 equiv 0

(43)

By solving the latter system of FPDEs presented in (43)we obtain the following system of FODEs

119863120572119905(1198881(119905) + 119888

2(119905)) + (119888

1(119905) + 119888

2(119905)) = 0

119863120573

119905(1198871(119905) minus 119887

2(119905)) minus (119887

1(119905) minus 119887

2(119905)) = 0

(1198881(119905) + 119888

2(119905)) (119887

1(119905) minus 119887

2(119905)) minus 1 = 0

(44)

together with the initial conditions

1198871(0) = 1 119887

2(0) = 0

1198881(0) = 1 119888

2(119905) = 0

(45)

6 International Journal of Mathematics and Mathematical Sciences

By solving (44) and (45) by considering and applyingDefinition 5 which is discussed in Section 2 we obtain thefollowing results

1198881(119905) + 119888

2(119905) = 119864

120572(minus119905120572)

1198871(119905) minus 119887

2(119905) = 119864

120573(119905120573)

119864120572(minus119905120572) 119864

120573(119905120573) = 1

(46)

By substituting (46) into (42) we obtain the exact solutionfor the system of inhomogeneous FPDEs presented in (39) asfollows

119906 (119909 119905) = 119890119909119864120572(minus119905120572)

V (119909 119905) = 119890minus119909119864120573(119905120573)

119864120572(minus119905120572) 119864

120573(119905120573) = 1

(47)

Thus this means that 119906(119909 119905) = 1V(119909 119905) If we now put120572 120573 rarr 1 in (47) or solve (39) and (40) for 120572 = 120573 = 1we obtain the following exact solution for the correspondingsystem of inhomogeneous PDEs

119906 (119909 119905) = 119890119909minus119905

V (119909 119905) = 119890minus119909+119905(48)

Example 9 Consider the following system of nonlinearFPDEs

119863120572119905119906 = minus119906 minus V

119909119908119910minus V119910119908119909

119863120573

119905V = V minus 119906

119909119908119910minus 119906119910119908119909

119863120574

119905119908 = 119908 minus 119906

119909V119910minus 119906119910V119909

(49)

where 0 lt 120572 and 120573 120574 le 1 subject to the initial conditions

119906 (119909 119910 0) = 119890119909+119910 V (119909 119910 0) = 119890119909minus119910

119908 (119909 119910 0) = 119890minus119909+119910(50)

To solve (49) and (50) by using the new homo-separationof variables method we choose the initial approximations for(49) as follows

1199060(119909 119910 119905) = 119906 (119909 119910 0) 119887

1(119905)

+ (119863119909119906 (119909 119910 0) + 119863

119910119906 (119909 119910 0)) 119887

2(119905)

V0(119909 119910 119905) = V (119909 119910 0) 119888

1(119905)

+ (119863119909V (119909 119910 0) + 119863

119910V (119909 119910 0)) 119888

2(119905)

1199080(119909 119910 119905) = 119908 (119909 119910 0) 119889

1(119905)

+ (119863119909119908 (119909 119910 0) + 119863

119910119908 (119909 119910 0)) 119889

2(119905)

(51)

By substituting the initial conditions given by (50) intothese initial approximations we get the following initialapproximations

1199060(119909 119910 119905) = 119890119909+119910119887

1(119905) + 2119890

119909+1199101198872(119905)

V0(119909 119910 119905) = 119890119909minus119910119888

1(119905)

1199080(119909 119910 119905) = 119890minus119909+119910119889

1(119905)

(52)

By considering and following the approach whichresulted in (27) and substituting the results presented by (52)into the system of nonlinear FPDEs that we would like tosolve which is presented by (49) we get

1198631205721199051199061= 119890minus119909+119910 (119863120572

1199051198871(119905) + 2119863

120572

1199051198872(119905) + 119887

1(119905) + 2119887

2(119905)) equiv 0

119863120573

119905119863119905V1= 119890119909minus119910 (119863

120573

1199051198881(119905) minus 119888

1(119905)) equiv 0

119863120574

1199051199081= 119890minus119909+119910 (119863

120574

1199051198891(119905) minus 119889

1(119905)) equiv 0

(53)By solving the latter system of FPDEs presented in (53)

we obtain the following system of FODEs119863120572119905(1198871(119905) + 2119887

2(119905)) + (119887

1(119905) + 2119887

2(119905)) = 0

119863120573

1199051198881(119905) minus 119888

1(119905) = 0

119863120574

1199051198891(119905) minus 119889

1(119905) = 0

(54)

together with the initial conditions1198871(0) = 1 119887

2(0) = 0

1198881(0) = 1 119888

2(119905) = 0

1198891(0) = 1 119889

2(119905) = 0

(55)

By solving (54) and (55) by considering and applyingDefinition 5 which is discussed in Section 2 of this paper weobtain the following results

1198871(119905) minus 2119887

2(119905) = 119864

120572(minus119905120572)

1198881(119905) = 119864

120573(119905120573)

1198891(0) = 119864

120574(119905120574)

(56)

By substituting (56) into (52) we obtain the exact solutionfor the system of nonlinear FPDEs in (49) as follows

119906 (119909 119910 119905) = 119890119909+119910119864120572(minus119905120572)

V (119909 119910 119905) = 119890119909minus119910119864120573(119905120573)

119908 (119909 119910 119905) = 119890minus119909+119910119864120574(119905120574)

(57)

If we now put 120572 120573 120574 rarr 1 in (57) or solve (49) and (50)for 120572 = 120573 = 120574 = 1 we obtain the following exact solution forthe corresponding system of nonlinear PDEs

119906 (119909 119910 119905) = 119890119909+119910minus119905

V (119909 119910 119905) = 119890119909minus119910+119905

119908 (119909 119910 t) = 119890minus119909+119910+119905

(58)

International Journal of Mathematics and Mathematical Sciences 7

5 Conclusions

In this paper a novel approach was introduced and utilizedto solve linear and nonlinear systems of FPDEs The newtechnique was coined by the authors of this paper as thehomo-separation of variables method In this research workit was demonstrated through different examples how this newmethod can be used for solving various systems of FPDEs

When compared with the existing published methods itis easy to notice that the new method has many advantagesIt is straightforward easy to understand and fast requiringmuch less computations to perform a limited number ofsteps of the simple procedure that can be applied to find theexact solution of a wide range of types of FPDE equationsrsquosystems Furthermore there is no need for using linearizationor restrictive assumptions when employing this newmethod

The basic principle of the current method is to performan elegant combination (which is simple and smart) of twoexisting techniques These two techniques are the popularhomotopy perturbation method (HPM) that was originallyproposed by He [25 26] and another popular approachcalled separation of variables Finally the resulting homo-separation of variables method which is analytical can beused to solve various systems of PDEs with integer andfractional order for example with respect to time

Acknowledgments

The authors would like to thank Maryam Shabani andMehran Shafaiee for their help and support as well asthe inspiration the authors got through friendly scientificdiscussions

References

[1] R L Magin Fractional Calculus in Bioengineering BegellHouse West Redding Conn USA 2006

[2] R L Magin and M Ovadia ldquoModeling the cardiac tissueelectrode interface using fractional calculusrdquo JVCJournal ofVibration and Control vol 14 no 9-10 pp 1431ndash1442 2008

[3] J Keener and J SneydMathematical Physiology Springer NewYork NY USA 2004

[4] A A M Arafa S Z Rida and M Khalil ldquoFractional modelingdynamics of HIV and CD4+ T-cells during primary infectionrdquoNonlinear Biomedical Physics vol 6 no 1 article 1 2012

[5] I S Jesus J A Tenreiro MacHado and J Boaventure CunhaldquoFractional electrical impedances in botanical elementsrdquo Jour-nal of Vibration and Control vol 14 no 9-10 pp 1389ndash14022008

[6] K S Cole ldquoSketches of general and comparative demographyrdquoin Cold Spring Harbor Symposia on Quantitative Biology vol 1pp 107ndash116 Cold Spring Harbor Laboratory Press 1933

[7] V D Djordjevi J Jari B Fabry J J Fredberg andD StamenovildquoFractional derivatives embody essential features of cell rheo-logical behaviorrdquo Annals of Biomedical Engineering vol 31 no6 pp 692ndash699 2003

[8] C Coussot S Kalyanam R Yapp and M Insana ldquoFractionalderivative models for ultrasonic characterization of polymer

and breast tissue viscoelasticityrdquo IEEETransactions onUltrason-ics Ferroelectrics and Frequency Control vol 56 no 4 pp 715ndash725 2009

[9] F Mainardi ldquoFractional diffusive waves in viscoelastic solidsrdquoin Nonlinear Waves in Solids pp 93ndash97 Fairfield 1995

[10] V G Gupta and B Sharma ldquoApplication of Sumudu transformin reaction-diffusion systems and nonlinear wavesrdquo AppliedMathematical Sciences vol 4 no 9ndash12 pp 435ndash446 2010

[11] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego Calif USA 1999

[12] S Momani and Z Odibat ldquoNumerical approach to differentialequations of fractional orderrdquo Journal of Computational andApplied Mathematics vol 207 no 1 pp 96ndash110 2007

[13] O Abdulaziz I Hashim and S Momani ldquoSolving systemsof fractional differential equations by homotopy-perturbationmethodrdquo Physics Letters A vol 372 no 4 pp 451ndash459 2008

[14] M Ganjiani and H Ganjiani ldquoSolution of coupled systemof nonlinear differential equations using homotopy analysismethodrdquo Nonlinear Dynamics of Nonlinear Dynamics andChaos in Engineering Systems vol 56 no 1-2 pp 159ndash167 2009

[15] H Jafari M Nazari D Baleanu and C M Khalique ldquoA newapproach for solving a system of fractional partial differentialequationsrdquo Computers amp Mathematics with Applications 2012

[16] S Kumar A Yildirim and L Wei ldquoA fractional model of thediffusion equation and its analytical solution using Laplacetransformrdquo Scientia Iranica vol 19 no 4 pp 1117ndash1123 2012

[17] D Kumar J Singh and S Rathore ldquoSumudu decompositionmethod for nonlinear equationsrdquo International MathematicalForum vol 7 no 9ndash12 pp 515ndash521 2012

[18] J Singh D Kumar and Sushila ldquoHomotopy perturbationsumudu transform method for nonlinear equationsrdquo Advancesin Theoretical and Applied Mechanics vol 4 no 4 pp 165ndash1752011

[19] A Karbalaie M M Montazeri and H H Muhammed ldquoNewapproach to find the exact solution of fractional partial differ-ential equationrdquo WSEAS Transactions on Mathematics vol 11no 10 pp 908ndash917 2012

[20] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Publishers Yverdon Switzerland 1993

[21] A A M Arafa S Z Rida and H Mohamed ldquoHomotopyanalysis method for solving biological population modelrdquoCommunications in Theoretical Physics vol 56 no 5 pp 797ndash800 2011

[22] J Peng and K Li ldquoA note on property of the Mittag-Lefflerfunctionrdquo Journal of Mathematical Analysis and Applicationsvol 370 no 2 pp 635ndash638 2010

[23] J-F Cheng and Y-M Chu ldquoSolution to the linear fractionaldifferential equation using Adomian decomposition methodrdquoMathematical Problems in Engineering vol 2011 Article ID587068 14 pages 2011

[24] Z M Odibat ldquoAnalytic study on linear systems of fractionaldifferential equationsrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1171ndash1183 2010

[25] J He ldquoApplication of homotopy perturbation method to non-linear wave equationsrdquo Chaos Solitons amp Fractals vol 26 no 3pp 695ndash700 2005

8 International Journal of Mathematics and Mathematical Sciences

[26] J H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 6 pp 207ndash208 2005

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

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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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

2 International Journal of Mathematics and Mathematical Sciences

method which gives a new method called the homotopyperturbation Sumudu transform method [18]

Recently the homotopy perturbation method and theAdomian decomposition method are frequently used forsolving nonlinear FPDE problems

According to the combinational methods mentionedabove it is possible to notice that HPM has a strong potentialto be combined with another method to produce a moreefficient approachThemain reason is thatHPM is an efficientmethod for solving PDEs and ODEs (ordinary differentialequations) with integer or fractional order Recently Kar-balaie et al [19] found the exact solution of one-dimensionalFPDEs by using truncated versions of modified HPMThere-fore by getting inspiration of the ideas methods and toolsof previous works the novel approach called the homo-separation of variables method is developed and utilized tofind the exact solutions of systems of FPDEs The methodspresented in Karbalaie et al [19] are extended and developedfurther to achieve methods that can solve n-dimensionalsystems of FPDEs This new approach is constructed by asmart combination of HPM and the separation of variablesmethod By using this method the system of FPDEs to besolved is changed into a system of FODEs Consequently theoriginal problem is simplified considerably which makes itmore straightforward and easier to solve

The structure of the current paper is as follows Foreasy reader-friendly reference some basic definitions andproperties of fractional differential equations are presentedin Section 2 In Section 3 the homo-separation of variablesmethod is described After that in Section 4 three examplesare presented to show the simplicity and efficiency (at thesame time) of the homo-separation of variables methodFinally in Section 5 relevant conclusions are drawn

2 Basic Definitions and Properties

In this section to make it easier to introduce the new ap-proach of this researchwork a number of good-to-know con-cepts and properties within the field of fractional differentialequations are defined and presented briefly

Definition 1 A real function 119891(119905) 119905 gt 0 is said to be in thespace 119862

120590 120590 isin R if there exists a real number 119901 gt 120590 such

that 119891(119905) = 1199051199011198911(119905) where 119891

1(119905) isin 119862[0infin) and it is said to be

in the space 119862119898120590if 119891119898 isin 119862

120590 119898 isin N

Definition 2 The left sided Riemann-Liouville fractionalintegral of order 120572 ge 0 of a function 119891 isin 119862

120590 120590 ge minus1 is

defined as

119869120572119905119891 (119905)

def=

1

Γ (120572)int119905

0

(119905 minus 120591)120572minus1119891 (120591) 119889120591 (1)

where 120572 gt 0 119905 gt 0 and Γ(120572) = intinfin0119903120572minus1119890minus119903119889119903 is the gamma

functionProperties of the operator 119869120572

119905(for119891 isin 119862

120583 120583 ge minus1 120572 120573 ge

0 120574 ge minus1) can be found in [20] and are defined as follows

(1) 1198690119905119891(119905) = 119891(119905)

(2) 119869120572119905119869120573

119905119891(119905) = 119869

120572+120573

119905119891(119905)

(3) 119869120572119905119905120574 = (Γ(120574 + 1)Γ(120572 + 120574 + 1))119905120572+120574

Definition 3 The left sided Caputo fractional derivative of119891 (119891 isin 119862119899

120583 119899 isin N cup 0) in Caputo sense is defined by [11]

as follows

119863120572119905119891 (119905)

def=

1

Γ (119899 minus 120572)int119905

0

(119905 minus 120591)119899minus120572minus1119891(119899) (120591) 119889120591 119899 minus 1 lt 120572 le 119899

119863119899119905119891 (119905) 120572 = 119899

(2)

Note that according to [21] (1) and (2) become

119869120572119905119891 (119909 119905) =

1

Γ (120572)int119905

0

(119905 minus 120591)120572minus1119891 (119909 120591) 119889120591 120572 gt 0 119905 gt 0

119863120572119905119891 (119909 119905) =

1

Γ (119899 minus 120572)int119905

0

(119905 minus 120591)119899minus120572minus1119891(119899) (119909 120591) 119889120591

119899 minus 1 lt 120572 le 119899

(3)

Definition 4 The single-parameter and the two-parametervariants of the Mittag-leffler function are denoted by 119864

120572(119905)

and 119864120572120573(119905) respectivelyThese two variants are relevant to be

considered here because of their connection with fractionalcalculus and are defined as

119864120572(119905) =

infin

sum119895=0

119905119895

Γ (120572119895 + 1) 120572 gt 0 119905 isin C

119864120572120573(119905) =

infin

sum119895=0

119905119895

Γ (120572119895 + 120573) 120572 120573 gt 0 119905 isin C

(4)

The 119896th derivatives of these two variants are

119864(119896)120572(119905) =

119889119896

119889119905119896119864120572(119905)

=infin

sum119895=0

(119896 + 119895)119905119895

119895Γ (120572119895 + 120572119896 + 1) 119896 = 0 1 2

119864(119896)120572120573(119905) =

119889119896

119889119905119896119864120572120573(119905)

=infin

sum119895=0

(119896 + 119895)119905119895

119895Γ (120572119895 + 120572119896 + 120573) 119896 = 0 1 2

(5)

Some special cases and properties of the Mittag-Lefflerfunctions are the following

(1) 1198641205721(119905) = 119864

120572(119905)

(2) 11986411(119905) = 119864

1(119905) = 119890119905

(3) 11986421(1199052) = cosh(119905)

(4) 11986422(1199052) = sinh(119905)119905

International Journal of Mathematics and Mathematical Sciences 3

(5) 119864120572120573(119905) + 119864

120572120573(minus119905) = 2119864

2120572120573(1199052)

(6) 119864120572120573(119905) minus 119864

120572120573(minus119905) = 2119905119864

2120572120572+120573(1199052)

(7) 119863120572119905(119905120572minus1119864

120572120572(119886119905120572)) = 119905120572minus1119864

120572120572(119886119905120572)

Further properties of the Mittag-Leffler functions canbe found in [21] These functions are generalizations ofthe exponential function therefore most linear differentialequations of fractional order (FPDEs) have solutions that areexpressed in terms of these functions

Important Note According to [22] 119864120572(119886119905120572) cannot satisfy the

following semigroup property

119864120572(119886(119905 + 119904)

120572) = 119864120572(119886119905120572) 119864

120572(119886119904120572) 119905 119904 ge 0 119886 isin R (6)

for any noninteger 120572 gt 0 The above property is not trueunless in the special cases where 120572 = 1 or = 0 Unfortunatelythis misunderstanding can be found in a number of publica-tions which can be misleading and confusing

Definition 5 Consider the following relaxation-oscillationequation which is discussed in [23]

119863120572119905119906 (119905) + 119860119906 (119905) = 119891 (119905) 119905 gt 0 (7)

subject to the initial condition

120597119894

120597119905119894119906(119905)

100381610038161003816100381610038161003816100381610038161003816119905=0= 119906119894 (0) = 119887

119894119894 = 1 2 119899 minus 1 (8)

where 119887119894(119894 = 1 2 119899minus1) are real constants and 119899minus1 lt 120572 le

119899 The solution of (7) is obtained according to [23] as follows

119906 (119905) = int119905

0

(119905 minus 120591)120572minus1119864120572120572(minus119860(119905 minus 120591)

120572) 119891 (120591) 119889120591

+119899minus1

sum119895=0

119887119895119863120572minus119895minus1

119905(119905120572minus1119864

120572120572(minus119860119905120572))

(9)

where 119864120572120572

is the Mittag-Leffler function It is easy to realizethat if 0 lt 120572 le 1 then the solution of (7) becomes as follows

119906 (119905) = int119905

0

(119905 minus 120591)120572minus1119864120572120572(minus119860(119905 minus 120591)

120572) 119891 (120591) 119889120591

+ 119863120572minus1119905

(119905120572minus1119864120572120572(minus119860119905120572))

(10)

which will be used to solve the systems of FPDEs in theexamples discussed in this paper

Definition 6 According to [24] consider the following linearsystem of fractional differential equations with its initialcondition

119863120572119905x (119905) = 119860x (119905) 0 lt 119905 le 119886

x (0) = x0

(11)

where x isin R119899 119886 gt 0 the matrix 119860 isin R119899 times R119899 and 119863120572119905is the

Caputo fractional derivative of order 120572 where 0 lt 120572 le 1Thegeneral analytical solution for (11) is given by

x (119905) = 1198881u(1)119864120572(1205821119905120572) + 119888

2u(2)119864120572(1205822t120572)

+ sdot sdot sdot + 119888119899u(119899)119864120572(120582119899119905120572)

(12)

where 1198881 1198882 119888

119899are arbitrary constants 119864

120572is the Mittag-

Leffler function and 1205821 1205822 120582

119899and u(1) u(2) u(119899) are

the eigenvalues and the corresponding eigenvectors of thecharacteristic equation (119860minus120582

119894I) u(119894) = 0 where 119894 = 1 2 119899

and I is an 119899 times 119899 identity matrix

3 The Homo-Separation of Variables Method

This approach is achieved by a novel combination of HPMand separation of variables In this section the algorithmof this new method is presented briefly by considering thefollowing system of FPDEs

119863120572119894

119905119906119894= 119891119894(119909 119905) +L

119894(1199061 1199062 119906

119899)

+N119894(1199061 1199062 119906

119899) 119894 = 1 2 119899

(13)

subject to the initial conditions

119906119894(119909 0) = 119892

119894(119909) (14)

where 119909 isin R119899minus1 and for 119894 = 1 2 119899 the termsL119894represent linear operators the terms N

119894are nonlinear

operators while the terms 119891119894represent known analytical

functions In addition119863120572119894119905= 120597120572119894120597119905120572119894 is the Caputo fractional

derivative of order 120572119894 where 0 lt 120572

119894le 1 for 119894 = 1 2 119899

According to the homotopy perturbation technique(HPM) we can construct the following homotopies

(119901 minus 1) (119863120572119894

119905119881119894minus 119863120572119894

1199051199061198940) + 119901 (119863

120572119894

119905119881119894minus L119894minus N119894minus 119891119894(119909 119905))

= 0 119894 = 1 2 119899

(15)

where L119894= L119894(1198811 1198812 119881

119899) N119894= N119894(1198811 1198812 119881

119899) and

the homotopy parameter denoted by119901 is considered as small(119901 isin [0 1]) In addition for 119894 = 1 2 119899 119906

1198940= 1199061198940(119909 119905) is

an initial approximation of the solution of (13) which satisfiesthe initial condition in (14)

We can assume that the solution of (15) can be expressedas a power series in 119901 as given below in (16)

1198811=infin

sum119895=0

1199011198951198811119895= 11988110+ 11990111988111+ 1199012119881

12+ 1199013119881

13+ sdot sdot sdot

1198812=infin

sum119895=0

1199011198951198812119895= 11988120+ 11990111988121+ 1199012119881

22+ 1199013119881

23+ sdot sdot sdot

119881119899=infin

sum119895=0

119901119895119881119899119895= 1198811198990+ 1199011198811198991+ 1199012119881

1198992+ 1199013119881

1198993+ sdot sdot sdot

(16)

4 International Journal of Mathematics and Mathematical Sciences

By substituting (16) into (15) and equating the terms withidentical powers of119901 a set of equations is obtained as follows

1199010 119863120572119894

1199051198811198940minus 119863120572119894

1199051199061198940= 0 (17)

1199011 119863120572119894

1199051198811198941+119863120572119894

11990511990610

minusL119894(11988110 11988120 119881

1198990) minusM

1198940minus 119891119894= 0

(18)

119901119895 119863120572119894

119905119881119894119895minusL119894(1198811119895minus1

1198812119895minus1

119881119899119895minus1

) minusM119894119895minus1

= 0

(19)

where the terms M119894119895(119894 = 1 2 119899 and 119895 = 0 1 ) are

obtained from (17)ndash(19) by equating the coefficients of thenonlinear operators N

119894119895(119894 = 1 2 119899 and 119895 = 0 1 )

with the identical powers 119895 of 119901 that is the terms containing119901119895

In case the 119901-parameter is considered as small the bestapproximate solution of (13) can be readily obtained asfollows

119906119894(119909 119905) = lim

119901rarr1

infin

sum119895=0

119901119895119881119894119895(119909 119905)

= 1198811198940(119909 119905) + 119881

1198941(119909 119905) + 119881

1198942(119909 119905) + sdot sdot sdot

119894 = 1 2 119899

(20)

where 119909 = (1199091 1199092 119909

119899minus1)

If in (20) there exists some 119881119894119873(119909 119905) = 0 where 119873 ge 1

and 119894 = 1 2 119899 then the exact solution can be written inthe following form

119906119894(119909 119905) = 119881

1198940(119909 119905) + 119901119881

1198941(119909 119905) + sdot sdot sdot + 119901

119873minus1119881119894119873minus1

(119909 119905)

119894 = 1 2 119899

(21)

For simplicity we assume that 1198811198941(119909 119905) equiv 0 in (21) which

means that the exact solution in (13) is simplified as follows119906119894(119909 119905) = 119881

1198940(119909 119905) 119894 = 1 2 119899 (22)

Now when solving (17) we obtain the following result1198811198940(119909 119905) = 119906

1198940(119909 119905) 119894 = 1 2 119899 (23)

By using (22) and (23) we have119906119894(119909 119905) = 119881

1198940(119909 119905) = 119906

1198940(119909 119905) 119894 = 1 2 119899 (24)

The core of the new method proposed in this researchwork is to formulate the initial approximation of (13) in theform of separation of variables as follows119906119894(119909 119905) = 119906

1198940(119909 119905) = 119906

119894(119909 0) 119888

1198941(119905)

+ (119899

sum119895=1

119863119909119895119906119894(119909 0)) 119888

1198942(119905)

=119892119894(119909) 1198881198941(119905)+(

119899

sum119895=1

119863119909119895119892119894(119909)) 119888

1198942(119905) 119894=1 2 119899

(25)

The task now is to find the terms 1198881198941(119905) and 119888

1198942(119905) to obtain

the exact solution of the system of FPDEs in (13) Since (25)satisfies the initial conditions in (14) we get

1198881198941(0) = 1 119888

1198942(0) = 0 119894 = 1 2 119899 (26)

By substituting (25) into (18) we get

119863120572119894

1199051198811198941= minus 119863

120572119894

119905(119892119894(119909) 1198881198941(119905) + (

119899

sum119895=1

119863119909119895119892119894(119909)) 119888

1198942(119905))

minus L119894minusM1198940+ 119891119894(119909 119905) equiv 0

(27)

where 119894 = 1 2 119899 and

L119894= L119894(1198921(119909) 11988811(119905) + (

119899

sum119895=1

1198631199091198951198921(119909)) 119888

12(119905)

(119892119899(119909) 1198881198991(119905) + (

119899

sum119895=1

119863119909119895119892119899(119909)) 119888

1198992(119905)))

M1198940= M1198940(1198921(119909) 11988811(119905) + (

119899

sum119895=1

1198631199091198951198921(119909)) 119888

12(119905)

(119892119899(119909) 1198881198991(119905) + (

119899

sum119895=1

119863119909119895119892119899(119909)) 119888

1198992(119905)))

(28)

Obviously our goal is finally achieved and the system ofFPDEs is changed into a system of FODEs Consequently theproblem at hand is simplified considerably as it is well knownthat solving a system of FODEs is generally more straightfor-ward andmuch easier than solving the corresponding systemof FPDEs

Finally the target unknowns 1198881198941(119905) and 119888

1198942(119905) can be

obtained by utilizing and solving the system of FODEs andthe initial conditions in (26)

4 Applications

In this section we illustrate the applicability simplicity andefficiency of the homo-separation of variables method forsolving systems of linear and nonlinear FPDEs

Example 7 Consider the following system of linear FPDEs

119863120572119905119906 = V119909minus V minus 119906

119863120572119905V = 119906119909minus V minus 119906

(29)

where 0 lt 120572 le 1 subject to the initial conditions119906 (119909 0) = sinh (119909) V (119909 0) = cosh (119909) (30)

To solve (29) and (30) by using the proposed homo-separation of variables method we choose the initial approx-imations for (29) as follows

1199060(119909 119905) = 119906 (119909 0) 119887

1(119905) + 119863

119909119906 (119909 0) 119887

2(119905)

V0(119909 119905) = V (119909 0) 119888

1(119905) + 119863

119909V (119909 0) 119888

2(119905)

(31)

International Journal of Mathematics and Mathematical Sciences 5

By substituting the two initial conditions given by (30)into these two initial approximations we get the followinginitial approximations

1199060(119909 119905) = 119887

1(119905) sinh (119909) + 119887

2(119905) cosh (119909)

V0(119909 119905) = 119888

1(119905) cosh (119909) + 119888

2(119905) sinh (119909)

(32)

By considering and following the approach whichresulted in (27) and substituting the results presented by (32)into the system of linear FPDEs in (29) we get

1198631199051199061= sinh (119909) (119863120572

1199051198871(119905) + 119887

1(119905) minus 119888

1(119905) + 119888

2(119905))

+ cosh (119909) (1198631205721199051198872(119905) + 119887

2(119905) + 119888

1(119905) minus 119888

2(119905)) equiv 0

119863119905V1= cosh (119909) (119863120572

1199051198881(119905) + 119888

1(119905) minus 119887

1(119905) + 119887

2(119905))

+ sinh (119909) (1198631205721199051198882(119905) + 119888

2(119905) + 119887

1(119905) minus 119887

2(119905)) equiv 0

(33)

By solving the latter system of linear FPDEs presented in(33) we obtain the following system of linear FODEs

1198631205721199051198871(119905) + 119887

1(119905) minus 119888

1(119905) + 119888

2(119905) = 0

1198631205721199051198872(119905) + 119887

2(119905) + 119888

1(119905) minus 119888

2(119905) = 0

1198631205721199051198881(119905) + 119888

1(119905) minus 119887

1(119905) + 119887

2(119905) = 0

1198631205721199051198882(119905) + 119888

2(119905) + 119887

1(119905) minus 119887

2(119905) = 0

(34)

together with the initial conditions

1198872(0) = 1 119887

2(0) = 0

1198881(0) = 1 119888

1(0) = 0

(35)

By solving (34) and (35) by considering and applyingDefinition 6 which is discussed in Section 2 we obtain thefollowing results

1198871(119905) =

1

2(119864120572(119905120572) + 119864

120572(minus119905120572))

1198872(119905) =

1

2(minus119864120572(119905120572) + 119864

120572(minus119905120572))

1198881(119905) =

1

2(119864120572(119905120572) + 119864

120572(minus119905120572))

1198882(119905) =

1

2(minus119864120572(119905120572) + 119864

120572(minus119905120572))

(36)

By substituting (36) into (32) and applying Definition 4in particular properties (2) (5) and (6) which are discussedin Section 2 we obtain the exact solution for the system oflinear FPDEs in (29) as follows

119906 (119909 119905) = 11986421205721

(1199052120572) sinh (119909) minus 1199051205721198642120572120572+1

(1199052120572) cosh (119909)

V (119909 119905) = 11986421205721

(1199052120572) cosh (119909) minus 1199051205721198642120572120572+1

(1199052120572) sinh (119909) (37)

If we now put 120572 rarr 1 in (37) or solve (29) and (30)for 120572 = 1 we obtain the following exact solution for thecorresponding system of linear PDEs

119906 (119909 119905) = sinh (119909 minus 119905)

V (119909 119905) = cosh (119909 minus 119905) (38)

Example 8 Consider the following system of inhomoge-neous FPDEs

119863120572119905119906 + 119906119909V + 119906 = 1

119863120573

119905V minus V119909119906 minus V = 1

(39)

where 0 lt 120572 and 120573 le 1 subject to the initial conditions

119906 (119909 0) = 119890119909 V (119909 0) = 119890minus119909 (40)

To solve (39) and (40) by using the new homo-separationof variables method we choose the initial approximations for(39) as follows

1199060(119909 119905) = 119906 (119909 0) 119888

1(119905) + 119863

119909119906 (119909 0) 119888

2(119905)

V0(119909 119905) = V (119909 0) 119887

1(119905) + 119863

119909V (119909 0) 119887

2(119905)

(41)

By substituting the two initial conditions given by (40)into these two initial approximations we get the followinginitial approximations

1199060(119909 119905) = 119890

1199091198881(119905) + 119890

1199091198882(119905)

V0(119909 119905) = 119890

minus1199091198871(119905) minus 119890

minus1199091198872(119905)

(42)

By considering and following the approach whichresulted in (27) and substituting the results presented by (42)into the system of inhomogeneous FPDEs that we would liketo solve which is presented by (39) we get

1198631205721199051199061= 119890119909 (119863120572

119905(1198881(119905) + 119888

2(119905)) + (119888

1(119905) + 119888

2(119905)))

+ (1198881(119905) + 119888

2(119905)) (119887

1(119905) minus 119887

2(119905)) minus 1 equiv 0

119863120573

119905V1= 119890119909 (119863

120573

119905(1198871(119905) minus 119887

2(119905)) minus (119887

1(119905) minus 119887

2(119905)))

+ (1198881(119905) + 119888

2(119905)) (119887

1(119905) minus 119887

2(119905)) minus 1 equiv 0

(43)

By solving the latter system of FPDEs presented in (43)we obtain the following system of FODEs

119863120572119905(1198881(119905) + 119888

2(119905)) + (119888

1(119905) + 119888

2(119905)) = 0

119863120573

119905(1198871(119905) minus 119887

2(119905)) minus (119887

1(119905) minus 119887

2(119905)) = 0

(1198881(119905) + 119888

2(119905)) (119887

1(119905) minus 119887

2(119905)) minus 1 = 0

(44)

together with the initial conditions

1198871(0) = 1 119887

2(0) = 0

1198881(0) = 1 119888

2(119905) = 0

(45)

6 International Journal of Mathematics and Mathematical Sciences

By solving (44) and (45) by considering and applyingDefinition 5 which is discussed in Section 2 we obtain thefollowing results

1198881(119905) + 119888

2(119905) = 119864

120572(minus119905120572)

1198871(119905) minus 119887

2(119905) = 119864

120573(119905120573)

119864120572(minus119905120572) 119864

120573(119905120573) = 1

(46)

By substituting (46) into (42) we obtain the exact solutionfor the system of inhomogeneous FPDEs presented in (39) asfollows

119906 (119909 119905) = 119890119909119864120572(minus119905120572)

V (119909 119905) = 119890minus119909119864120573(119905120573)

119864120572(minus119905120572) 119864

120573(119905120573) = 1

(47)

Thus this means that 119906(119909 119905) = 1V(119909 119905) If we now put120572 120573 rarr 1 in (47) or solve (39) and (40) for 120572 = 120573 = 1we obtain the following exact solution for the correspondingsystem of inhomogeneous PDEs

119906 (119909 119905) = 119890119909minus119905

V (119909 119905) = 119890minus119909+119905(48)

Example 9 Consider the following system of nonlinearFPDEs

119863120572119905119906 = minus119906 minus V

119909119908119910minus V119910119908119909

119863120573

119905V = V minus 119906

119909119908119910minus 119906119910119908119909

119863120574

119905119908 = 119908 minus 119906

119909V119910minus 119906119910V119909

(49)

where 0 lt 120572 and 120573 120574 le 1 subject to the initial conditions

119906 (119909 119910 0) = 119890119909+119910 V (119909 119910 0) = 119890119909minus119910

119908 (119909 119910 0) = 119890minus119909+119910(50)

To solve (49) and (50) by using the new homo-separationof variables method we choose the initial approximations for(49) as follows

1199060(119909 119910 119905) = 119906 (119909 119910 0) 119887

1(119905)

+ (119863119909119906 (119909 119910 0) + 119863

119910119906 (119909 119910 0)) 119887

2(119905)

V0(119909 119910 119905) = V (119909 119910 0) 119888

1(119905)

+ (119863119909V (119909 119910 0) + 119863

119910V (119909 119910 0)) 119888

2(119905)

1199080(119909 119910 119905) = 119908 (119909 119910 0) 119889

1(119905)

+ (119863119909119908 (119909 119910 0) + 119863

119910119908 (119909 119910 0)) 119889

2(119905)

(51)

By substituting the initial conditions given by (50) intothese initial approximations we get the following initialapproximations

1199060(119909 119910 119905) = 119890119909+119910119887

1(119905) + 2119890

119909+1199101198872(119905)

V0(119909 119910 119905) = 119890119909minus119910119888

1(119905)

1199080(119909 119910 119905) = 119890minus119909+119910119889

1(119905)

(52)

By considering and following the approach whichresulted in (27) and substituting the results presented by (52)into the system of nonlinear FPDEs that we would like tosolve which is presented by (49) we get

1198631205721199051199061= 119890minus119909+119910 (119863120572

1199051198871(119905) + 2119863

120572

1199051198872(119905) + 119887

1(119905) + 2119887

2(119905)) equiv 0

119863120573

119905119863119905V1= 119890119909minus119910 (119863

120573

1199051198881(119905) minus 119888

1(119905)) equiv 0

119863120574

1199051199081= 119890minus119909+119910 (119863

120574

1199051198891(119905) minus 119889

1(119905)) equiv 0

(53)By solving the latter system of FPDEs presented in (53)

we obtain the following system of FODEs119863120572119905(1198871(119905) + 2119887

2(119905)) + (119887

1(119905) + 2119887

2(119905)) = 0

119863120573

1199051198881(119905) minus 119888

1(119905) = 0

119863120574

1199051198891(119905) minus 119889

1(119905) = 0

(54)

together with the initial conditions1198871(0) = 1 119887

2(0) = 0

1198881(0) = 1 119888

2(119905) = 0

1198891(0) = 1 119889

2(119905) = 0

(55)

By solving (54) and (55) by considering and applyingDefinition 5 which is discussed in Section 2 of this paper weobtain the following results

1198871(119905) minus 2119887

2(119905) = 119864

120572(minus119905120572)

1198881(119905) = 119864

120573(119905120573)

1198891(0) = 119864

120574(119905120574)

(56)

By substituting (56) into (52) we obtain the exact solutionfor the system of nonlinear FPDEs in (49) as follows

119906 (119909 119910 119905) = 119890119909+119910119864120572(minus119905120572)

V (119909 119910 119905) = 119890119909minus119910119864120573(119905120573)

119908 (119909 119910 119905) = 119890minus119909+119910119864120574(119905120574)

(57)

If we now put 120572 120573 120574 rarr 1 in (57) or solve (49) and (50)for 120572 = 120573 = 120574 = 1 we obtain the following exact solution forthe corresponding system of nonlinear PDEs

119906 (119909 119910 119905) = 119890119909+119910minus119905

V (119909 119910 119905) = 119890119909minus119910+119905

119908 (119909 119910 t) = 119890minus119909+119910+119905

(58)

International Journal of Mathematics and Mathematical Sciences 7

5 Conclusions

In this paper a novel approach was introduced and utilizedto solve linear and nonlinear systems of FPDEs The newtechnique was coined by the authors of this paper as thehomo-separation of variables method In this research workit was demonstrated through different examples how this newmethod can be used for solving various systems of FPDEs

When compared with the existing published methods itis easy to notice that the new method has many advantagesIt is straightforward easy to understand and fast requiringmuch less computations to perform a limited number ofsteps of the simple procedure that can be applied to find theexact solution of a wide range of types of FPDE equationsrsquosystems Furthermore there is no need for using linearizationor restrictive assumptions when employing this newmethod

The basic principle of the current method is to performan elegant combination (which is simple and smart) of twoexisting techniques These two techniques are the popularhomotopy perturbation method (HPM) that was originallyproposed by He [25 26] and another popular approachcalled separation of variables Finally the resulting homo-separation of variables method which is analytical can beused to solve various systems of PDEs with integer andfractional order for example with respect to time

Acknowledgments

The authors would like to thank Maryam Shabani andMehran Shafaiee for their help and support as well asthe inspiration the authors got through friendly scientificdiscussions

References

[1] R L Magin Fractional Calculus in Bioengineering BegellHouse West Redding Conn USA 2006

[2] R L Magin and M Ovadia ldquoModeling the cardiac tissueelectrode interface using fractional calculusrdquo JVCJournal ofVibration and Control vol 14 no 9-10 pp 1431ndash1442 2008

[3] J Keener and J SneydMathematical Physiology Springer NewYork NY USA 2004

[4] A A M Arafa S Z Rida and M Khalil ldquoFractional modelingdynamics of HIV and CD4+ T-cells during primary infectionrdquoNonlinear Biomedical Physics vol 6 no 1 article 1 2012

[5] I S Jesus J A Tenreiro MacHado and J Boaventure CunhaldquoFractional electrical impedances in botanical elementsrdquo Jour-nal of Vibration and Control vol 14 no 9-10 pp 1389ndash14022008

[6] K S Cole ldquoSketches of general and comparative demographyrdquoin Cold Spring Harbor Symposia on Quantitative Biology vol 1pp 107ndash116 Cold Spring Harbor Laboratory Press 1933

[7] V D Djordjevi J Jari B Fabry J J Fredberg andD StamenovildquoFractional derivatives embody essential features of cell rheo-logical behaviorrdquo Annals of Biomedical Engineering vol 31 no6 pp 692ndash699 2003

[8] C Coussot S Kalyanam R Yapp and M Insana ldquoFractionalderivative models for ultrasonic characterization of polymer

and breast tissue viscoelasticityrdquo IEEETransactions onUltrason-ics Ferroelectrics and Frequency Control vol 56 no 4 pp 715ndash725 2009

[9] F Mainardi ldquoFractional diffusive waves in viscoelastic solidsrdquoin Nonlinear Waves in Solids pp 93ndash97 Fairfield 1995

[10] V G Gupta and B Sharma ldquoApplication of Sumudu transformin reaction-diffusion systems and nonlinear wavesrdquo AppliedMathematical Sciences vol 4 no 9ndash12 pp 435ndash446 2010

[11] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego Calif USA 1999

[12] S Momani and Z Odibat ldquoNumerical approach to differentialequations of fractional orderrdquo Journal of Computational andApplied Mathematics vol 207 no 1 pp 96ndash110 2007

[13] O Abdulaziz I Hashim and S Momani ldquoSolving systemsof fractional differential equations by homotopy-perturbationmethodrdquo Physics Letters A vol 372 no 4 pp 451ndash459 2008

[14] M Ganjiani and H Ganjiani ldquoSolution of coupled systemof nonlinear differential equations using homotopy analysismethodrdquo Nonlinear Dynamics of Nonlinear Dynamics andChaos in Engineering Systems vol 56 no 1-2 pp 159ndash167 2009

[15] H Jafari M Nazari D Baleanu and C M Khalique ldquoA newapproach for solving a system of fractional partial differentialequationsrdquo Computers amp Mathematics with Applications 2012

[16] S Kumar A Yildirim and L Wei ldquoA fractional model of thediffusion equation and its analytical solution using Laplacetransformrdquo Scientia Iranica vol 19 no 4 pp 1117ndash1123 2012

[17] D Kumar J Singh and S Rathore ldquoSumudu decompositionmethod for nonlinear equationsrdquo International MathematicalForum vol 7 no 9ndash12 pp 515ndash521 2012

[18] J Singh D Kumar and Sushila ldquoHomotopy perturbationsumudu transform method for nonlinear equationsrdquo Advancesin Theoretical and Applied Mechanics vol 4 no 4 pp 165ndash1752011

[19] A Karbalaie M M Montazeri and H H Muhammed ldquoNewapproach to find the exact solution of fractional partial differ-ential equationrdquo WSEAS Transactions on Mathematics vol 11no 10 pp 908ndash917 2012

[20] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Publishers Yverdon Switzerland 1993

[21] A A M Arafa S Z Rida and H Mohamed ldquoHomotopyanalysis method for solving biological population modelrdquoCommunications in Theoretical Physics vol 56 no 5 pp 797ndash800 2011

[22] J Peng and K Li ldquoA note on property of the Mittag-Lefflerfunctionrdquo Journal of Mathematical Analysis and Applicationsvol 370 no 2 pp 635ndash638 2010

[23] J-F Cheng and Y-M Chu ldquoSolution to the linear fractionaldifferential equation using Adomian decomposition methodrdquoMathematical Problems in Engineering vol 2011 Article ID587068 14 pages 2011

[24] Z M Odibat ldquoAnalytic study on linear systems of fractionaldifferential equationsrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1171ndash1183 2010

[25] J He ldquoApplication of homotopy perturbation method to non-linear wave equationsrdquo Chaos Solitons amp Fractals vol 26 no 3pp 695ndash700 2005

8 International Journal of Mathematics and Mathematical Sciences

[26] J H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 6 pp 207ndash208 2005

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Stochastic AnalysisInternational Journal of

International Journal of Mathematics and Mathematical Sciences 3

(5) 119864120572120573(119905) + 119864

120572120573(minus119905) = 2119864

2120572120573(1199052)

(6) 119864120572120573(119905) minus 119864

120572120573(minus119905) = 2119905119864

2120572120572+120573(1199052)

(7) 119863120572119905(119905120572minus1119864

120572120572(119886119905120572)) = 119905120572minus1119864

120572120572(119886119905120572)

Further properties of the Mittag-Leffler functions canbe found in [21] These functions are generalizations ofthe exponential function therefore most linear differentialequations of fractional order (FPDEs) have solutions that areexpressed in terms of these functions

Important Note According to [22] 119864120572(119886119905120572) cannot satisfy the

following semigroup property

119864120572(119886(119905 + 119904)

120572) = 119864120572(119886119905120572) 119864

120572(119886119904120572) 119905 119904 ge 0 119886 isin R (6)

for any noninteger 120572 gt 0 The above property is not trueunless in the special cases where 120572 = 1 or = 0 Unfortunatelythis misunderstanding can be found in a number of publica-tions which can be misleading and confusing

Definition 5 Consider the following relaxation-oscillationequation which is discussed in [23]

119863120572119905119906 (119905) + 119860119906 (119905) = 119891 (119905) 119905 gt 0 (7)

subject to the initial condition

120597119894

120597119905119894119906(119905)

100381610038161003816100381610038161003816100381610038161003816119905=0= 119906119894 (0) = 119887

119894119894 = 1 2 119899 minus 1 (8)

where 119887119894(119894 = 1 2 119899minus1) are real constants and 119899minus1 lt 120572 le

119899 The solution of (7) is obtained according to [23] as follows

119906 (119905) = int119905

0

(119905 minus 120591)120572minus1119864120572120572(minus119860(119905 minus 120591)

120572) 119891 (120591) 119889120591

+119899minus1

sum119895=0

119887119895119863120572minus119895minus1

119905(119905120572minus1119864

120572120572(minus119860119905120572))

(9)

where 119864120572120572

is the Mittag-Leffler function It is easy to realizethat if 0 lt 120572 le 1 then the solution of (7) becomes as follows

119906 (119905) = int119905

0

(119905 minus 120591)120572minus1119864120572120572(minus119860(119905 minus 120591)

120572) 119891 (120591) 119889120591

+ 119863120572minus1119905

(119905120572minus1119864120572120572(minus119860119905120572))

(10)

which will be used to solve the systems of FPDEs in theexamples discussed in this paper

Definition 6 According to [24] consider the following linearsystem of fractional differential equations with its initialcondition

119863120572119905x (119905) = 119860x (119905) 0 lt 119905 le 119886

x (0) = x0

(11)

where x isin R119899 119886 gt 0 the matrix 119860 isin R119899 times R119899 and 119863120572119905is the

Caputo fractional derivative of order 120572 where 0 lt 120572 le 1Thegeneral analytical solution for (11) is given by

x (119905) = 1198881u(1)119864120572(1205821119905120572) + 119888

2u(2)119864120572(1205822t120572)

+ sdot sdot sdot + 119888119899u(119899)119864120572(120582119899119905120572)

(12)

where 1198881 1198882 119888

119899are arbitrary constants 119864

120572is the Mittag-

Leffler function and 1205821 1205822 120582

119899and u(1) u(2) u(119899) are

the eigenvalues and the corresponding eigenvectors of thecharacteristic equation (119860minus120582

119894I) u(119894) = 0 where 119894 = 1 2 119899

and I is an 119899 times 119899 identity matrix

3 The Homo-Separation of Variables Method

This approach is achieved by a novel combination of HPMand separation of variables In this section the algorithmof this new method is presented briefly by considering thefollowing system of FPDEs

119863120572119894

119905119906119894= 119891119894(119909 119905) +L

119894(1199061 1199062 119906

119899)

+N119894(1199061 1199062 119906

119899) 119894 = 1 2 119899

(13)

subject to the initial conditions

119906119894(119909 0) = 119892

119894(119909) (14)

where 119909 isin R119899minus1 and for 119894 = 1 2 119899 the termsL119894represent linear operators the terms N

119894are nonlinear

operators while the terms 119891119894represent known analytical

functions In addition119863120572119894119905= 120597120572119894120597119905120572119894 is the Caputo fractional

derivative of order 120572119894 where 0 lt 120572

119894le 1 for 119894 = 1 2 119899

According to the homotopy perturbation technique(HPM) we can construct the following homotopies

(119901 minus 1) (119863120572119894

119905119881119894minus 119863120572119894

1199051199061198940) + 119901 (119863

120572119894

119905119881119894minus L119894minus N119894minus 119891119894(119909 119905))

= 0 119894 = 1 2 119899

(15)

where L119894= L119894(1198811 1198812 119881

119899) N119894= N119894(1198811 1198812 119881

119899) and

the homotopy parameter denoted by119901 is considered as small(119901 isin [0 1]) In addition for 119894 = 1 2 119899 119906

1198940= 1199061198940(119909 119905) is

an initial approximation of the solution of (13) which satisfiesthe initial condition in (14)

We can assume that the solution of (15) can be expressedas a power series in 119901 as given below in (16)

1198811=infin

sum119895=0

1199011198951198811119895= 11988110+ 11990111988111+ 1199012119881

12+ 1199013119881

13+ sdot sdot sdot

1198812=infin

sum119895=0

1199011198951198812119895= 11988120+ 11990111988121+ 1199012119881

22+ 1199013119881

23+ sdot sdot sdot

119881119899=infin

sum119895=0

119901119895119881119899119895= 1198811198990+ 1199011198811198991+ 1199012119881

1198992+ 1199013119881

1198993+ sdot sdot sdot

(16)

4 International Journal of Mathematics and Mathematical Sciences

By substituting (16) into (15) and equating the terms withidentical powers of119901 a set of equations is obtained as follows

1199010 119863120572119894

1199051198811198940minus 119863120572119894

1199051199061198940= 0 (17)

1199011 119863120572119894

1199051198811198941+119863120572119894

11990511990610

minusL119894(11988110 11988120 119881

1198990) minusM

1198940minus 119891119894= 0

(18)

119901119895 119863120572119894

119905119881119894119895minusL119894(1198811119895minus1

1198812119895minus1

119881119899119895minus1

) minusM119894119895minus1

= 0

(19)

where the terms M119894119895(119894 = 1 2 119899 and 119895 = 0 1 ) are

obtained from (17)ndash(19) by equating the coefficients of thenonlinear operators N

119894119895(119894 = 1 2 119899 and 119895 = 0 1 )

with the identical powers 119895 of 119901 that is the terms containing119901119895

In case the 119901-parameter is considered as small the bestapproximate solution of (13) can be readily obtained asfollows

119906119894(119909 119905) = lim

119901rarr1

infin

sum119895=0

119901119895119881119894119895(119909 119905)

= 1198811198940(119909 119905) + 119881

1198941(119909 119905) + 119881

1198942(119909 119905) + sdot sdot sdot

119894 = 1 2 119899

(20)

where 119909 = (1199091 1199092 119909

119899minus1)

If in (20) there exists some 119881119894119873(119909 119905) = 0 where 119873 ge 1

and 119894 = 1 2 119899 then the exact solution can be written inthe following form

119906119894(119909 119905) = 119881

1198940(119909 119905) + 119901119881

1198941(119909 119905) + sdot sdot sdot + 119901

119873minus1119881119894119873minus1

(119909 119905)

119894 = 1 2 119899

(21)

For simplicity we assume that 1198811198941(119909 119905) equiv 0 in (21) which

means that the exact solution in (13) is simplified as follows119906119894(119909 119905) = 119881

1198940(119909 119905) 119894 = 1 2 119899 (22)

Now when solving (17) we obtain the following result1198811198940(119909 119905) = 119906

1198940(119909 119905) 119894 = 1 2 119899 (23)

By using (22) and (23) we have119906119894(119909 119905) = 119881

1198940(119909 119905) = 119906

1198940(119909 119905) 119894 = 1 2 119899 (24)

The core of the new method proposed in this researchwork is to formulate the initial approximation of (13) in theform of separation of variables as follows119906119894(119909 119905) = 119906

1198940(119909 119905) = 119906

119894(119909 0) 119888

1198941(119905)

+ (119899

sum119895=1

119863119909119895119906119894(119909 0)) 119888

1198942(119905)

=119892119894(119909) 1198881198941(119905)+(

119899

sum119895=1

119863119909119895119892119894(119909)) 119888

1198942(119905) 119894=1 2 119899

(25)

The task now is to find the terms 1198881198941(119905) and 119888

1198942(119905) to obtain

the exact solution of the system of FPDEs in (13) Since (25)satisfies the initial conditions in (14) we get

1198881198941(0) = 1 119888

1198942(0) = 0 119894 = 1 2 119899 (26)

By substituting (25) into (18) we get

119863120572119894

1199051198811198941= minus 119863

120572119894

119905(119892119894(119909) 1198881198941(119905) + (

119899

sum119895=1

119863119909119895119892119894(119909)) 119888

1198942(119905))

minus L119894minusM1198940+ 119891119894(119909 119905) equiv 0

(27)

where 119894 = 1 2 119899 and

L119894= L119894(1198921(119909) 11988811(119905) + (

119899

sum119895=1

1198631199091198951198921(119909)) 119888

12(119905)

(119892119899(119909) 1198881198991(119905) + (

119899

sum119895=1

119863119909119895119892119899(119909)) 119888

1198992(119905)))

M1198940= M1198940(1198921(119909) 11988811(119905) + (

119899

sum119895=1

1198631199091198951198921(119909)) 119888

12(119905)

(119892119899(119909) 1198881198991(119905) + (

119899

sum119895=1

119863119909119895119892119899(119909)) 119888

1198992(119905)))

(28)

Obviously our goal is finally achieved and the system ofFPDEs is changed into a system of FODEs Consequently theproblem at hand is simplified considerably as it is well knownthat solving a system of FODEs is generally more straightfor-ward andmuch easier than solving the corresponding systemof FPDEs

Finally the target unknowns 1198881198941(119905) and 119888

1198942(119905) can be

obtained by utilizing and solving the system of FODEs andthe initial conditions in (26)

4 Applications

In this section we illustrate the applicability simplicity andefficiency of the homo-separation of variables method forsolving systems of linear and nonlinear FPDEs

Example 7 Consider the following system of linear FPDEs

119863120572119905119906 = V119909minus V minus 119906

119863120572119905V = 119906119909minus V minus 119906

(29)

where 0 lt 120572 le 1 subject to the initial conditions119906 (119909 0) = sinh (119909) V (119909 0) = cosh (119909) (30)

To solve (29) and (30) by using the proposed homo-separation of variables method we choose the initial approx-imations for (29) as follows

1199060(119909 119905) = 119906 (119909 0) 119887

1(119905) + 119863

119909119906 (119909 0) 119887

2(119905)

V0(119909 119905) = V (119909 0) 119888

1(119905) + 119863

119909V (119909 0) 119888

2(119905)

(31)

International Journal of Mathematics and Mathematical Sciences 5

By substituting the two initial conditions given by (30)into these two initial approximations we get the followinginitial approximations

1199060(119909 119905) = 119887

1(119905) sinh (119909) + 119887

2(119905) cosh (119909)

V0(119909 119905) = 119888

1(119905) cosh (119909) + 119888

2(119905) sinh (119909)

(32)

By considering and following the approach whichresulted in (27) and substituting the results presented by (32)into the system of linear FPDEs in (29) we get

1198631199051199061= sinh (119909) (119863120572

1199051198871(119905) + 119887

1(119905) minus 119888

1(119905) + 119888

2(119905))

+ cosh (119909) (1198631205721199051198872(119905) + 119887

2(119905) + 119888

1(119905) minus 119888

2(119905)) equiv 0

119863119905V1= cosh (119909) (119863120572

1199051198881(119905) + 119888

1(119905) minus 119887

1(119905) + 119887

2(119905))

+ sinh (119909) (1198631205721199051198882(119905) + 119888

2(119905) + 119887

1(119905) minus 119887

2(119905)) equiv 0

(33)

By solving the latter system of linear FPDEs presented in(33) we obtain the following system of linear FODEs

1198631205721199051198871(119905) + 119887

1(119905) minus 119888

1(119905) + 119888

2(119905) = 0

1198631205721199051198872(119905) + 119887

2(119905) + 119888

1(119905) minus 119888

2(119905) = 0

1198631205721199051198881(119905) + 119888

1(119905) minus 119887

1(119905) + 119887

2(119905) = 0

1198631205721199051198882(119905) + 119888

2(119905) + 119887

1(119905) minus 119887

2(119905) = 0

(34)

together with the initial conditions

1198872(0) = 1 119887

2(0) = 0

1198881(0) = 1 119888

1(0) = 0

(35)

By solving (34) and (35) by considering and applyingDefinition 6 which is discussed in Section 2 we obtain thefollowing results

1198871(119905) =

1

2(119864120572(119905120572) + 119864

120572(minus119905120572))

1198872(119905) =

1

2(minus119864120572(119905120572) + 119864

120572(minus119905120572))

1198881(119905) =

1

2(119864120572(119905120572) + 119864

120572(minus119905120572))

1198882(119905) =

1

2(minus119864120572(119905120572) + 119864

120572(minus119905120572))

(36)

By substituting (36) into (32) and applying Definition 4in particular properties (2) (5) and (6) which are discussedin Section 2 we obtain the exact solution for the system oflinear FPDEs in (29) as follows

119906 (119909 119905) = 11986421205721

(1199052120572) sinh (119909) minus 1199051205721198642120572120572+1

(1199052120572) cosh (119909)

V (119909 119905) = 11986421205721

(1199052120572) cosh (119909) minus 1199051205721198642120572120572+1

(1199052120572) sinh (119909) (37)

If we now put 120572 rarr 1 in (37) or solve (29) and (30)for 120572 = 1 we obtain the following exact solution for thecorresponding system of linear PDEs

119906 (119909 119905) = sinh (119909 minus 119905)

V (119909 119905) = cosh (119909 minus 119905) (38)

Example 8 Consider the following system of inhomoge-neous FPDEs

119863120572119905119906 + 119906119909V + 119906 = 1

119863120573

119905V minus V119909119906 minus V = 1

(39)

where 0 lt 120572 and 120573 le 1 subject to the initial conditions

119906 (119909 0) = 119890119909 V (119909 0) = 119890minus119909 (40)

To solve (39) and (40) by using the new homo-separationof variables method we choose the initial approximations for(39) as follows

1199060(119909 119905) = 119906 (119909 0) 119888

1(119905) + 119863

119909119906 (119909 0) 119888

2(119905)

V0(119909 119905) = V (119909 0) 119887

1(119905) + 119863

119909V (119909 0) 119887

2(119905)

(41)

By substituting the two initial conditions given by (40)into these two initial approximations we get the followinginitial approximations

1199060(119909 119905) = 119890

1199091198881(119905) + 119890

1199091198882(119905)

V0(119909 119905) = 119890

minus1199091198871(119905) minus 119890

minus1199091198872(119905)

(42)

By considering and following the approach whichresulted in (27) and substituting the results presented by (42)into the system of inhomogeneous FPDEs that we would liketo solve which is presented by (39) we get

1198631205721199051199061= 119890119909 (119863120572

119905(1198881(119905) + 119888

2(119905)) + (119888

1(119905) + 119888

2(119905)))

+ (1198881(119905) + 119888

2(119905)) (119887

1(119905) minus 119887

2(119905)) minus 1 equiv 0

119863120573

119905V1= 119890119909 (119863

120573

119905(1198871(119905) minus 119887

2(119905)) minus (119887

1(119905) minus 119887

2(119905)))

+ (1198881(119905) + 119888

2(119905)) (119887

1(119905) minus 119887

2(119905)) minus 1 equiv 0

(43)

By solving the latter system of FPDEs presented in (43)we obtain the following system of FODEs

119863120572119905(1198881(119905) + 119888

2(119905)) + (119888

1(119905) + 119888

2(119905)) = 0

119863120573

119905(1198871(119905) minus 119887

2(119905)) minus (119887

1(119905) minus 119887

2(119905)) = 0

(1198881(119905) + 119888

2(119905)) (119887

1(119905) minus 119887

2(119905)) minus 1 = 0

(44)

together with the initial conditions

1198871(0) = 1 119887

2(0) = 0

1198881(0) = 1 119888

2(119905) = 0

(45)

6 International Journal of Mathematics and Mathematical Sciences

By solving (44) and (45) by considering and applyingDefinition 5 which is discussed in Section 2 we obtain thefollowing results

1198881(119905) + 119888

2(119905) = 119864

120572(minus119905120572)

1198871(119905) minus 119887

2(119905) = 119864

120573(119905120573)

119864120572(minus119905120572) 119864

120573(119905120573) = 1

(46)

By substituting (46) into (42) we obtain the exact solutionfor the system of inhomogeneous FPDEs presented in (39) asfollows

119906 (119909 119905) = 119890119909119864120572(minus119905120572)

V (119909 119905) = 119890minus119909119864120573(119905120573)

119864120572(minus119905120572) 119864

120573(119905120573) = 1

(47)

Thus this means that 119906(119909 119905) = 1V(119909 119905) If we now put120572 120573 rarr 1 in (47) or solve (39) and (40) for 120572 = 120573 = 1we obtain the following exact solution for the correspondingsystem of inhomogeneous PDEs

119906 (119909 119905) = 119890119909minus119905

V (119909 119905) = 119890minus119909+119905(48)

Example 9 Consider the following system of nonlinearFPDEs

119863120572119905119906 = minus119906 minus V

119909119908119910minus V119910119908119909

119863120573

119905V = V minus 119906

119909119908119910minus 119906119910119908119909

119863120574

119905119908 = 119908 minus 119906

119909V119910minus 119906119910V119909

(49)

where 0 lt 120572 and 120573 120574 le 1 subject to the initial conditions

119906 (119909 119910 0) = 119890119909+119910 V (119909 119910 0) = 119890119909minus119910

119908 (119909 119910 0) = 119890minus119909+119910(50)

To solve (49) and (50) by using the new homo-separationof variables method we choose the initial approximations for(49) as follows

1199060(119909 119910 119905) = 119906 (119909 119910 0) 119887

1(119905)

+ (119863119909119906 (119909 119910 0) + 119863

119910119906 (119909 119910 0)) 119887

2(119905)

V0(119909 119910 119905) = V (119909 119910 0) 119888

1(119905)

+ (119863119909V (119909 119910 0) + 119863

119910V (119909 119910 0)) 119888

2(119905)

1199080(119909 119910 119905) = 119908 (119909 119910 0) 119889

1(119905)

+ (119863119909119908 (119909 119910 0) + 119863

119910119908 (119909 119910 0)) 119889

2(119905)

(51)

By substituting the initial conditions given by (50) intothese initial approximations we get the following initialapproximations

1199060(119909 119910 119905) = 119890119909+119910119887

1(119905) + 2119890

119909+1199101198872(119905)

V0(119909 119910 119905) = 119890119909minus119910119888

1(119905)

1199080(119909 119910 119905) = 119890minus119909+119910119889

1(119905)

(52)

By considering and following the approach whichresulted in (27) and substituting the results presented by (52)into the system of nonlinear FPDEs that we would like tosolve which is presented by (49) we get

1198631205721199051199061= 119890minus119909+119910 (119863120572

1199051198871(119905) + 2119863

120572

1199051198872(119905) + 119887

1(119905) + 2119887

2(119905)) equiv 0

119863120573

119905119863119905V1= 119890119909minus119910 (119863

120573

1199051198881(119905) minus 119888

1(119905)) equiv 0

119863120574

1199051199081= 119890minus119909+119910 (119863

120574

1199051198891(119905) minus 119889

1(119905)) equiv 0

(53)By solving the latter system of FPDEs presented in (53)

we obtain the following system of FODEs119863120572119905(1198871(119905) + 2119887

2(119905)) + (119887

1(119905) + 2119887

2(119905)) = 0

119863120573

1199051198881(119905) minus 119888

1(119905) = 0

119863120574

1199051198891(119905) minus 119889

1(119905) = 0

(54)

together with the initial conditions1198871(0) = 1 119887

2(0) = 0

1198881(0) = 1 119888

2(119905) = 0

1198891(0) = 1 119889

2(119905) = 0

(55)

By solving (54) and (55) by considering and applyingDefinition 5 which is discussed in Section 2 of this paper weobtain the following results

1198871(119905) minus 2119887

2(119905) = 119864

120572(minus119905120572)

1198881(119905) = 119864

120573(119905120573)

1198891(0) = 119864

120574(119905120574)

(56)

By substituting (56) into (52) we obtain the exact solutionfor the system of nonlinear FPDEs in (49) as follows

119906 (119909 119910 119905) = 119890119909+119910119864120572(minus119905120572)

V (119909 119910 119905) = 119890119909minus119910119864120573(119905120573)

119908 (119909 119910 119905) = 119890minus119909+119910119864120574(119905120574)

(57)

If we now put 120572 120573 120574 rarr 1 in (57) or solve (49) and (50)for 120572 = 120573 = 120574 = 1 we obtain the following exact solution forthe corresponding system of nonlinear PDEs

119906 (119909 119910 119905) = 119890119909+119910minus119905

V (119909 119910 119905) = 119890119909minus119910+119905

119908 (119909 119910 t) = 119890minus119909+119910+119905

(58)

International Journal of Mathematics and Mathematical Sciences 7

5 Conclusions

In this paper a novel approach was introduced and utilizedto solve linear and nonlinear systems of FPDEs The newtechnique was coined by the authors of this paper as thehomo-separation of variables method In this research workit was demonstrated through different examples how this newmethod can be used for solving various systems of FPDEs

When compared with the existing published methods itis easy to notice that the new method has many advantagesIt is straightforward easy to understand and fast requiringmuch less computations to perform a limited number ofsteps of the simple procedure that can be applied to find theexact solution of a wide range of types of FPDE equationsrsquosystems Furthermore there is no need for using linearizationor restrictive assumptions when employing this newmethod

The basic principle of the current method is to performan elegant combination (which is simple and smart) of twoexisting techniques These two techniques are the popularhomotopy perturbation method (HPM) that was originallyproposed by He [25 26] and another popular approachcalled separation of variables Finally the resulting homo-separation of variables method which is analytical can beused to solve various systems of PDEs with integer andfractional order for example with respect to time

Acknowledgments

The authors would like to thank Maryam Shabani andMehran Shafaiee for their help and support as well asthe inspiration the authors got through friendly scientificdiscussions

References

[1] R L Magin Fractional Calculus in Bioengineering BegellHouse West Redding Conn USA 2006

[2] R L Magin and M Ovadia ldquoModeling the cardiac tissueelectrode interface using fractional calculusrdquo JVCJournal ofVibration and Control vol 14 no 9-10 pp 1431ndash1442 2008

[3] J Keener and J SneydMathematical Physiology Springer NewYork NY USA 2004

[4] A A M Arafa S Z Rida and M Khalil ldquoFractional modelingdynamics of HIV and CD4+ T-cells during primary infectionrdquoNonlinear Biomedical Physics vol 6 no 1 article 1 2012

[5] I S Jesus J A Tenreiro MacHado and J Boaventure CunhaldquoFractional electrical impedances in botanical elementsrdquo Jour-nal of Vibration and Control vol 14 no 9-10 pp 1389ndash14022008

[6] K S Cole ldquoSketches of general and comparative demographyrdquoin Cold Spring Harbor Symposia on Quantitative Biology vol 1pp 107ndash116 Cold Spring Harbor Laboratory Press 1933

[7] V D Djordjevi J Jari B Fabry J J Fredberg andD StamenovildquoFractional derivatives embody essential features of cell rheo-logical behaviorrdquo Annals of Biomedical Engineering vol 31 no6 pp 692ndash699 2003

[8] C Coussot S Kalyanam R Yapp and M Insana ldquoFractionalderivative models for ultrasonic characterization of polymer

and breast tissue viscoelasticityrdquo IEEETransactions onUltrason-ics Ferroelectrics and Frequency Control vol 56 no 4 pp 715ndash725 2009

[9] F Mainardi ldquoFractional diffusive waves in viscoelastic solidsrdquoin Nonlinear Waves in Solids pp 93ndash97 Fairfield 1995

[10] V G Gupta and B Sharma ldquoApplication of Sumudu transformin reaction-diffusion systems and nonlinear wavesrdquo AppliedMathematical Sciences vol 4 no 9ndash12 pp 435ndash446 2010

[11] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego Calif USA 1999

[12] S Momani and Z Odibat ldquoNumerical approach to differentialequations of fractional orderrdquo Journal of Computational andApplied Mathematics vol 207 no 1 pp 96ndash110 2007

[13] O Abdulaziz I Hashim and S Momani ldquoSolving systemsof fractional differential equations by homotopy-perturbationmethodrdquo Physics Letters A vol 372 no 4 pp 451ndash459 2008

[14] M Ganjiani and H Ganjiani ldquoSolution of coupled systemof nonlinear differential equations using homotopy analysismethodrdquo Nonlinear Dynamics of Nonlinear Dynamics andChaos in Engineering Systems vol 56 no 1-2 pp 159ndash167 2009

[15] H Jafari M Nazari D Baleanu and C M Khalique ldquoA newapproach for solving a system of fractional partial differentialequationsrdquo Computers amp Mathematics with Applications 2012

[16] S Kumar A Yildirim and L Wei ldquoA fractional model of thediffusion equation and its analytical solution using Laplacetransformrdquo Scientia Iranica vol 19 no 4 pp 1117ndash1123 2012

[17] D Kumar J Singh and S Rathore ldquoSumudu decompositionmethod for nonlinear equationsrdquo International MathematicalForum vol 7 no 9ndash12 pp 515ndash521 2012

[18] J Singh D Kumar and Sushila ldquoHomotopy perturbationsumudu transform method for nonlinear equationsrdquo Advancesin Theoretical and Applied Mechanics vol 4 no 4 pp 165ndash1752011

[19] A Karbalaie M M Montazeri and H H Muhammed ldquoNewapproach to find the exact solution of fractional partial differ-ential equationrdquo WSEAS Transactions on Mathematics vol 11no 10 pp 908ndash917 2012

[20] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Publishers Yverdon Switzerland 1993

[21] A A M Arafa S Z Rida and H Mohamed ldquoHomotopyanalysis method for solving biological population modelrdquoCommunications in Theoretical Physics vol 56 no 5 pp 797ndash800 2011

[22] J Peng and K Li ldquoA note on property of the Mittag-Lefflerfunctionrdquo Journal of Mathematical Analysis and Applicationsvol 370 no 2 pp 635ndash638 2010

[23] J-F Cheng and Y-M Chu ldquoSolution to the linear fractionaldifferential equation using Adomian decomposition methodrdquoMathematical Problems in Engineering vol 2011 Article ID587068 14 pages 2011

[24] Z M Odibat ldquoAnalytic study on linear systems of fractionaldifferential equationsrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1171ndash1183 2010

[25] J He ldquoApplication of homotopy perturbation method to non-linear wave equationsrdquo Chaos Solitons amp Fractals vol 26 no 3pp 695ndash700 2005

8 International Journal of Mathematics and Mathematical Sciences

[26] J H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 6 pp 207ndash208 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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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

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Journal of Mathematics and Mathematical Sciences

By substituting (16) into (15) and equating the terms withidentical powers of119901 a set of equations is obtained as follows

1199010 119863120572119894

1199051198811198940minus 119863120572119894

1199051199061198940= 0 (17)

1199011 119863120572119894

1199051198811198941+119863120572119894

11990511990610

minusL119894(11988110 11988120 119881

1198990) minusM

1198940minus 119891119894= 0

(18)

119901119895 119863120572119894

119905119881119894119895minusL119894(1198811119895minus1

1198812119895minus1

119881119899119895minus1

) minusM119894119895minus1

= 0

(19)

where the terms M119894119895(119894 = 1 2 119899 and 119895 = 0 1 ) are

obtained from (17)ndash(19) by equating the coefficients of thenonlinear operators N

119894119895(119894 = 1 2 119899 and 119895 = 0 1 )

with the identical powers 119895 of 119901 that is the terms containing119901119895

In case the 119901-parameter is considered as small the bestapproximate solution of (13) can be readily obtained asfollows

119906119894(119909 119905) = lim

119901rarr1

infin

sum119895=0

119901119895119881119894119895(119909 119905)

= 1198811198940(119909 119905) + 119881

1198941(119909 119905) + 119881

1198942(119909 119905) + sdot sdot sdot

119894 = 1 2 119899

(20)

where 119909 = (1199091 1199092 119909

119899minus1)

If in (20) there exists some 119881119894119873(119909 119905) = 0 where 119873 ge 1

and 119894 = 1 2 119899 then the exact solution can be written inthe following form

119906119894(119909 119905) = 119881

1198940(119909 119905) + 119901119881

1198941(119909 119905) + sdot sdot sdot + 119901

119873minus1119881119894119873minus1

(119909 119905)

119894 = 1 2 119899

(21)

For simplicity we assume that 1198811198941(119909 119905) equiv 0 in (21) which

means that the exact solution in (13) is simplified as follows119906119894(119909 119905) = 119881

1198940(119909 119905) 119894 = 1 2 119899 (22)

Now when solving (17) we obtain the following result1198811198940(119909 119905) = 119906

1198940(119909 119905) 119894 = 1 2 119899 (23)

By using (22) and (23) we have119906119894(119909 119905) = 119881

1198940(119909 119905) = 119906

1198940(119909 119905) 119894 = 1 2 119899 (24)

The core of the new method proposed in this researchwork is to formulate the initial approximation of (13) in theform of separation of variables as follows119906119894(119909 119905) = 119906

1198940(119909 119905) = 119906

119894(119909 0) 119888

1198941(119905)

+ (119899

sum119895=1

119863119909119895119906119894(119909 0)) 119888

1198942(119905)

=119892119894(119909) 1198881198941(119905)+(

119899

sum119895=1

119863119909119895119892119894(119909)) 119888

1198942(119905) 119894=1 2 119899

(25)

The task now is to find the terms 1198881198941(119905) and 119888

1198942(119905) to obtain

the exact solution of the system of FPDEs in (13) Since (25)satisfies the initial conditions in (14) we get

1198881198941(0) = 1 119888

1198942(0) = 0 119894 = 1 2 119899 (26)

By substituting (25) into (18) we get

119863120572119894

1199051198811198941= minus 119863

120572119894

119905(119892119894(119909) 1198881198941(119905) + (

119899

sum119895=1

119863119909119895119892119894(119909)) 119888

1198942(119905))

minus L119894minusM1198940+ 119891119894(119909 119905) equiv 0

(27)

where 119894 = 1 2 119899 and

L119894= L119894(1198921(119909) 11988811(119905) + (

119899

sum119895=1

1198631199091198951198921(119909)) 119888

12(119905)

(119892119899(119909) 1198881198991(119905) + (

119899

sum119895=1

119863119909119895119892119899(119909)) 119888

1198992(119905)))

M1198940= M1198940(1198921(119909) 11988811(119905) + (

119899

sum119895=1

1198631199091198951198921(119909)) 119888

12(119905)

(119892119899(119909) 1198881198991(119905) + (

119899

sum119895=1

119863119909119895119892119899(119909)) 119888

1198992(119905)))

(28)

Obviously our goal is finally achieved and the system ofFPDEs is changed into a system of FODEs Consequently theproblem at hand is simplified considerably as it is well knownthat solving a system of FODEs is generally more straightfor-ward andmuch easier than solving the corresponding systemof FPDEs

Finally the target unknowns 1198881198941(119905) and 119888

1198942(119905) can be

obtained by utilizing and solving the system of FODEs andthe initial conditions in (26)

4 Applications

In this section we illustrate the applicability simplicity andefficiency of the homo-separation of variables method forsolving systems of linear and nonlinear FPDEs

Example 7 Consider the following system of linear FPDEs

119863120572119905119906 = V119909minus V minus 119906

119863120572119905V = 119906119909minus V minus 119906

(29)

where 0 lt 120572 le 1 subject to the initial conditions119906 (119909 0) = sinh (119909) V (119909 0) = cosh (119909) (30)

To solve (29) and (30) by using the proposed homo-separation of variables method we choose the initial approx-imations for (29) as follows

1199060(119909 119905) = 119906 (119909 0) 119887

1(119905) + 119863

119909119906 (119909 0) 119887

2(119905)

V0(119909 119905) = V (119909 0) 119888

1(119905) + 119863

119909V (119909 0) 119888

2(119905)

(31)

International Journal of Mathematics and Mathematical Sciences 5

By substituting the two initial conditions given by (30)into these two initial approximations we get the followinginitial approximations

1199060(119909 119905) = 119887

1(119905) sinh (119909) + 119887

2(119905) cosh (119909)

V0(119909 119905) = 119888

1(119905) cosh (119909) + 119888

2(119905) sinh (119909)

(32)

By considering and following the approach whichresulted in (27) and substituting the results presented by (32)into the system of linear FPDEs in (29) we get

1198631199051199061= sinh (119909) (119863120572

1199051198871(119905) + 119887

1(119905) minus 119888

1(119905) + 119888

2(119905))

+ cosh (119909) (1198631205721199051198872(119905) + 119887

2(119905) + 119888

1(119905) minus 119888

2(119905)) equiv 0

119863119905V1= cosh (119909) (119863120572

1199051198881(119905) + 119888

1(119905) minus 119887

1(119905) + 119887

2(119905))

+ sinh (119909) (1198631205721199051198882(119905) + 119888

2(119905) + 119887

1(119905) minus 119887

2(119905)) equiv 0

(33)

By solving the latter system of linear FPDEs presented in(33) we obtain the following system of linear FODEs

1198631205721199051198871(119905) + 119887

1(119905) minus 119888

1(119905) + 119888

2(119905) = 0

1198631205721199051198872(119905) + 119887

2(119905) + 119888

1(119905) minus 119888

2(119905) = 0

1198631205721199051198881(119905) + 119888

1(119905) minus 119887

1(119905) + 119887

2(119905) = 0

1198631205721199051198882(119905) + 119888

2(119905) + 119887

1(119905) minus 119887

2(119905) = 0

(34)

together with the initial conditions

1198872(0) = 1 119887

2(0) = 0

1198881(0) = 1 119888

1(0) = 0

(35)

By solving (34) and (35) by considering and applyingDefinition 6 which is discussed in Section 2 we obtain thefollowing results

1198871(119905) =

1

2(119864120572(119905120572) + 119864

120572(minus119905120572))

1198872(119905) =

1

2(minus119864120572(119905120572) + 119864

120572(minus119905120572))

1198881(119905) =

1

2(119864120572(119905120572) + 119864

120572(minus119905120572))

1198882(119905) =

1

2(minus119864120572(119905120572) + 119864

120572(minus119905120572))

(36)

By substituting (36) into (32) and applying Definition 4in particular properties (2) (5) and (6) which are discussedin Section 2 we obtain the exact solution for the system oflinear FPDEs in (29) as follows

119906 (119909 119905) = 11986421205721

(1199052120572) sinh (119909) minus 1199051205721198642120572120572+1

(1199052120572) cosh (119909)

V (119909 119905) = 11986421205721

(1199052120572) cosh (119909) minus 1199051205721198642120572120572+1

(1199052120572) sinh (119909) (37)

If we now put 120572 rarr 1 in (37) or solve (29) and (30)for 120572 = 1 we obtain the following exact solution for thecorresponding system of linear PDEs

119906 (119909 119905) = sinh (119909 minus 119905)

V (119909 119905) = cosh (119909 minus 119905) (38)

Example 8 Consider the following system of inhomoge-neous FPDEs

119863120572119905119906 + 119906119909V + 119906 = 1

119863120573

119905V minus V119909119906 minus V = 1

(39)

where 0 lt 120572 and 120573 le 1 subject to the initial conditions

119906 (119909 0) = 119890119909 V (119909 0) = 119890minus119909 (40)

To solve (39) and (40) by using the new homo-separationof variables method we choose the initial approximations for(39) as follows

1199060(119909 119905) = 119906 (119909 0) 119888

1(119905) + 119863

119909119906 (119909 0) 119888

2(119905)

V0(119909 119905) = V (119909 0) 119887

1(119905) + 119863

119909V (119909 0) 119887

2(119905)

(41)

By substituting the two initial conditions given by (40)into these two initial approximations we get the followinginitial approximations

1199060(119909 119905) = 119890

1199091198881(119905) + 119890

1199091198882(119905)

V0(119909 119905) = 119890

minus1199091198871(119905) minus 119890

minus1199091198872(119905)

(42)

By considering and following the approach whichresulted in (27) and substituting the results presented by (42)into the system of inhomogeneous FPDEs that we would liketo solve which is presented by (39) we get

1198631205721199051199061= 119890119909 (119863120572

119905(1198881(119905) + 119888

2(119905)) + (119888

1(119905) + 119888

2(119905)))

+ (1198881(119905) + 119888

2(119905)) (119887

1(119905) minus 119887

2(119905)) minus 1 equiv 0

119863120573

119905V1= 119890119909 (119863

120573

119905(1198871(119905) minus 119887

2(119905)) minus (119887

1(119905) minus 119887

2(119905)))

+ (1198881(119905) + 119888

2(119905)) (119887

1(119905) minus 119887

2(119905)) minus 1 equiv 0

(43)

By solving the latter system of FPDEs presented in (43)we obtain the following system of FODEs

119863120572119905(1198881(119905) + 119888

2(119905)) + (119888

1(119905) + 119888

2(119905)) = 0

119863120573

119905(1198871(119905) minus 119887

2(119905)) minus (119887

1(119905) minus 119887

2(119905)) = 0

(1198881(119905) + 119888

2(119905)) (119887

1(119905) minus 119887

2(119905)) minus 1 = 0

(44)

together with the initial conditions

1198871(0) = 1 119887

2(0) = 0

1198881(0) = 1 119888

2(119905) = 0

(45)

6 International Journal of Mathematics and Mathematical Sciences

By solving (44) and (45) by considering and applyingDefinition 5 which is discussed in Section 2 we obtain thefollowing results

1198881(119905) + 119888

2(119905) = 119864

120572(minus119905120572)

1198871(119905) minus 119887

2(119905) = 119864

120573(119905120573)

119864120572(minus119905120572) 119864

120573(119905120573) = 1

(46)

By substituting (46) into (42) we obtain the exact solutionfor the system of inhomogeneous FPDEs presented in (39) asfollows

119906 (119909 119905) = 119890119909119864120572(minus119905120572)

V (119909 119905) = 119890minus119909119864120573(119905120573)

119864120572(minus119905120572) 119864

120573(119905120573) = 1

(47)

Thus this means that 119906(119909 119905) = 1V(119909 119905) If we now put120572 120573 rarr 1 in (47) or solve (39) and (40) for 120572 = 120573 = 1we obtain the following exact solution for the correspondingsystem of inhomogeneous PDEs

119906 (119909 119905) = 119890119909minus119905

V (119909 119905) = 119890minus119909+119905(48)

Example 9 Consider the following system of nonlinearFPDEs

119863120572119905119906 = minus119906 minus V

119909119908119910minus V119910119908119909

119863120573

119905V = V minus 119906

119909119908119910minus 119906119910119908119909

119863120574

119905119908 = 119908 minus 119906

119909V119910minus 119906119910V119909

(49)

where 0 lt 120572 and 120573 120574 le 1 subject to the initial conditions

119906 (119909 119910 0) = 119890119909+119910 V (119909 119910 0) = 119890119909minus119910

119908 (119909 119910 0) = 119890minus119909+119910(50)

To solve (49) and (50) by using the new homo-separationof variables method we choose the initial approximations for(49) as follows

1199060(119909 119910 119905) = 119906 (119909 119910 0) 119887

1(119905)

+ (119863119909119906 (119909 119910 0) + 119863

119910119906 (119909 119910 0)) 119887

2(119905)

V0(119909 119910 119905) = V (119909 119910 0) 119888

1(119905)

+ (119863119909V (119909 119910 0) + 119863

119910V (119909 119910 0)) 119888

2(119905)

1199080(119909 119910 119905) = 119908 (119909 119910 0) 119889

1(119905)

+ (119863119909119908 (119909 119910 0) + 119863

119910119908 (119909 119910 0)) 119889

2(119905)

(51)

By substituting the initial conditions given by (50) intothese initial approximations we get the following initialapproximations

1199060(119909 119910 119905) = 119890119909+119910119887

1(119905) + 2119890

119909+1199101198872(119905)

V0(119909 119910 119905) = 119890119909minus119910119888

1(119905)

1199080(119909 119910 119905) = 119890minus119909+119910119889

1(119905)

(52)

By considering and following the approach whichresulted in (27) and substituting the results presented by (52)into the system of nonlinear FPDEs that we would like tosolve which is presented by (49) we get

1198631205721199051199061= 119890minus119909+119910 (119863120572

1199051198871(119905) + 2119863

120572

1199051198872(119905) + 119887

1(119905) + 2119887

2(119905)) equiv 0

119863120573

119905119863119905V1= 119890119909minus119910 (119863

120573

1199051198881(119905) minus 119888

1(119905)) equiv 0

119863120574

1199051199081= 119890minus119909+119910 (119863

120574

1199051198891(119905) minus 119889

1(119905)) equiv 0

(53)By solving the latter system of FPDEs presented in (53)

we obtain the following system of FODEs119863120572119905(1198871(119905) + 2119887

2(119905)) + (119887

1(119905) + 2119887

2(119905)) = 0

119863120573

1199051198881(119905) minus 119888

1(119905) = 0

119863120574

1199051198891(119905) minus 119889

1(119905) = 0

(54)

together with the initial conditions1198871(0) = 1 119887

2(0) = 0

1198881(0) = 1 119888

2(119905) = 0

1198891(0) = 1 119889

2(119905) = 0

(55)

By solving (54) and (55) by considering and applyingDefinition 5 which is discussed in Section 2 of this paper weobtain the following results

1198871(119905) minus 2119887

2(119905) = 119864

120572(minus119905120572)

1198881(119905) = 119864

120573(119905120573)

1198891(0) = 119864

120574(119905120574)

(56)

By substituting (56) into (52) we obtain the exact solutionfor the system of nonlinear FPDEs in (49) as follows

119906 (119909 119910 119905) = 119890119909+119910119864120572(minus119905120572)

V (119909 119910 119905) = 119890119909minus119910119864120573(119905120573)

119908 (119909 119910 119905) = 119890minus119909+119910119864120574(119905120574)

(57)

If we now put 120572 120573 120574 rarr 1 in (57) or solve (49) and (50)for 120572 = 120573 = 120574 = 1 we obtain the following exact solution forthe corresponding system of nonlinear PDEs

119906 (119909 119910 119905) = 119890119909+119910minus119905

V (119909 119910 119905) = 119890119909minus119910+119905

119908 (119909 119910 t) = 119890minus119909+119910+119905

(58)

International Journal of Mathematics and Mathematical Sciences 7

5 Conclusions

In this paper a novel approach was introduced and utilizedto solve linear and nonlinear systems of FPDEs The newtechnique was coined by the authors of this paper as thehomo-separation of variables method In this research workit was demonstrated through different examples how this newmethod can be used for solving various systems of FPDEs

When compared with the existing published methods itis easy to notice that the new method has many advantagesIt is straightforward easy to understand and fast requiringmuch less computations to perform a limited number ofsteps of the simple procedure that can be applied to find theexact solution of a wide range of types of FPDE equationsrsquosystems Furthermore there is no need for using linearizationor restrictive assumptions when employing this newmethod

The basic principle of the current method is to performan elegant combination (which is simple and smart) of twoexisting techniques These two techniques are the popularhomotopy perturbation method (HPM) that was originallyproposed by He [25 26] and another popular approachcalled separation of variables Finally the resulting homo-separation of variables method which is analytical can beused to solve various systems of PDEs with integer andfractional order for example with respect to time

Acknowledgments

The authors would like to thank Maryam Shabani andMehran Shafaiee for their help and support as well asthe inspiration the authors got through friendly scientificdiscussions

References

[1] R L Magin Fractional Calculus in Bioengineering BegellHouse West Redding Conn USA 2006

[2] R L Magin and M Ovadia ldquoModeling the cardiac tissueelectrode interface using fractional calculusrdquo JVCJournal ofVibration and Control vol 14 no 9-10 pp 1431ndash1442 2008

[3] J Keener and J SneydMathematical Physiology Springer NewYork NY USA 2004

[4] A A M Arafa S Z Rida and M Khalil ldquoFractional modelingdynamics of HIV and CD4+ T-cells during primary infectionrdquoNonlinear Biomedical Physics vol 6 no 1 article 1 2012

[5] I S Jesus J A Tenreiro MacHado and J Boaventure CunhaldquoFractional electrical impedances in botanical elementsrdquo Jour-nal of Vibration and Control vol 14 no 9-10 pp 1389ndash14022008

[6] K S Cole ldquoSketches of general and comparative demographyrdquoin Cold Spring Harbor Symposia on Quantitative Biology vol 1pp 107ndash116 Cold Spring Harbor Laboratory Press 1933

[7] V D Djordjevi J Jari B Fabry J J Fredberg andD StamenovildquoFractional derivatives embody essential features of cell rheo-logical behaviorrdquo Annals of Biomedical Engineering vol 31 no6 pp 692ndash699 2003

[8] C Coussot S Kalyanam R Yapp and M Insana ldquoFractionalderivative models for ultrasonic characterization of polymer

and breast tissue viscoelasticityrdquo IEEETransactions onUltrason-ics Ferroelectrics and Frequency Control vol 56 no 4 pp 715ndash725 2009

[9] F Mainardi ldquoFractional diffusive waves in viscoelastic solidsrdquoin Nonlinear Waves in Solids pp 93ndash97 Fairfield 1995

[10] V G Gupta and B Sharma ldquoApplication of Sumudu transformin reaction-diffusion systems and nonlinear wavesrdquo AppliedMathematical Sciences vol 4 no 9ndash12 pp 435ndash446 2010

[11] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego Calif USA 1999

[12] S Momani and Z Odibat ldquoNumerical approach to differentialequations of fractional orderrdquo Journal of Computational andApplied Mathematics vol 207 no 1 pp 96ndash110 2007

[13] O Abdulaziz I Hashim and S Momani ldquoSolving systemsof fractional differential equations by homotopy-perturbationmethodrdquo Physics Letters A vol 372 no 4 pp 451ndash459 2008

[14] M Ganjiani and H Ganjiani ldquoSolution of coupled systemof nonlinear differential equations using homotopy analysismethodrdquo Nonlinear Dynamics of Nonlinear Dynamics andChaos in Engineering Systems vol 56 no 1-2 pp 159ndash167 2009

[15] H Jafari M Nazari D Baleanu and C M Khalique ldquoA newapproach for solving a system of fractional partial differentialequationsrdquo Computers amp Mathematics with Applications 2012

[16] S Kumar A Yildirim and L Wei ldquoA fractional model of thediffusion equation and its analytical solution using Laplacetransformrdquo Scientia Iranica vol 19 no 4 pp 1117ndash1123 2012

[17] D Kumar J Singh and S Rathore ldquoSumudu decompositionmethod for nonlinear equationsrdquo International MathematicalForum vol 7 no 9ndash12 pp 515ndash521 2012

[18] J Singh D Kumar and Sushila ldquoHomotopy perturbationsumudu transform method for nonlinear equationsrdquo Advancesin Theoretical and Applied Mechanics vol 4 no 4 pp 165ndash1752011

[19] A Karbalaie M M Montazeri and H H Muhammed ldquoNewapproach to find the exact solution of fractional partial differ-ential equationrdquo WSEAS Transactions on Mathematics vol 11no 10 pp 908ndash917 2012

[20] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Publishers Yverdon Switzerland 1993

[21] A A M Arafa S Z Rida and H Mohamed ldquoHomotopyanalysis method for solving biological population modelrdquoCommunications in Theoretical Physics vol 56 no 5 pp 797ndash800 2011

[22] J Peng and K Li ldquoA note on property of the Mittag-Lefflerfunctionrdquo Journal of Mathematical Analysis and Applicationsvol 370 no 2 pp 635ndash638 2010

[23] J-F Cheng and Y-M Chu ldquoSolution to the linear fractionaldifferential equation using Adomian decomposition methodrdquoMathematical Problems in Engineering vol 2011 Article ID587068 14 pages 2011

[24] Z M Odibat ldquoAnalytic study on linear systems of fractionaldifferential equationsrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1171ndash1183 2010

[25] J He ldquoApplication of homotopy perturbation method to non-linear wave equationsrdquo Chaos Solitons amp Fractals vol 26 no 3pp 695ndash700 2005

8 International Journal of Mathematics and Mathematical Sciences

[26] J H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 6 pp 207ndash208 2005

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

International Journal of Mathematics and Mathematical Sciences 5

By substituting the two initial conditions given by (30)into these two initial approximations we get the followinginitial approximations

1199060(119909 119905) = 119887

1(119905) sinh (119909) + 119887

2(119905) cosh (119909)

V0(119909 119905) = 119888

1(119905) cosh (119909) + 119888

2(119905) sinh (119909)

(32)

By considering and following the approach whichresulted in (27) and substituting the results presented by (32)into the system of linear FPDEs in (29) we get

1198631199051199061= sinh (119909) (119863120572

1199051198871(119905) + 119887

1(119905) minus 119888

1(119905) + 119888

2(119905))

+ cosh (119909) (1198631205721199051198872(119905) + 119887

2(119905) + 119888

1(119905) minus 119888

2(119905)) equiv 0

119863119905V1= cosh (119909) (119863120572

1199051198881(119905) + 119888

1(119905) minus 119887

1(119905) + 119887

2(119905))

+ sinh (119909) (1198631205721199051198882(119905) + 119888

2(119905) + 119887

1(119905) minus 119887

2(119905)) equiv 0

(33)

By solving the latter system of linear FPDEs presented in(33) we obtain the following system of linear FODEs

1198631205721199051198871(119905) + 119887

1(119905) minus 119888

1(119905) + 119888

2(119905) = 0

1198631205721199051198872(119905) + 119887

2(119905) + 119888

1(119905) minus 119888

2(119905) = 0

1198631205721199051198881(119905) + 119888

1(119905) minus 119887

1(119905) + 119887

2(119905) = 0

1198631205721199051198882(119905) + 119888

2(119905) + 119887

1(119905) minus 119887

2(119905) = 0

(34)

together with the initial conditions

1198872(0) = 1 119887

2(0) = 0

1198881(0) = 1 119888

1(0) = 0

(35)

By solving (34) and (35) by considering and applyingDefinition 6 which is discussed in Section 2 we obtain thefollowing results

1198871(119905) =

1

2(119864120572(119905120572) + 119864

120572(minus119905120572))

1198872(119905) =

1

2(minus119864120572(119905120572) + 119864

120572(minus119905120572))

1198881(119905) =

1

2(119864120572(119905120572) + 119864

120572(minus119905120572))

1198882(119905) =

1

2(minus119864120572(119905120572) + 119864

120572(minus119905120572))

(36)

By substituting (36) into (32) and applying Definition 4in particular properties (2) (5) and (6) which are discussedin Section 2 we obtain the exact solution for the system oflinear FPDEs in (29) as follows

119906 (119909 119905) = 11986421205721

(1199052120572) sinh (119909) minus 1199051205721198642120572120572+1

(1199052120572) cosh (119909)

V (119909 119905) = 11986421205721

(1199052120572) cosh (119909) minus 1199051205721198642120572120572+1

(1199052120572) sinh (119909) (37)

If we now put 120572 rarr 1 in (37) or solve (29) and (30)for 120572 = 1 we obtain the following exact solution for thecorresponding system of linear PDEs

119906 (119909 119905) = sinh (119909 minus 119905)

V (119909 119905) = cosh (119909 minus 119905) (38)

Example 8 Consider the following system of inhomoge-neous FPDEs

119863120572119905119906 + 119906119909V + 119906 = 1

119863120573

119905V minus V119909119906 minus V = 1

(39)

where 0 lt 120572 and 120573 le 1 subject to the initial conditions

119906 (119909 0) = 119890119909 V (119909 0) = 119890minus119909 (40)

To solve (39) and (40) by using the new homo-separationof variables method we choose the initial approximations for(39) as follows

1199060(119909 119905) = 119906 (119909 0) 119888

1(119905) + 119863

119909119906 (119909 0) 119888

2(119905)

V0(119909 119905) = V (119909 0) 119887

1(119905) + 119863

119909V (119909 0) 119887

2(119905)

(41)

By substituting the two initial conditions given by (40)into these two initial approximations we get the followinginitial approximations

1199060(119909 119905) = 119890

1199091198881(119905) + 119890

1199091198882(119905)

V0(119909 119905) = 119890

minus1199091198871(119905) minus 119890

minus1199091198872(119905)

(42)

By considering and following the approach whichresulted in (27) and substituting the results presented by (42)into the system of inhomogeneous FPDEs that we would liketo solve which is presented by (39) we get

1198631205721199051199061= 119890119909 (119863120572

119905(1198881(119905) + 119888

2(119905)) + (119888

1(119905) + 119888

2(119905)))

+ (1198881(119905) + 119888

2(119905)) (119887

1(119905) minus 119887

2(119905)) minus 1 equiv 0

119863120573

119905V1= 119890119909 (119863

120573

119905(1198871(119905) minus 119887

2(119905)) minus (119887

1(119905) minus 119887

2(119905)))

+ (1198881(119905) + 119888

2(119905)) (119887

1(119905) minus 119887

2(119905)) minus 1 equiv 0

(43)

By solving the latter system of FPDEs presented in (43)we obtain the following system of FODEs

119863120572119905(1198881(119905) + 119888

2(119905)) + (119888

1(119905) + 119888

2(119905)) = 0

119863120573

119905(1198871(119905) minus 119887

2(119905)) minus (119887

1(119905) minus 119887

2(119905)) = 0

(1198881(119905) + 119888

2(119905)) (119887

1(119905) minus 119887

2(119905)) minus 1 = 0

(44)

together with the initial conditions

1198871(0) = 1 119887

2(0) = 0

1198881(0) = 1 119888

2(119905) = 0

(45)

6 International Journal of Mathematics and Mathematical Sciences

By solving (44) and (45) by considering and applyingDefinition 5 which is discussed in Section 2 we obtain thefollowing results

1198881(119905) + 119888

2(119905) = 119864

120572(minus119905120572)

1198871(119905) minus 119887

2(119905) = 119864

120573(119905120573)

119864120572(minus119905120572) 119864

120573(119905120573) = 1

(46)

By substituting (46) into (42) we obtain the exact solutionfor the system of inhomogeneous FPDEs presented in (39) asfollows

119906 (119909 119905) = 119890119909119864120572(minus119905120572)

V (119909 119905) = 119890minus119909119864120573(119905120573)

119864120572(minus119905120572) 119864

120573(119905120573) = 1

(47)

Thus this means that 119906(119909 119905) = 1V(119909 119905) If we now put120572 120573 rarr 1 in (47) or solve (39) and (40) for 120572 = 120573 = 1we obtain the following exact solution for the correspondingsystem of inhomogeneous PDEs

119906 (119909 119905) = 119890119909minus119905

V (119909 119905) = 119890minus119909+119905(48)

Example 9 Consider the following system of nonlinearFPDEs

119863120572119905119906 = minus119906 minus V

119909119908119910minus V119910119908119909

119863120573

119905V = V minus 119906

119909119908119910minus 119906119910119908119909

119863120574

119905119908 = 119908 minus 119906

119909V119910minus 119906119910V119909

(49)

where 0 lt 120572 and 120573 120574 le 1 subject to the initial conditions

119906 (119909 119910 0) = 119890119909+119910 V (119909 119910 0) = 119890119909minus119910

119908 (119909 119910 0) = 119890minus119909+119910(50)

To solve (49) and (50) by using the new homo-separationof variables method we choose the initial approximations for(49) as follows

1199060(119909 119910 119905) = 119906 (119909 119910 0) 119887

1(119905)

+ (119863119909119906 (119909 119910 0) + 119863

119910119906 (119909 119910 0)) 119887

2(119905)

V0(119909 119910 119905) = V (119909 119910 0) 119888

1(119905)

+ (119863119909V (119909 119910 0) + 119863

119910V (119909 119910 0)) 119888

2(119905)

1199080(119909 119910 119905) = 119908 (119909 119910 0) 119889

1(119905)

+ (119863119909119908 (119909 119910 0) + 119863

119910119908 (119909 119910 0)) 119889

2(119905)

(51)

By substituting the initial conditions given by (50) intothese initial approximations we get the following initialapproximations

1199060(119909 119910 119905) = 119890119909+119910119887

1(119905) + 2119890

119909+1199101198872(119905)

V0(119909 119910 119905) = 119890119909minus119910119888

1(119905)

1199080(119909 119910 119905) = 119890minus119909+119910119889

1(119905)

(52)

By considering and following the approach whichresulted in (27) and substituting the results presented by (52)into the system of nonlinear FPDEs that we would like tosolve which is presented by (49) we get

1198631205721199051199061= 119890minus119909+119910 (119863120572

1199051198871(119905) + 2119863

120572

1199051198872(119905) + 119887

1(119905) + 2119887

2(119905)) equiv 0

119863120573

119905119863119905V1= 119890119909minus119910 (119863

120573

1199051198881(119905) minus 119888

1(119905)) equiv 0

119863120574

1199051199081= 119890minus119909+119910 (119863

120574

1199051198891(119905) minus 119889

1(119905)) equiv 0

(53)By solving the latter system of FPDEs presented in (53)

we obtain the following system of FODEs119863120572119905(1198871(119905) + 2119887

2(119905)) + (119887

1(119905) + 2119887

2(119905)) = 0

119863120573

1199051198881(119905) minus 119888

1(119905) = 0

119863120574

1199051198891(119905) minus 119889

1(119905) = 0

(54)

together with the initial conditions1198871(0) = 1 119887

2(0) = 0

1198881(0) = 1 119888

2(119905) = 0

1198891(0) = 1 119889

2(119905) = 0

(55)

By solving (54) and (55) by considering and applyingDefinition 5 which is discussed in Section 2 of this paper weobtain the following results

1198871(119905) minus 2119887

2(119905) = 119864

120572(minus119905120572)

1198881(119905) = 119864

120573(119905120573)

1198891(0) = 119864

120574(119905120574)

(56)

By substituting (56) into (52) we obtain the exact solutionfor the system of nonlinear FPDEs in (49) as follows

119906 (119909 119910 119905) = 119890119909+119910119864120572(minus119905120572)

V (119909 119910 119905) = 119890119909minus119910119864120573(119905120573)

119908 (119909 119910 119905) = 119890minus119909+119910119864120574(119905120574)

(57)

If we now put 120572 120573 120574 rarr 1 in (57) or solve (49) and (50)for 120572 = 120573 = 120574 = 1 we obtain the following exact solution forthe corresponding system of nonlinear PDEs

119906 (119909 119910 119905) = 119890119909+119910minus119905

V (119909 119910 119905) = 119890119909minus119910+119905

119908 (119909 119910 t) = 119890minus119909+119910+119905

(58)

International Journal of Mathematics and Mathematical Sciences 7

5 Conclusions

In this paper a novel approach was introduced and utilizedto solve linear and nonlinear systems of FPDEs The newtechnique was coined by the authors of this paper as thehomo-separation of variables method In this research workit was demonstrated through different examples how this newmethod can be used for solving various systems of FPDEs

When compared with the existing published methods itis easy to notice that the new method has many advantagesIt is straightforward easy to understand and fast requiringmuch less computations to perform a limited number ofsteps of the simple procedure that can be applied to find theexact solution of a wide range of types of FPDE equationsrsquosystems Furthermore there is no need for using linearizationor restrictive assumptions when employing this newmethod

The basic principle of the current method is to performan elegant combination (which is simple and smart) of twoexisting techniques These two techniques are the popularhomotopy perturbation method (HPM) that was originallyproposed by He [25 26] and another popular approachcalled separation of variables Finally the resulting homo-separation of variables method which is analytical can beused to solve various systems of PDEs with integer andfractional order for example with respect to time

Acknowledgments

The authors would like to thank Maryam Shabani andMehran Shafaiee for their help and support as well asthe inspiration the authors got through friendly scientificdiscussions

References

[1] R L Magin Fractional Calculus in Bioengineering BegellHouse West Redding Conn USA 2006

[2] R L Magin and M Ovadia ldquoModeling the cardiac tissueelectrode interface using fractional calculusrdquo JVCJournal ofVibration and Control vol 14 no 9-10 pp 1431ndash1442 2008

[3] J Keener and J SneydMathematical Physiology Springer NewYork NY USA 2004

[4] A A M Arafa S Z Rida and M Khalil ldquoFractional modelingdynamics of HIV and CD4+ T-cells during primary infectionrdquoNonlinear Biomedical Physics vol 6 no 1 article 1 2012

[5] I S Jesus J A Tenreiro MacHado and J Boaventure CunhaldquoFractional electrical impedances in botanical elementsrdquo Jour-nal of Vibration and Control vol 14 no 9-10 pp 1389ndash14022008

[6] K S Cole ldquoSketches of general and comparative demographyrdquoin Cold Spring Harbor Symposia on Quantitative Biology vol 1pp 107ndash116 Cold Spring Harbor Laboratory Press 1933

[7] V D Djordjevi J Jari B Fabry J J Fredberg andD StamenovildquoFractional derivatives embody essential features of cell rheo-logical behaviorrdquo Annals of Biomedical Engineering vol 31 no6 pp 692ndash699 2003

[8] C Coussot S Kalyanam R Yapp and M Insana ldquoFractionalderivative models for ultrasonic characterization of polymer

and breast tissue viscoelasticityrdquo IEEETransactions onUltrason-ics Ferroelectrics and Frequency Control vol 56 no 4 pp 715ndash725 2009

[9] F Mainardi ldquoFractional diffusive waves in viscoelastic solidsrdquoin Nonlinear Waves in Solids pp 93ndash97 Fairfield 1995

[10] V G Gupta and B Sharma ldquoApplication of Sumudu transformin reaction-diffusion systems and nonlinear wavesrdquo AppliedMathematical Sciences vol 4 no 9ndash12 pp 435ndash446 2010

[11] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego Calif USA 1999

[12] S Momani and Z Odibat ldquoNumerical approach to differentialequations of fractional orderrdquo Journal of Computational andApplied Mathematics vol 207 no 1 pp 96ndash110 2007

[13] O Abdulaziz I Hashim and S Momani ldquoSolving systemsof fractional differential equations by homotopy-perturbationmethodrdquo Physics Letters A vol 372 no 4 pp 451ndash459 2008

[14] M Ganjiani and H Ganjiani ldquoSolution of coupled systemof nonlinear differential equations using homotopy analysismethodrdquo Nonlinear Dynamics of Nonlinear Dynamics andChaos in Engineering Systems vol 56 no 1-2 pp 159ndash167 2009

[15] H Jafari M Nazari D Baleanu and C M Khalique ldquoA newapproach for solving a system of fractional partial differentialequationsrdquo Computers amp Mathematics with Applications 2012

[16] S Kumar A Yildirim and L Wei ldquoA fractional model of thediffusion equation and its analytical solution using Laplacetransformrdquo Scientia Iranica vol 19 no 4 pp 1117ndash1123 2012

[17] D Kumar J Singh and S Rathore ldquoSumudu decompositionmethod for nonlinear equationsrdquo International MathematicalForum vol 7 no 9ndash12 pp 515ndash521 2012

[18] J Singh D Kumar and Sushila ldquoHomotopy perturbationsumudu transform method for nonlinear equationsrdquo Advancesin Theoretical and Applied Mechanics vol 4 no 4 pp 165ndash1752011

[19] A Karbalaie M M Montazeri and H H Muhammed ldquoNewapproach to find the exact solution of fractional partial differ-ential equationrdquo WSEAS Transactions on Mathematics vol 11no 10 pp 908ndash917 2012

[20] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Publishers Yverdon Switzerland 1993

[21] A A M Arafa S Z Rida and H Mohamed ldquoHomotopyanalysis method for solving biological population modelrdquoCommunications in Theoretical Physics vol 56 no 5 pp 797ndash800 2011

[22] J Peng and K Li ldquoA note on property of the Mittag-Lefflerfunctionrdquo Journal of Mathematical Analysis and Applicationsvol 370 no 2 pp 635ndash638 2010

[23] J-F Cheng and Y-M Chu ldquoSolution to the linear fractionaldifferential equation using Adomian decomposition methodrdquoMathematical Problems in Engineering vol 2011 Article ID587068 14 pages 2011

[24] Z M Odibat ldquoAnalytic study on linear systems of fractionaldifferential equationsrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1171ndash1183 2010

[25] J He ldquoApplication of homotopy perturbation method to non-linear wave equationsrdquo Chaos Solitons amp Fractals vol 26 no 3pp 695ndash700 2005

8 International Journal of Mathematics and Mathematical Sciences

[26] J H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 6 pp 207ndash208 2005

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 International Journal of Mathematics and Mathematical Sciences

By solving (44) and (45) by considering and applyingDefinition 5 which is discussed in Section 2 we obtain thefollowing results

1198881(119905) + 119888

2(119905) = 119864

120572(minus119905120572)

1198871(119905) minus 119887

2(119905) = 119864

120573(119905120573)

119864120572(minus119905120572) 119864

120573(119905120573) = 1

(46)

By substituting (46) into (42) we obtain the exact solutionfor the system of inhomogeneous FPDEs presented in (39) asfollows

119906 (119909 119905) = 119890119909119864120572(minus119905120572)

V (119909 119905) = 119890minus119909119864120573(119905120573)

119864120572(minus119905120572) 119864

120573(119905120573) = 1

(47)

Thus this means that 119906(119909 119905) = 1V(119909 119905) If we now put120572 120573 rarr 1 in (47) or solve (39) and (40) for 120572 = 120573 = 1we obtain the following exact solution for the correspondingsystem of inhomogeneous PDEs

119906 (119909 119905) = 119890119909minus119905

V (119909 119905) = 119890minus119909+119905(48)

Example 9 Consider the following system of nonlinearFPDEs

119863120572119905119906 = minus119906 minus V

119909119908119910minus V119910119908119909

119863120573

119905V = V minus 119906

119909119908119910minus 119906119910119908119909

119863120574

119905119908 = 119908 minus 119906

119909V119910minus 119906119910V119909

(49)

where 0 lt 120572 and 120573 120574 le 1 subject to the initial conditions

119906 (119909 119910 0) = 119890119909+119910 V (119909 119910 0) = 119890119909minus119910

119908 (119909 119910 0) = 119890minus119909+119910(50)

To solve (49) and (50) by using the new homo-separationof variables method we choose the initial approximations for(49) as follows

1199060(119909 119910 119905) = 119906 (119909 119910 0) 119887

1(119905)

+ (119863119909119906 (119909 119910 0) + 119863

119910119906 (119909 119910 0)) 119887

2(119905)

V0(119909 119910 119905) = V (119909 119910 0) 119888

1(119905)

+ (119863119909V (119909 119910 0) + 119863

119910V (119909 119910 0)) 119888

2(119905)

1199080(119909 119910 119905) = 119908 (119909 119910 0) 119889

1(119905)

+ (119863119909119908 (119909 119910 0) + 119863

119910119908 (119909 119910 0)) 119889

2(119905)

(51)

By substituting the initial conditions given by (50) intothese initial approximations we get the following initialapproximations

1199060(119909 119910 119905) = 119890119909+119910119887

1(119905) + 2119890

119909+1199101198872(119905)

V0(119909 119910 119905) = 119890119909minus119910119888

1(119905)

1199080(119909 119910 119905) = 119890minus119909+119910119889

1(119905)

(52)

By considering and following the approach whichresulted in (27) and substituting the results presented by (52)into the system of nonlinear FPDEs that we would like tosolve which is presented by (49) we get

1198631205721199051199061= 119890minus119909+119910 (119863120572

1199051198871(119905) + 2119863

120572

1199051198872(119905) + 119887

1(119905) + 2119887

2(119905)) equiv 0

119863120573

119905119863119905V1= 119890119909minus119910 (119863

120573

1199051198881(119905) minus 119888

1(119905)) equiv 0

119863120574

1199051199081= 119890minus119909+119910 (119863

120574

1199051198891(119905) minus 119889

1(119905)) equiv 0

(53)By solving the latter system of FPDEs presented in (53)

we obtain the following system of FODEs119863120572119905(1198871(119905) + 2119887

2(119905)) + (119887

1(119905) + 2119887

2(119905)) = 0

119863120573

1199051198881(119905) minus 119888

1(119905) = 0

119863120574

1199051198891(119905) minus 119889

1(119905) = 0

(54)

together with the initial conditions1198871(0) = 1 119887

2(0) = 0

1198881(0) = 1 119888

2(119905) = 0

1198891(0) = 1 119889

2(119905) = 0

(55)

By solving (54) and (55) by considering and applyingDefinition 5 which is discussed in Section 2 of this paper weobtain the following results

1198871(119905) minus 2119887

2(119905) = 119864

120572(minus119905120572)

1198881(119905) = 119864

120573(119905120573)

1198891(0) = 119864

120574(119905120574)

(56)

By substituting (56) into (52) we obtain the exact solutionfor the system of nonlinear FPDEs in (49) as follows

119906 (119909 119910 119905) = 119890119909+119910119864120572(minus119905120572)

V (119909 119910 119905) = 119890119909minus119910119864120573(119905120573)

119908 (119909 119910 119905) = 119890minus119909+119910119864120574(119905120574)

(57)

If we now put 120572 120573 120574 rarr 1 in (57) or solve (49) and (50)for 120572 = 120573 = 120574 = 1 we obtain the following exact solution forthe corresponding system of nonlinear PDEs

119906 (119909 119910 119905) = 119890119909+119910minus119905

V (119909 119910 119905) = 119890119909minus119910+119905

119908 (119909 119910 t) = 119890minus119909+119910+119905

(58)

International Journal of Mathematics and Mathematical Sciences 7

5 Conclusions

In this paper a novel approach was introduced and utilizedto solve linear and nonlinear systems of FPDEs The newtechnique was coined by the authors of this paper as thehomo-separation of variables method In this research workit was demonstrated through different examples how this newmethod can be used for solving various systems of FPDEs

When compared with the existing published methods itis easy to notice that the new method has many advantagesIt is straightforward easy to understand and fast requiringmuch less computations to perform a limited number ofsteps of the simple procedure that can be applied to find theexact solution of a wide range of types of FPDE equationsrsquosystems Furthermore there is no need for using linearizationor restrictive assumptions when employing this newmethod

The basic principle of the current method is to performan elegant combination (which is simple and smart) of twoexisting techniques These two techniques are the popularhomotopy perturbation method (HPM) that was originallyproposed by He [25 26] and another popular approachcalled separation of variables Finally the resulting homo-separation of variables method which is analytical can beused to solve various systems of PDEs with integer andfractional order for example with respect to time

Acknowledgments

The authors would like to thank Maryam Shabani andMehran Shafaiee for their help and support as well asthe inspiration the authors got through friendly scientificdiscussions

References

[1] R L Magin Fractional Calculus in Bioengineering BegellHouse West Redding Conn USA 2006

[2] R L Magin and M Ovadia ldquoModeling the cardiac tissueelectrode interface using fractional calculusrdquo JVCJournal ofVibration and Control vol 14 no 9-10 pp 1431ndash1442 2008

[3] J Keener and J SneydMathematical Physiology Springer NewYork NY USA 2004

[4] A A M Arafa S Z Rida and M Khalil ldquoFractional modelingdynamics of HIV and CD4+ T-cells during primary infectionrdquoNonlinear Biomedical Physics vol 6 no 1 article 1 2012

[5] I S Jesus J A Tenreiro MacHado and J Boaventure CunhaldquoFractional electrical impedances in botanical elementsrdquo Jour-nal of Vibration and Control vol 14 no 9-10 pp 1389ndash14022008

[6] K S Cole ldquoSketches of general and comparative demographyrdquoin Cold Spring Harbor Symposia on Quantitative Biology vol 1pp 107ndash116 Cold Spring Harbor Laboratory Press 1933

[7] V D Djordjevi J Jari B Fabry J J Fredberg andD StamenovildquoFractional derivatives embody essential features of cell rheo-logical behaviorrdquo Annals of Biomedical Engineering vol 31 no6 pp 692ndash699 2003

[8] C Coussot S Kalyanam R Yapp and M Insana ldquoFractionalderivative models for ultrasonic characterization of polymer

and breast tissue viscoelasticityrdquo IEEETransactions onUltrason-ics Ferroelectrics and Frequency Control vol 56 no 4 pp 715ndash725 2009

[9] F Mainardi ldquoFractional diffusive waves in viscoelastic solidsrdquoin Nonlinear Waves in Solids pp 93ndash97 Fairfield 1995

[10] V G Gupta and B Sharma ldquoApplication of Sumudu transformin reaction-diffusion systems and nonlinear wavesrdquo AppliedMathematical Sciences vol 4 no 9ndash12 pp 435ndash446 2010

[11] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego Calif USA 1999

[12] S Momani and Z Odibat ldquoNumerical approach to differentialequations of fractional orderrdquo Journal of Computational andApplied Mathematics vol 207 no 1 pp 96ndash110 2007

[13] O Abdulaziz I Hashim and S Momani ldquoSolving systemsof fractional differential equations by homotopy-perturbationmethodrdquo Physics Letters A vol 372 no 4 pp 451ndash459 2008

[14] M Ganjiani and H Ganjiani ldquoSolution of coupled systemof nonlinear differential equations using homotopy analysismethodrdquo Nonlinear Dynamics of Nonlinear Dynamics andChaos in Engineering Systems vol 56 no 1-2 pp 159ndash167 2009

[15] H Jafari M Nazari D Baleanu and C M Khalique ldquoA newapproach for solving a system of fractional partial differentialequationsrdquo Computers amp Mathematics with Applications 2012

[16] S Kumar A Yildirim and L Wei ldquoA fractional model of thediffusion equation and its analytical solution using Laplacetransformrdquo Scientia Iranica vol 19 no 4 pp 1117ndash1123 2012

[17] D Kumar J Singh and S Rathore ldquoSumudu decompositionmethod for nonlinear equationsrdquo International MathematicalForum vol 7 no 9ndash12 pp 515ndash521 2012

[18] J Singh D Kumar and Sushila ldquoHomotopy perturbationsumudu transform method for nonlinear equationsrdquo Advancesin Theoretical and Applied Mechanics vol 4 no 4 pp 165ndash1752011

[19] A Karbalaie M M Montazeri and H H Muhammed ldquoNewapproach to find the exact solution of fractional partial differ-ential equationrdquo WSEAS Transactions on Mathematics vol 11no 10 pp 908ndash917 2012

[20] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Publishers Yverdon Switzerland 1993

[21] A A M Arafa S Z Rida and H Mohamed ldquoHomotopyanalysis method for solving biological population modelrdquoCommunications in Theoretical Physics vol 56 no 5 pp 797ndash800 2011

[22] J Peng and K Li ldquoA note on property of the Mittag-Lefflerfunctionrdquo Journal of Mathematical Analysis and Applicationsvol 370 no 2 pp 635ndash638 2010

[23] J-F Cheng and Y-M Chu ldquoSolution to the linear fractionaldifferential equation using Adomian decomposition methodrdquoMathematical Problems in Engineering vol 2011 Article ID587068 14 pages 2011

[24] Z M Odibat ldquoAnalytic study on linear systems of fractionaldifferential equationsrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1171ndash1183 2010

[25] J He ldquoApplication of homotopy perturbation method to non-linear wave equationsrdquo Chaos Solitons amp Fractals vol 26 no 3pp 695ndash700 2005

8 International Journal of Mathematics and Mathematical Sciences

[26] J H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 6 pp 207ndash208 2005

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

International Journal of Mathematics and Mathematical Sciences 7

5 Conclusions

In this paper a novel approach was introduced and utilizedto solve linear and nonlinear systems of FPDEs The newtechnique was coined by the authors of this paper as thehomo-separation of variables method In this research workit was demonstrated through different examples how this newmethod can be used for solving various systems of FPDEs

When compared with the existing published methods itis easy to notice that the new method has many advantagesIt is straightforward easy to understand and fast requiringmuch less computations to perform a limited number ofsteps of the simple procedure that can be applied to find theexact solution of a wide range of types of FPDE equationsrsquosystems Furthermore there is no need for using linearizationor restrictive assumptions when employing this newmethod

The basic principle of the current method is to performan elegant combination (which is simple and smart) of twoexisting techniques These two techniques are the popularhomotopy perturbation method (HPM) that was originallyproposed by He [25 26] and another popular approachcalled separation of variables Finally the resulting homo-separation of variables method which is analytical can beused to solve various systems of PDEs with integer andfractional order for example with respect to time

Acknowledgments

The authors would like to thank Maryam Shabani andMehran Shafaiee for their help and support as well asthe inspiration the authors got through friendly scientificdiscussions

References

[1] R L Magin Fractional Calculus in Bioengineering BegellHouse West Redding Conn USA 2006

[2] R L Magin and M Ovadia ldquoModeling the cardiac tissueelectrode interface using fractional calculusrdquo JVCJournal ofVibration and Control vol 14 no 9-10 pp 1431ndash1442 2008

[3] J Keener and J SneydMathematical Physiology Springer NewYork NY USA 2004

[4] A A M Arafa S Z Rida and M Khalil ldquoFractional modelingdynamics of HIV and CD4+ T-cells during primary infectionrdquoNonlinear Biomedical Physics vol 6 no 1 article 1 2012

[5] I S Jesus J A Tenreiro MacHado and J Boaventure CunhaldquoFractional electrical impedances in botanical elementsrdquo Jour-nal of Vibration and Control vol 14 no 9-10 pp 1389ndash14022008

[6] K S Cole ldquoSketches of general and comparative demographyrdquoin Cold Spring Harbor Symposia on Quantitative Biology vol 1pp 107ndash116 Cold Spring Harbor Laboratory Press 1933

[7] V D Djordjevi J Jari B Fabry J J Fredberg andD StamenovildquoFractional derivatives embody essential features of cell rheo-logical behaviorrdquo Annals of Biomedical Engineering vol 31 no6 pp 692ndash699 2003

[8] C Coussot S Kalyanam R Yapp and M Insana ldquoFractionalderivative models for ultrasonic characterization of polymer

and breast tissue viscoelasticityrdquo IEEETransactions onUltrason-ics Ferroelectrics and Frequency Control vol 56 no 4 pp 715ndash725 2009

[9] F Mainardi ldquoFractional diffusive waves in viscoelastic solidsrdquoin Nonlinear Waves in Solids pp 93ndash97 Fairfield 1995

[10] V G Gupta and B Sharma ldquoApplication of Sumudu transformin reaction-diffusion systems and nonlinear wavesrdquo AppliedMathematical Sciences vol 4 no 9ndash12 pp 435ndash446 2010

[11] I Podlubny Fractional Differential Equations An Introductionto Fractional Derivatives Fractional Differential Equations toMethods of Their Solution and Some of Their Applications vol198 ofMathematics in Science and Engineering Academic PressSan Diego Calif USA 1999

[12] S Momani and Z Odibat ldquoNumerical approach to differentialequations of fractional orderrdquo Journal of Computational andApplied Mathematics vol 207 no 1 pp 96ndash110 2007

[13] O Abdulaziz I Hashim and S Momani ldquoSolving systemsof fractional differential equations by homotopy-perturbationmethodrdquo Physics Letters A vol 372 no 4 pp 451ndash459 2008

[14] M Ganjiani and H Ganjiani ldquoSolution of coupled systemof nonlinear differential equations using homotopy analysismethodrdquo Nonlinear Dynamics of Nonlinear Dynamics andChaos in Engineering Systems vol 56 no 1-2 pp 159ndash167 2009

[15] H Jafari M Nazari D Baleanu and C M Khalique ldquoA newapproach for solving a system of fractional partial differentialequationsrdquo Computers amp Mathematics with Applications 2012

[16] S Kumar A Yildirim and L Wei ldquoA fractional model of thediffusion equation and its analytical solution using Laplacetransformrdquo Scientia Iranica vol 19 no 4 pp 1117ndash1123 2012

[17] D Kumar J Singh and S Rathore ldquoSumudu decompositionmethod for nonlinear equationsrdquo International MathematicalForum vol 7 no 9ndash12 pp 515ndash521 2012

[18] J Singh D Kumar and Sushila ldquoHomotopy perturbationsumudu transform method for nonlinear equationsrdquo Advancesin Theoretical and Applied Mechanics vol 4 no 4 pp 165ndash1752011

[19] A Karbalaie M M Montazeri and H H Muhammed ldquoNewapproach to find the exact solution of fractional partial differ-ential equationrdquo WSEAS Transactions on Mathematics vol 11no 10 pp 908ndash917 2012

[20] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Publishers Yverdon Switzerland 1993

[21] A A M Arafa S Z Rida and H Mohamed ldquoHomotopyanalysis method for solving biological population modelrdquoCommunications in Theoretical Physics vol 56 no 5 pp 797ndash800 2011

[22] J Peng and K Li ldquoA note on property of the Mittag-Lefflerfunctionrdquo Journal of Mathematical Analysis and Applicationsvol 370 no 2 pp 635ndash638 2010

[23] J-F Cheng and Y-M Chu ldquoSolution to the linear fractionaldifferential equation using Adomian decomposition methodrdquoMathematical Problems in Engineering vol 2011 Article ID587068 14 pages 2011

[24] Z M Odibat ldquoAnalytic study on linear systems of fractionaldifferential equationsrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1171ndash1183 2010

[25] J He ldquoApplication of homotopy perturbation method to non-linear wave equationsrdquo Chaos Solitons amp Fractals vol 26 no 3pp 695ndash700 2005

8 International Journal of Mathematics and Mathematical Sciences

[26] J H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 6 pp 207ndash208 2005

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 International Journal of Mathematics and Mathematical Sciences

[26] J H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 6 pp 207ndash208 2005

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