Research ArticleA Spherically Symmetric Model for the Tumor Growth

Saeed M Ali Ashfaque H Bokhari M Yousuf and F D Zaman

Department of Mathematics and Statistics King Fahd University of Petroleum ampMinerals Dhahran 31261 Saudi Arabia

Correspondence should be addressed to F D Zaman fzamankfupmedusa

Received 25 May 2013 Accepted 19 December 2013 Published 19 January 2014

Academic Editor Renat Zhdanov

Copyright copy 2014 Saeed M Ali et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The nonlinear tumor equation in spherical coordinates assuming that both the diffusivity and the killing rate are functions ofconcentration of tumor cell is studiedA complete classificationwith regard to the diffusivity andnet killing rate is obtained using Liesymmetry analysis The reduction of the nonlinear governing equation is carried out in some interesting cases and exact solutionsare obtained

1 Introduction

The tumor growth has been usually modeled as a reaction-diffusion process in the literature Jones et al [1] have givena simple tumor model based upon this idea A model de-scribing the growth of the tumor in brain taking into accountdiffusion or motility as well as proliferation of tumor cells hasbeen developed in a series of papers [2 3] In continuation ofthis approach Tracqui et al [4] suggest a model which takesinto account treatment and thus killing rate of tumor cellsalong with the above factors The governing equation in thiscase is

119906119905= 119863nabla

2119906 + 119875119906 minus 119896119906 (1)

where 119906 is the concentration of tumor cells119863 is the diffusioncoefficient 119875 is the proliferation rate and 119896 is the killing rateAssuming complete radial summery Moyo and Leach [3]have studied this model with 119870(119909 119905) = 119875 minus 119896 being variableThe resulting governing equation reduces to the simple form

119906119905= 119906119909119909minus 119870119906 (2)

where 119906(119909 119905) = 119903119906(119903 119905) They have performed Lie symmetryanalysis and presented some exact solutions based uponthis approach Consequently Bokhari et al [5] used Liesymmetry analysis to obtain a number of invariant reductionsand exact solutions in the case of killing rate 119870(119906) beingfunction of 119906 The present study is based upon the factthat the diffusivity is not necessarily a constant and may

depend upon the concentration of tumor cells Moreoverthe net killing rate 119870 is also taken to be 119906-dependent Thisintroduces nonlinearity in the governing equation Keepingthese assumptions in mind (1) becomes

119906119905= nabla sdot (119863 (119906) nabla119906) minus 119870 (119906) 119906 (3)

which in spherical coordinates and with radial symmetryassumption becomes

1

1199092

120597

120597119909

(1199092119863 (119906) 119906

119909) minus 119870 (119906) 119906 = 119906

119905 (4)

where 119863(119906) is the diffusivity of the medium and 119870(119906) is thenet killing rate We present a classification of the functions119863(119906) and 119870(119906) using Lie symmetry analysis The Lie sym-metry approach first proposed by Lie [6] has been usedto classify nonlinear differential equations find appropriatesimilarity transformations and find exact solutions Onemayrefer to [7ndash10] for a good account of thismethod Some recentstudies in nonlinear diffusion equations using this approachcan be found in [1 6]

2 Symmetry Analysis of the Tumor Equation

In this section we perform the symmetry analysis of (4) Tothis end we use the Lie symmetrymethod [10] which is basedupon finding Lie point symmetries of the PDEs that leavethem invariant In order to find the Lie symmetry generatorsof (4) and obtain closed-form solutions for all 119870(119906) we

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 726837 7 pageshttpdxdoiorg1011552014726837

2 Journal of Applied Mathematics

consider the one-parameter Lie point transformation thatleaves (4) invariant These transformation formulae [7 9] aregiven as follows

119909119894= 119909119894+ 120598120585119894(119909 119910 119905 119906) + 119874 (120598

2) 119894 = 1 4 (5)

where 120585119894 = (120597119909119894120597120598)|120598=0

defines the symmetry generator [9]associated with (2) given by

119881 = 120585

120597

120597119909

+ 120578

120597

120597119910

+ 120591

120597

120597119905

+ 120601

120597

120597119906

(6)

Requiring invariance of (4) with respect to the prolongedsymmetry generator

V(2) = 119881 +2

sum

119868=0

120601119869 120597

120597119906119869

+

3

sum

119868119869=0

120601119868119869 120597

120597119906119868119869

119868 119869 = 0 1 (7)

with 0 representing 119905 and 1 representing 119909

120585 (119863 (119906) 119906119909119909+ 1198631199061199062

119909minus 119896 (119906) 119906 minus 119906

119905)

+ 120601 (119909119863119906119906119909119909+ 1199091198631199061199061199062

119909+ 2119863119906119906119909minus 119909119896 (119906) minus 119909119906119896

119906)

+ 120601119909(2119909119863119906119906119909+ 2119863 (119906)) minus 119909120601

119905+ 119909119863 (119906) 120601

119909119909= 0

(8)

and the coefficients 120601119869 and120601119869119870 of the derivativeswith respectto dependent variables in (8) are to be evaluated using theexpressions

120601119869= 119863119894(120601 minus 120585

119895119906119895119894) + 120585119895119906119895119894

120601119869119870= 119863119894119863119895(120601 minus 120585

119895119906119895119894) + 120585119896119906119896119894119895

(9)

Using (9) into (8) and comparing terms involving derivativesof the dependent variable 119906 we obtain the following systemof differential equations

120585119906= 0 = 120591

119909= 120591119906= 120601119906119906 (10)

119863119906120601 minus 2119863120585

119909+ 119863120591119905= 0 (11)

minus 119909119870120601 minus 119909119906119870119906120601 + 2119863120601

119909minus 119909120601119905+ 119909119906119870120601

119906

minus 119909119906119870120591119905+ 119909119863120601

119909119909= 0

(12)

minus 2119863120585 + 2119909119863119906120601 minus 2119909119863120585

119909+ 2119909119863120591

119905+ 1199092120585119905

+21198631199092120601119909119906minus 1198631199092120585119909119909= 0

(13)

To determine the unknown functions 120585 120591 and 120601 we solvethe above coupled system of differential equations by firstconsidering (11) Differentiating this equation twice withrespect to 119906 leads to the following expression

120601119906119906= (

119863

119863119906

)

119906119906

(2120585119909minus 120591119905) (14)

Using (10) into (14) reduces to

(

119863

119863119906

)

119906119906

(2120585119909minus 120591119905) = 0 (15)

We proceed from the above equation to obtain a completeclassification of both119863 and119870 as shown in the next section

3 Classification

In order to perform a complete classification of solution of(4) we notice that the following three cases arise from (15)

(I) 2120585119909minus 120591119905= 0

(II) (119863119863119906)119906119906= 0

(III) 2120585119909minus 120591119905= 0 = (119863119863

119906)119906119906

For obtaining a complete classification we consider all thethree cases one by one Since procedure of classification inall the three cases is similar we present a complete analysisin the first case and briefly state the results in the remainingcases To begin the classification we proceed as follows

31 Case I (2120585119909minus120591119905=0) In this case the system of determining

equations given by (10)ndash(13) becomes

120585119906= 0 = 120591

119909= 120591119906= 120601 (16)

2120585119909= 120591119905 (17)

minus119909119906119870120591119905= 0 (18)

minus2119863120585 + 119909119863120591119905+ 1199092120585119905= 0 (19)

From (18) three subcases arise

(a) 120591119905= 0

(b) 119870(119906) = 0(c) 120591119905= 0 = 119870(119906)

We first consider (a)

311 Subcase (a) (120591119905=0) Using these conditions arising in

this case we deduce that

120585119909= 0 997904rArr 120585 = 120585 (119905) (20)

and (19) becomes

minus2119863120585 + 1199092120585119905= 0 (21)

Note that (21) is a separable DE and can be easily solved tofind 120585 given by 120585 = 119888

2exp(minus2119863119909minus2119905)

But 120585 depend only on 119905 so that two possibilities arise fromexpression of 120585

(a1) 1198882= 0 and119863 = 0

(a2) 119863 = 0 and 1198882= 0

Subsubcase (a1)This subcase leads to the fact that 119863(119906) and119870(119906) are arbitrary and the infinitesimals are

120585 = 0 120591 = 1198881 120601 = 0 (22)

Only one symmetry generator is associated with above infin-itesimals which is X = (120597120597119905)

Subsubcase (a2) As a result of this subcase 119863(119906) = 0 and119870(119906) are arbitrary and the infinitesimals are

120585 = 1198882 120591 = 119888

1 120601 = 0 (23)

Journal of Applied Mathematics 3

Table 1 Commutator table of the tumor equation

[XiXj] X1 X2

X1 0 minus2X2

X2 2X2 0

In this subcase we have two generators which are

X1 =120597

120597119909

X2 =120597

120597119905

(24)

312 Subcase (b) (119870(119906)=0) Using these conditions arising inthis case into (13) it becomes

minus2119863120585 + 2119909119863120585119909+ 1199092120585119905+ 3119863119909

2120585119909119909= 0 (25)

Differentiating (25) with respect to 119906 we get

31199092120585119909119909+ 2119909120585

119909minus 2120585 = 0 (26)

The above equation is a Cauchy Euler differential equationits solution given by

120585 = 119860 (119905) 119909 + 119861 (119905) 119909minus(23)

(27)

To require consistency of 120585 found above we use (27) into(25) This suggests that (25) is satisfied when the followingdifferential constraint is met

1198601199051199093+ 11986111990511990943

= 0 (28)

Differentiating (28) four times with respect to 119909 we get119861(119905) =1198882and hence 119860(119905) = 119888

1 Therefore

120585 = 1198881119909 + 1198882119909minus(23)

(29)

Using (29) into (17) we conclude that

120591 = 21198881119905 minus

4

3

1198882119909minus(53)

+ 1198883 (30)

From (16) and (30) we infer that 1198882= 0 and consequently the

general expressions of 120585 120591 120601119863 and119870 are

120585 = 1198881119909 120591 = 2119888

1119905 + 1198883 120601 = 0

119870 (119906) = 0 119863 (119906) is arbitrary(31)

The two symmetry generators associatedwith above infinites-imals are given by

1198831= 119909

120597

120597119909

+ 2119905

120597

120597119905

1198832=

120597

120597119905

(32)

The commutation relation for each of the above symmetrygenerators is listed in Table 1

313 Subcase (c) (120591119905=0=119870(119906)) Following the procedure

adopted in earlier cases we obtain the same symmetry gen-erators in cases (a) and (b)

32 Case II ((119863119863119906)119906119906=0) Solving equation (119863119863

119906)119906119906= 0

for119863(119906) instantly yields

119863 (119906) = (119887119906)1119887 (33)

where 119887 is a constant Using (33) into (11) we obtain

120601 = (119887119906) (2120585119909minus 120591119905) (34)

Using (34) and (33) into (13) we obtain a differential relationin the 120585 given by

minus 2(119887119906)1119887120585 + 2119909(119887119906)

1119887120585119909+ 1199092120585119905

+ 1199092(4119887 minus 1) (119887119906)

1119887120585119909119909= 0

(35)

Differentiating (35) with respect to 119906 we obtain

minus2120585 + 2119909120585119909+ 1199092(4119887 minus 1) 120585

119909119909= 0 (36)

From (36) two cases arise

(a) 119887 = 14(b) 119887 = 14

321 Subcase (a) (119887 = 14) This subcase gives that

120585 = 119860 (119905) 119909 + 119861 (119905) 119909minus2119898

1 where 119898

1= 4119887 minus 1 (37)

Using (37) into (35) yields

1198601199051199093+ 119861119905119909minus(2119898

1)+2

= 0 (38)

Differentiating (38) with respect to 119909 4 times we concludethat 119861(119905) = 119888

2and hence 119860(119905) = 119888

1 Therefore

120585 = 1198881119909 + 1198882119909minus2119898

1 where 119898

1= 4119887 minus 1 (39)

As a result of (39) we obtain

120601 = 119887119906(21198881minus

4

1198981

1198882119909minus(2119898

1)minus120591119905) (40)

Using (39) and (40) into (12) we obtain

minus 1198871199062119870119906(21198881minus

4

1198981

1198882119909minus(2119898

1)minus1

minus 120591119905) + 119887119906120591

119905119905minus 119906119870120591

119905

minus

8

1198981

1198882(119887119906)(1119887)+1

(minus

2

1198981

minus 1)119909minus(2119898

1)minus3

minus

4

1198981

1198882(119887119906)(1119887)+1

(minus

2

1198981minus 1

)(minus

2

1198981minus 2

)119909minus(2119898

1)minus3

= 0

(41)

Differentiating (41) with respect to 119909 yields

minus 1198871199062119870119906

4

1198981

(

minus2

1198981

minus 1) 1198882119909(minus2119898

1)minus2

minus

8

1198981

1198882(119887119906)(1119887)+1

times (minus

2

1198981

minus 1)(minus

2

1198981

minus 3)119909minus(2119898

1)minus4

minus

4

1198981

1198882(119887119906)(1119887)+1

times (minus

2

1198981

minus 1)(minus

2

1198981

minus 1)(minus

2

1198981

minus 3)119909minus(2119898

1)minus4

= 0

(42)

4 Journal of Applied Mathematics

The above equation holds if 1198882= 0 and substituting this value

into (41) gives

minus119887119906119870119906(21198881minus 120591119905) + 119887120591

119905119905minus 119870120591119905= 0 (43)

Differentiating (43) with respect to 119905

minus119887119906119870119906minus 119870 =

minus119887120591119905119905119905

120591119905119905

(44)

The equality in (44) holds if

minus119887119906119870119906minus 119870 =

minus119887120591119905119905119905

120591119905119905

= 120574 (45)

where 120574 is 119886 constantIn accordance with (45) we conclude that

120591 (119905) =

11988831198872

1205742

exp(120574

119887

119905) + 1198884119905 + 1198885

119870 (119906) = 120574 + 1205821199061119887

(46)

Using (46) into (43) yields

minus211988811205821199061119887minus 1205741198884= 0 (47)

Differentiating (47) with respect to 119906 leads to

minus21198881

1

119887

120582119906(1119887)minus1

= 0 (48)

From (48) three cases arise namely

(a1) 1198881= 0 and 120582 = 0

(a2) 1198881= 0 and 120582 = 0

(a3) 1198881= 0 and 120582 = 0

Subsubcase (a1) (119863(119906) = (119887119906)1119887 and119870(119906) = 120574+1205821199061119887)Usingthese conditions arising in this case into (47) we obtain 119888

4= 0

and hence the infinitesimals are determined

120585 = 0 120591 =

1198883119887

120574

exp(120574

119887

119905) + 1198885

120601 = minus1198883119887119906 exp(

120574

119887

119905)

(49)

The two symmetry generators associated with the above in-finitesimals are given by

X1 =119887

120574

exp (120574

119887

119905)

120597

120597119905

minus 119887119906 exp(120574

119887

119905)

120597

120597119906

X2 =120597

120597119905

(50)

The commutation relation for these generators is given inTable 2

Table 2 Commutator table of the tumor equation

[XiXj] X1 X2

X1 0 minus

120574

119887

X1

X2120574

119887

X1 0

Table 3 Commutator table of the tumor equation

[XiXj] X1 X2 X3

X1 0 0 minus

120574

119887

X1

1198832

0 0 0

1198833

120574

119887

X1 0 0

Subsubcase (a2) (119863(119906) = (119887119906)1119887 and 119870(119906) = 120574) Similarly

using these conditions arising in this case into (47) we obtain1198884= 0 and hence the infinitesimals are determined

120585 = 1198881119909 120591 =

1198883119887

120574

exp (120574

119887

119905) + 1198885

120601 = 119887119906 (21198881minus 1198883exp(

120574

119887

119905))

(51)

The three symmetry generators associated with the aboveinfinitesimals are given by

X1 =119887

120574

exp(120574

119887

119905)

120597

120597119905

minus 119887119906 exp (120574

119887

119905)

120597

120597119906

X2 = 119909120597

120597119909

+ 2119887119906

120597

120597119906

X3 =120597

120597119905

(52)

The commutation relation for these generators is given inTable 3

Subsubcase (a3) (119863(119906) = (119887119906)1119887 and 119870(119906) = 120574) using theseconditions arising in this case into (47) we obtain 119888

4= 0 and

hence the infinitesimals are

120585 = 0 120591 =

1198883119887

120574

exp(120574

119887

119905) + 1198885

120601 = minus1198883119887119906 exp(

120574

119887

119905)

(53)

The two symmetry generators associated with the above in-finitesimals are given by

X1 =119887

120574

exp(120574

119887

119905)

120597

120597119905

minus 119887119906 exp (120574

119887

119905)

120597

120597119906

X2 =120597

120597119905

(54)

The commutation relation for these generators is given inTable 4

322 Subcase (b) (119887=14) In accordance with this subcase(35) becomes

2119909120585119909minus 2120585 = 0 then 120585 = 119909120573 (119905) (55)

Journal of Applied Mathematics 5

Table 4 Commutator table of the tumor equation

[XiXj] X1 X2

X1 0 minus

120574

119887

X1

X2120574

119887

X1 0

Using (33) into (35) gives 120573 = 1198881and hence

120585 = 1198881119909 120601 =

1

4

119906 (21198881minus 120591119905) (56)

Using (56) into (12) yields

minus

1

4

119906119870119906(21198881minus 120591119905) +

1

4

120591119905119905minus 119870120591119905= 0 (57)

Differentiating (57) with respect to 119905

minus

1

4

119906119870119906minus 119870 =

minus (14) 120591119905119905119905

120591119905119905

(58)

The equality in (58) holds if

minus119887119906119870119906minus 119870 =

minus (14) 120591119905119905119905

120591119905119905

= 120574 where 120574 is a constant

(59)

In accordance with (59) we have

120591 (119905) =

1198883

161205742exp (4120574119905) + 119888

4119905 + 1198885 119870 (119906) = 120574 + 120582119906

4 (60)

Using (60) into (57) yields

minus211988811205821199064minus 1205741198884= 0 (61)

Differentiating (61) with respect to 119906 leads to

minus811988811205821199063= 0 (62)

From (62) three cases arise

(b1) 1198881= 0 and 120582 = 0

(b2) 1198881= 0 and 120582 = 0

(b3) 1198881= 0 and 120582 = 0

Subsubcase (b1) (119863(119906) = ((14)119906)4 and 119870(119906) = 120574 + 1205821199064) Inaccordance with this subsubcase and (61) we obtain 119888

4= 0

and hence the infinitesimals are determined as

120585 = 0 120591 =

1198883

4120574

exp (4120574119905) + 1198885

120601 = minus

1

4

1198883119906 exp (4120574119905)

(63)

The generators associated with this subsubcase are

X1=

1

4120574

exp (4120574119905) 120597120597119905

minus

1

4

119906 exp (4120574119905) 120597120597119906

X2 =120597

120597119905

(64)

Table 5 Commutator table of the tumor equation

[XiXj] X1 X2

X1 0 minus4120574X1

X2 4120574X1 0

Table 6 Commutator table of the tumor equation

[XiXj] X1 X2 X3

X1 0 0 minus4120574X1

1198832

0 0 0

1198833

4120574X1 0 0

The commutation relation for these generators is given inTable 5

Subsubcase (b2) (119863(119906) = ((14)119906)4 and 119870(119906) = 120574) This

subsubcase with (61) yields to 1198884= 0 and hence the infin-

itesimals are

120585 = 1198881119909 120591 =

1198883

4120574

exp (4120574119905) + 1198885

120601 =

1

4

119906 (21198881minus 1198883exp (4120574119905))

(65)

The generators associated with this subsubcase are

X1 =1

4120574

exp (4120574119905) 120597120597119905

minus

1

4

119906 exp (4120574119905) 120597120597119906

X2 = 119909120597

120597119909

+

1

2

119906

120597

120597119906

X3 =120597

120597119905

(66)

The commutation relation for these generators is given inTable 6

Subsubcase (b3) (119863(119906) = ((14)119906)4 and 119870(119906) = 120574) In ac-

cordance with this subsubcase and (61) we obtain 1198884= 0 and

hence the infinitesimals are

120585 = 0 120591 =

1198883

4120574

exp (4120574119905) + 1198885

120601 = minus

1

4

1198883119906 exp (4120574119905)

(67)

The generators corresponding to this subsubcase are

X1 =1

4120574

exp (4120574119905) 120597120597119905

minus

1

4

119906 exp (4120574119905) 120597120597119906

X2 =120597

120597119905

(68)

Remark 1 Case (III) represents the particular case of (I)and (II) So by similar manipulation as in the previous caseswe obtain the same symmetry generators

4 Some Reduction

In this section we present solutions of (4) via reductionsThese reductions are obtained by the similarity variablesobtained through symmetry generators

6 Journal of Applied Mathematics

Figure 1 Plot of solution (74) with 1198881= 1 1198882= 0

Case 1 (119870(119906) and 119863(119906) are arbitrary) In this case we haveonly one generator that is X = 120597120597119905 Thus the characteristicequation corresponding to this generator is

119889119909

0

=

119889119905

1

=

119889119906

0

(69)

Solving the above equation it is straightforward [11] to findthat it yields the similarity variables 119903 = 119909 and 119908 = 119906Replacing 119906 in (4) in terms of new variables it becomes

119903119863 (119908)119908119903119903+ 1199031198631199081199082

119903+ 2119863 (119908)119908

119903minus 119903119870 (119908)119908 = 0 (70)

Case 2 (119870(119906) = 0 and 119863(119906) are arbitrary) We consider thegenerator X

1= 119909(120597120597119909) + 2119905(120597120597119905) Thus the characteristic

equation associated with this generator is119889119909

119909

=

119889119905

2119905

=

119889119906

0

(71)

The similarity variables corresponding to the above equationbecome 119903 = 119909119905

minus(12) and 119908 = 119906 These new variables reduce(4) to a ODE of the form

1199032119863(119908)119908

119903119903+ 11990321198631199081199082

119903+ 2119903119863 (119908)119908

119903+

1

2

1199033119908119903= 0 (72)

Choosing 119863(119908) = 1 in the above equation then the solutionof the resulting equation is

119908 (119903) = 1198882+ 1198881(minus

1

119903

exp(minus1199032

4

) minus

1

2

radicΠErf( 1199032

)) (73)

Recasting the above equation in its original coordinates theexact solution of (4) becomes

119906 (119909 119905) = 1198882+ 1198881(minus

radic119905

119909

exp(minus1199092

4119905

) minus

1

2

radicΠErf( 119909

2radic119905

))

(74)

The graph of this solution is plotted in Figure 1

Case 3 (119870(119906) = 120574 + 1205821199061119887 and 119863(119906) = 119886(119887119906)1119887) We take thegenerator

X1 =119887

120574

exp(120574

119887

119905)

120597

120597119905

minus 119887119906 exp(120574

119887

119905)

120597

120597119906

(75)

Figure 2 Plot of solution (79) with 1198881= 0 120574 = 119888

2= 1

and its characteristic equation

119889119909

0

=

120574

119887

exp (minus120574

119887

119905) 119889119905 = exp(minus120574

119887

119905)

119889119906

minus119887119906

(76)

gives the similarity variables 119903 = 119909 and 119908 = exp(120574119905)119906 Inaccordance of these similarities (4) is transformed to theODE in the form

119903119908119903119903+

119903

119887

1

119908

1199082

119903+ 2119908119903minus 119903120582119887minus(1119887)

119908 = 0 (77)

Choosing 119887 = 120582 = 120574 = 1 in the above equation then itssolution becomes

119908 (119903) =

1198882

radic119903

radiccosh (radic2119903 + 1198941198881) (78)

Recasting the above equation in its original coordinates theexact solution of (4) becomes

119906 (119909 119905) = exp (minus119905)1198882

radic119909

radiccosh (radic2119909 + 1198941198881) (79)

The graph of this solution is plotted in Figure 2

Case 4 (119870(119906) = 120574 and 119863(119906) = 119886(119887119906)1119887) In this case we

consider the generator

X1 =119887

120574

exp(120574

119887

119905)

120597

120597119905

minus 119887119906 exp (120574

119887

119905)

120597

120597119906

(80)

The characteristic equation corresponding to this generatoris

119889119909

0

=

120574

119887

exp(minus120574

119887

119905) 119889119905 = exp(minus120574

119887

119905)

119889119906

minus119887119906

(81)

solving the above characteristic equation gives 119903 = 119909 with119908 = exp(120574119905)119906 These variables can be used to recast (4) to anODE

119903119908119903119903+

119903

119887

1

119908

1199082

119903+ 2119908119903= 0 (82)

Journal of Applied Mathematics 7

Figure 3 Plot of solution (84) with 119887 = 14 120574 = 1198881= 1198882= 1

Choosing 119887 = (14) in (82) then its solution becomes

119908 (119903) = (

minus119903

5(1198881minus 1198882119903)

)

minus15

(83)

Recasting the above equation in its original coordinates theexact solution of (4) becomes

119906 (119909 119905) = exp (minus120574119905) ( minus119909

5(1198881minus 1198882119909)

)

minus15

(84)

The graph of this solution is plotted in Figure 3

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to acknowledge the support providedby the Deanship of Research at King Fahd University ofPetroleum and Minerals (KFUPM) for funding this workthrough Project IN111008

References

[1] D S Jones M Plank and B D Saleem Differential Equationsand Mathematical Biology Mathematical amp ComputationalBiology Chapman and HallCRC New York NY USA 2011

[2] G C Cruywagen D E Woodward P Tracqui G T BartooJ D Murray and C A Ellsworth ldquoThe modelling of diffusivetumoursrdquo Journal of Biological Systems vol 3 no 4 pp 937ndash9451995

[3] S Moyo and P G L Leach ldquoSymmetry methods applied to amathematical model of a tumour of the brainrdquo Proceedings ofInstitute of Mathematics of NAS of Ukraine vol 50 pp 204ndash2102004

[4] P Tracqui G C Cruywagen D E Woodward G T Bartoo JD Murray and E C Alvord ldquoA mathematical model of gliomagrowth the effect of chemotherapy on spatio-temporal growthrdquoCell Proliferation vol 28 no 1 pp 17ndash31 1995

[5] A H Bokhari A H Kara and F D Zaman ldquoOn the solutionsand conservation laws of the model for tumor growth in thebrainrdquo Journal of Mathematical Analysis and Applications vol350 no 1 pp 256ndash261 2009

[6] S LieTheorie der Transoformationsgruppen vol 3 BG TubnerEd Liepzig Germany 1893

[7] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2002

[8] G W Bluman A F Cheviakov and S C Anco Applications ofSymmetry Methods to Partial Differential Equations SpringerNew York NY USA 2010

[9] G W Bluman and S Kumei Symmetries and DifferentialEquations Springer New York NY USA 1989

[10] B J Cantwell Introduction to Symmetry Analysis CambridgeUniversity Press Cambridge UK 2002

[11] A Ahmad A H Bokhari A H Kara and F D ZamanldquoSymmetry classifications and reductions of some classes of (2+1)-nonlinear heat equationrdquo Journal of Mathematical Analysisand Applications vol 339 no 1 pp 175ndash181 2008

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Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

consider the one-parameter Lie point transformation thatleaves (4) invariant These transformation formulae [7 9] aregiven as follows

119909119894= 119909119894+ 120598120585119894(119909 119910 119905 119906) + 119874 (120598

2) 119894 = 1 4 (5)

where 120585119894 = (120597119909119894120597120598)|120598=0

defines the symmetry generator [9]associated with (2) given by

119881 = 120585

120597

120597119909

+ 120578

120597

120597119910

+ 120591

120597

120597119905

+ 120601

120597

120597119906

(6)

Requiring invariance of (4) with respect to the prolongedsymmetry generator

V(2) = 119881 +2

sum

119868=0

120601119869 120597

120597119906119869

+

3

sum

119868119869=0

120601119868119869 120597

120597119906119868119869

119868 119869 = 0 1 (7)

with 0 representing 119905 and 1 representing 119909

120585 (119863 (119906) 119906119909119909+ 1198631199061199062

119909minus 119896 (119906) 119906 minus 119906

119905)

+ 120601 (119909119863119906119906119909119909+ 1199091198631199061199061199062

119909+ 2119863119906119906119909minus 119909119896 (119906) minus 119909119906119896

119906)

+ 120601119909(2119909119863119906119906119909+ 2119863 (119906)) minus 119909120601

119905+ 119909119863 (119906) 120601

119909119909= 0

(8)

and the coefficients 120601119869 and120601119869119870 of the derivativeswith respectto dependent variables in (8) are to be evaluated using theexpressions

120601119869= 119863119894(120601 minus 120585

119895119906119895119894) + 120585119895119906119895119894

120601119869119870= 119863119894119863119895(120601 minus 120585

119895119906119895119894) + 120585119896119906119896119894119895

(9)

Using (9) into (8) and comparing terms involving derivativesof the dependent variable 119906 we obtain the following systemof differential equations

120585119906= 0 = 120591

119909= 120591119906= 120601119906119906 (10)

119863119906120601 minus 2119863120585

119909+ 119863120591119905= 0 (11)

minus 119909119870120601 minus 119909119906119870119906120601 + 2119863120601

119909minus 119909120601119905+ 119909119906119870120601

119906

minus 119909119906119870120591119905+ 119909119863120601

119909119909= 0

(12)

minus 2119863120585 + 2119909119863119906120601 minus 2119909119863120585

119909+ 2119909119863120591

119905+ 1199092120585119905

+21198631199092120601119909119906minus 1198631199092120585119909119909= 0

(13)

To determine the unknown functions 120585 120591 and 120601 we solvethe above coupled system of differential equations by firstconsidering (11) Differentiating this equation twice withrespect to 119906 leads to the following expression

120601119906119906= (

119863

119863119906

)

119906119906

(2120585119909minus 120591119905) (14)

Using (10) into (14) reduces to

(

119863

119863119906

)

119906119906

(2120585119909minus 120591119905) = 0 (15)

We proceed from the above equation to obtain a completeclassification of both119863 and119870 as shown in the next section

3 Classification

In order to perform a complete classification of solution of(4) we notice that the following three cases arise from (15)

(I) 2120585119909minus 120591119905= 0

(II) (119863119863119906)119906119906= 0

(III) 2120585119909minus 120591119905= 0 = (119863119863

119906)119906119906

For obtaining a complete classification we consider all thethree cases one by one Since procedure of classification inall the three cases is similar we present a complete analysisin the first case and briefly state the results in the remainingcases To begin the classification we proceed as follows

31 Case I (2120585119909minus120591119905=0) In this case the system of determining

equations given by (10)ndash(13) becomes

120585119906= 0 = 120591

119909= 120591119906= 120601 (16)

2120585119909= 120591119905 (17)

minus119909119906119870120591119905= 0 (18)

minus2119863120585 + 119909119863120591119905+ 1199092120585119905= 0 (19)

From (18) three subcases arise

(a) 120591119905= 0

(b) 119870(119906) = 0(c) 120591119905= 0 = 119870(119906)

We first consider (a)

311 Subcase (a) (120591119905=0) Using these conditions arising in

this case we deduce that

120585119909= 0 997904rArr 120585 = 120585 (119905) (20)

and (19) becomes

minus2119863120585 + 1199092120585119905= 0 (21)

Note that (21) is a separable DE and can be easily solved tofind 120585 given by 120585 = 119888

2exp(minus2119863119909minus2119905)

But 120585 depend only on 119905 so that two possibilities arise fromexpression of 120585

(a1) 1198882= 0 and119863 = 0

(a2) 119863 = 0 and 1198882= 0

Subsubcase (a1)This subcase leads to the fact that 119863(119906) and119870(119906) are arbitrary and the infinitesimals are

120585 = 0 120591 = 1198881 120601 = 0 (22)

Only one symmetry generator is associated with above infin-itesimals which is X = (120597120597119905)

Subsubcase (a2) As a result of this subcase 119863(119906) = 0 and119870(119906) are arbitrary and the infinitesimals are

120585 = 1198882 120591 = 119888

1 120601 = 0 (23)

Journal of Applied Mathematics 3

Table 1 Commutator table of the tumor equation

[XiXj] X1 X2

X1 0 minus2X2

X2 2X2 0

In this subcase we have two generators which are

X1 =120597

120597119909

X2 =120597

120597119905

(24)

312 Subcase (b) (119870(119906)=0) Using these conditions arising inthis case into (13) it becomes

minus2119863120585 + 2119909119863120585119909+ 1199092120585119905+ 3119863119909

2120585119909119909= 0 (25)

Differentiating (25) with respect to 119906 we get

31199092120585119909119909+ 2119909120585

119909minus 2120585 = 0 (26)

The above equation is a Cauchy Euler differential equationits solution given by

120585 = 119860 (119905) 119909 + 119861 (119905) 119909minus(23)

(27)

To require consistency of 120585 found above we use (27) into(25) This suggests that (25) is satisfied when the followingdifferential constraint is met

1198601199051199093+ 11986111990511990943

= 0 (28)

Differentiating (28) four times with respect to 119909 we get119861(119905) =1198882and hence 119860(119905) = 119888

1 Therefore

120585 = 1198881119909 + 1198882119909minus(23)

(29)

Using (29) into (17) we conclude that

120591 = 21198881119905 minus

4

3

1198882119909minus(53)

+ 1198883 (30)

From (16) and (30) we infer that 1198882= 0 and consequently the

general expressions of 120585 120591 120601119863 and119870 are

120585 = 1198881119909 120591 = 2119888

1119905 + 1198883 120601 = 0

119870 (119906) = 0 119863 (119906) is arbitrary(31)

The two symmetry generators associatedwith above infinites-imals are given by

1198831= 119909

120597

120597119909

+ 2119905

120597

120597119905

1198832=

120597

120597119905

(32)

The commutation relation for each of the above symmetrygenerators is listed in Table 1

313 Subcase (c) (120591119905=0=119870(119906)) Following the procedure

adopted in earlier cases we obtain the same symmetry gen-erators in cases (a) and (b)

32 Case II ((119863119863119906)119906119906=0) Solving equation (119863119863

119906)119906119906= 0

for119863(119906) instantly yields

119863 (119906) = (119887119906)1119887 (33)

where 119887 is a constant Using (33) into (11) we obtain

120601 = (119887119906) (2120585119909minus 120591119905) (34)

Using (34) and (33) into (13) we obtain a differential relationin the 120585 given by

minus 2(119887119906)1119887120585 + 2119909(119887119906)

1119887120585119909+ 1199092120585119905

+ 1199092(4119887 minus 1) (119887119906)

1119887120585119909119909= 0

(35)

Differentiating (35) with respect to 119906 we obtain

minus2120585 + 2119909120585119909+ 1199092(4119887 minus 1) 120585

119909119909= 0 (36)

From (36) two cases arise

(a) 119887 = 14(b) 119887 = 14

321 Subcase (a) (119887 = 14) This subcase gives that

120585 = 119860 (119905) 119909 + 119861 (119905) 119909minus2119898

1 where 119898

1= 4119887 minus 1 (37)

Using (37) into (35) yields

1198601199051199093+ 119861119905119909minus(2119898

1)+2

= 0 (38)

Differentiating (38) with respect to 119909 4 times we concludethat 119861(119905) = 119888

2and hence 119860(119905) = 119888

1 Therefore

120585 = 1198881119909 + 1198882119909minus2119898

1 where 119898

1= 4119887 minus 1 (39)

As a result of (39) we obtain

120601 = 119887119906(21198881minus

4

1198981

1198882119909minus(2119898

1)minus120591119905) (40)

Using (39) and (40) into (12) we obtain

minus 1198871199062119870119906(21198881minus

4

1198981

1198882119909minus(2119898

1)minus1

minus 120591119905) + 119887119906120591

119905119905minus 119906119870120591

119905

minus

8

1198981

1198882(119887119906)(1119887)+1

(minus

2

1198981

minus 1)119909minus(2119898

1)minus3

minus

4

1198981

1198882(119887119906)(1119887)+1

(minus

2

1198981minus 1

)(minus

2

1198981minus 2

)119909minus(2119898

1)minus3

= 0

(41)

Differentiating (41) with respect to 119909 yields

minus 1198871199062119870119906

4

1198981

(

minus2

1198981

minus 1) 1198882119909(minus2119898

1)minus2

minus

8

1198981

1198882(119887119906)(1119887)+1

times (minus

2

1198981

minus 1)(minus

2

1198981

minus 3)119909minus(2119898

1)minus4

minus

4

1198981

1198882(119887119906)(1119887)+1

times (minus

2

1198981

minus 1)(minus

2

1198981

minus 1)(minus

2

1198981

minus 3)119909minus(2119898

1)minus4

= 0

(42)

4 Journal of Applied Mathematics

The above equation holds if 1198882= 0 and substituting this value

into (41) gives

minus119887119906119870119906(21198881minus 120591119905) + 119887120591

119905119905minus 119870120591119905= 0 (43)

Differentiating (43) with respect to 119905

minus119887119906119870119906minus 119870 =

minus119887120591119905119905119905

120591119905119905

(44)

The equality in (44) holds if

minus119887119906119870119906minus 119870 =

minus119887120591119905119905119905

120591119905119905

= 120574 (45)

where 120574 is 119886 constantIn accordance with (45) we conclude that

120591 (119905) =

11988831198872

1205742

exp(120574

119887

119905) + 1198884119905 + 1198885

119870 (119906) = 120574 + 1205821199061119887

(46)

Using (46) into (43) yields

minus211988811205821199061119887minus 1205741198884= 0 (47)

Differentiating (47) with respect to 119906 leads to

minus21198881

1

119887

120582119906(1119887)minus1

= 0 (48)

From (48) three cases arise namely

(a1) 1198881= 0 and 120582 = 0

(a2) 1198881= 0 and 120582 = 0

(a3) 1198881= 0 and 120582 = 0

Subsubcase (a1) (119863(119906) = (119887119906)1119887 and119870(119906) = 120574+1205821199061119887)Usingthese conditions arising in this case into (47) we obtain 119888

4= 0

and hence the infinitesimals are determined

120585 = 0 120591 =

1198883119887

120574

exp(120574

119887

119905) + 1198885

120601 = minus1198883119887119906 exp(

120574

119887

119905)

(49)

The two symmetry generators associated with the above in-finitesimals are given by

X1 =119887

120574

exp (120574

119887

119905)

120597

120597119905

minus 119887119906 exp(120574

119887

119905)

120597

120597119906

X2 =120597

120597119905

(50)

The commutation relation for these generators is given inTable 2

Table 2 Commutator table of the tumor equation

[XiXj] X1 X2

X1 0 minus

120574

119887

X1

X2120574

119887

X1 0

Table 3 Commutator table of the tumor equation

[XiXj] X1 X2 X3

X1 0 0 minus

120574

119887

X1

1198832

0 0 0

1198833

120574

119887

X1 0 0

Subsubcase (a2) (119863(119906) = (119887119906)1119887 and 119870(119906) = 120574) Similarly

using these conditions arising in this case into (47) we obtain1198884= 0 and hence the infinitesimals are determined

120585 = 1198881119909 120591 =

1198883119887

120574

exp (120574

119887

119905) + 1198885

120601 = 119887119906 (21198881minus 1198883exp(

120574

119887

119905))

(51)

The three symmetry generators associated with the aboveinfinitesimals are given by

X1 =119887

120574

exp(120574

119887

119905)

120597

120597119905

minus 119887119906 exp (120574

119887

119905)

120597

120597119906

X2 = 119909120597

120597119909

+ 2119887119906

120597

120597119906

X3 =120597

120597119905

(52)

The commutation relation for these generators is given inTable 3

Subsubcase (a3) (119863(119906) = (119887119906)1119887 and 119870(119906) = 120574) using theseconditions arising in this case into (47) we obtain 119888

4= 0 and

hence the infinitesimals are

120585 = 0 120591 =

1198883119887

120574

exp(120574

119887

119905) + 1198885

120601 = minus1198883119887119906 exp(

120574

119887

119905)

(53)

The two symmetry generators associated with the above in-finitesimals are given by

X1 =119887

120574

exp(120574

119887

119905)

120597

120597119905

minus 119887119906 exp (120574

119887

119905)

120597

120597119906

X2 =120597

120597119905

(54)

The commutation relation for these generators is given inTable 4

322 Subcase (b) (119887=14) In accordance with this subcase(35) becomes

2119909120585119909minus 2120585 = 0 then 120585 = 119909120573 (119905) (55)

Journal of Applied Mathematics 5

Table 4 Commutator table of the tumor equation

[XiXj] X1 X2

X1 0 minus

120574

119887

X1

X2120574

119887

X1 0

Using (33) into (35) gives 120573 = 1198881and hence

120585 = 1198881119909 120601 =

1

4

119906 (21198881minus 120591119905) (56)

Using (56) into (12) yields

minus

1

4

119906119870119906(21198881minus 120591119905) +

1

4

120591119905119905minus 119870120591119905= 0 (57)

Differentiating (57) with respect to 119905

minus

1

4

119906119870119906minus 119870 =

minus (14) 120591119905119905119905

120591119905119905

(58)

The equality in (58) holds if

minus119887119906119870119906minus 119870 =

minus (14) 120591119905119905119905

120591119905119905

= 120574 where 120574 is a constant

(59)

In accordance with (59) we have

120591 (119905) =

1198883

161205742exp (4120574119905) + 119888

4119905 + 1198885 119870 (119906) = 120574 + 120582119906

4 (60)

Using (60) into (57) yields

minus211988811205821199064minus 1205741198884= 0 (61)

Differentiating (61) with respect to 119906 leads to

minus811988811205821199063= 0 (62)

From (62) three cases arise

(b1) 1198881= 0 and 120582 = 0

(b2) 1198881= 0 and 120582 = 0

(b3) 1198881= 0 and 120582 = 0

Subsubcase (b1) (119863(119906) = ((14)119906)4 and 119870(119906) = 120574 + 1205821199064) Inaccordance with this subsubcase and (61) we obtain 119888

4= 0

and hence the infinitesimals are determined as

120585 = 0 120591 =

1198883

4120574

exp (4120574119905) + 1198885

120601 = minus

1

4

1198883119906 exp (4120574119905)

(63)

The generators associated with this subsubcase are

X1=

1

4120574

exp (4120574119905) 120597120597119905

minus

1

4

119906 exp (4120574119905) 120597120597119906

X2 =120597

120597119905

(64)

Table 5 Commutator table of the tumor equation

[XiXj] X1 X2

X1 0 minus4120574X1

X2 4120574X1 0

Table 6 Commutator table of the tumor equation

[XiXj] X1 X2 X3

X1 0 0 minus4120574X1

1198832

0 0 0

1198833

4120574X1 0 0

The commutation relation for these generators is given inTable 5

Subsubcase (b2) (119863(119906) = ((14)119906)4 and 119870(119906) = 120574) This

subsubcase with (61) yields to 1198884= 0 and hence the infin-

itesimals are

120585 = 1198881119909 120591 =

1198883

4120574

exp (4120574119905) + 1198885

120601 =

1

4

119906 (21198881minus 1198883exp (4120574119905))

(65)

The generators associated with this subsubcase are

X1 =1

4120574

exp (4120574119905) 120597120597119905

minus

1

4

119906 exp (4120574119905) 120597120597119906

X2 = 119909120597

120597119909

+

1

2

119906

120597

120597119906

X3 =120597

120597119905

(66)

The commutation relation for these generators is given inTable 6

Subsubcase (b3) (119863(119906) = ((14)119906)4 and 119870(119906) = 120574) In ac-

cordance with this subsubcase and (61) we obtain 1198884= 0 and

hence the infinitesimals are

120585 = 0 120591 =

1198883

4120574

exp (4120574119905) + 1198885

120601 = minus

1

4

1198883119906 exp (4120574119905)

(67)

The generators corresponding to this subsubcase are

X1 =1

4120574

exp (4120574119905) 120597120597119905

minus

1

4

119906 exp (4120574119905) 120597120597119906

X2 =120597

120597119905

(68)

Remark 1 Case (III) represents the particular case of (I)and (II) So by similar manipulation as in the previous caseswe obtain the same symmetry generators

4 Some Reduction

In this section we present solutions of (4) via reductionsThese reductions are obtained by the similarity variablesobtained through symmetry generators

6 Journal of Applied Mathematics

Figure 1 Plot of solution (74) with 1198881= 1 1198882= 0

Case 1 (119870(119906) and 119863(119906) are arbitrary) In this case we haveonly one generator that is X = 120597120597119905 Thus the characteristicequation corresponding to this generator is

119889119909

0

=

119889119905

1

=

119889119906

0

(69)

Solving the above equation it is straightforward [11] to findthat it yields the similarity variables 119903 = 119909 and 119908 = 119906Replacing 119906 in (4) in terms of new variables it becomes

119903119863 (119908)119908119903119903+ 1199031198631199081199082

119903+ 2119863 (119908)119908

119903minus 119903119870 (119908)119908 = 0 (70)

Case 2 (119870(119906) = 0 and 119863(119906) are arbitrary) We consider thegenerator X

1= 119909(120597120597119909) + 2119905(120597120597119905) Thus the characteristic

equation associated with this generator is119889119909

119909

=

119889119905

2119905

=

119889119906

0

(71)

The similarity variables corresponding to the above equationbecome 119903 = 119909119905

minus(12) and 119908 = 119906 These new variables reduce(4) to a ODE of the form

1199032119863(119908)119908

119903119903+ 11990321198631199081199082

119903+ 2119903119863 (119908)119908

119903+

1

2

1199033119908119903= 0 (72)

Choosing 119863(119908) = 1 in the above equation then the solutionof the resulting equation is

119908 (119903) = 1198882+ 1198881(minus

1

119903

exp(minus1199032

4

) minus

1

2

radicΠErf( 1199032

)) (73)

Recasting the above equation in its original coordinates theexact solution of (4) becomes

119906 (119909 119905) = 1198882+ 1198881(minus

radic119905

119909

exp(minus1199092

4119905

) minus

1

2

radicΠErf( 119909

2radic119905

))

(74)

The graph of this solution is plotted in Figure 1

Case 3 (119870(119906) = 120574 + 1205821199061119887 and 119863(119906) = 119886(119887119906)1119887) We take thegenerator

X1 =119887

120574

exp(120574

119887

119905)

120597

120597119905

minus 119887119906 exp(120574

119887

119905)

120597

120597119906

(75)

Figure 2 Plot of solution (79) with 1198881= 0 120574 = 119888

2= 1

and its characteristic equation

119889119909

0

=

120574

119887

exp (minus120574

119887

119905) 119889119905 = exp(minus120574

119887

119905)

119889119906

minus119887119906

(76)

gives the similarity variables 119903 = 119909 and 119908 = exp(120574119905)119906 Inaccordance of these similarities (4) is transformed to theODE in the form

119903119908119903119903+

119903

119887

1

119908

1199082

119903+ 2119908119903minus 119903120582119887minus(1119887)

119908 = 0 (77)

Choosing 119887 = 120582 = 120574 = 1 in the above equation then itssolution becomes

119908 (119903) =

1198882

radic119903

radiccosh (radic2119903 + 1198941198881) (78)

Recasting the above equation in its original coordinates theexact solution of (4) becomes

119906 (119909 119905) = exp (minus119905)1198882

radic119909

radiccosh (radic2119909 + 1198941198881) (79)

The graph of this solution is plotted in Figure 2

Case 4 (119870(119906) = 120574 and 119863(119906) = 119886(119887119906)1119887) In this case we

consider the generator

X1 =119887

120574

exp(120574

119887

119905)

120597

120597119905

minus 119887119906 exp (120574

119887

119905)

120597

120597119906

(80)

The characteristic equation corresponding to this generatoris

119889119909

0

=

120574

119887

exp(minus120574

119887

119905) 119889119905 = exp(minus120574

119887

119905)

119889119906

minus119887119906

(81)

solving the above characteristic equation gives 119903 = 119909 with119908 = exp(120574119905)119906 These variables can be used to recast (4) to anODE

119903119908119903119903+

119903

119887

1

119908

1199082

119903+ 2119908119903= 0 (82)

Journal of Applied Mathematics 7

Figure 3 Plot of solution (84) with 119887 = 14 120574 = 1198881= 1198882= 1

Choosing 119887 = (14) in (82) then its solution becomes

119908 (119903) = (

minus119903

5(1198881minus 1198882119903)

)

minus15

(83)

Recasting the above equation in its original coordinates theexact solution of (4) becomes

119906 (119909 119905) = exp (minus120574119905) ( minus119909

5(1198881minus 1198882119909)

)

minus15

(84)

The graph of this solution is plotted in Figure 3

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to acknowledge the support providedby the Deanship of Research at King Fahd University ofPetroleum and Minerals (KFUPM) for funding this workthrough Project IN111008

References

[1] D S Jones M Plank and B D Saleem Differential Equationsand Mathematical Biology Mathematical amp ComputationalBiology Chapman and HallCRC New York NY USA 2011

[2] G C Cruywagen D E Woodward P Tracqui G T BartooJ D Murray and C A Ellsworth ldquoThe modelling of diffusivetumoursrdquo Journal of Biological Systems vol 3 no 4 pp 937ndash9451995

[3] S Moyo and P G L Leach ldquoSymmetry methods applied to amathematical model of a tumour of the brainrdquo Proceedings ofInstitute of Mathematics of NAS of Ukraine vol 50 pp 204ndash2102004

[4] P Tracqui G C Cruywagen D E Woodward G T Bartoo JD Murray and E C Alvord ldquoA mathematical model of gliomagrowth the effect of chemotherapy on spatio-temporal growthrdquoCell Proliferation vol 28 no 1 pp 17ndash31 1995

[5] A H Bokhari A H Kara and F D Zaman ldquoOn the solutionsand conservation laws of the model for tumor growth in thebrainrdquo Journal of Mathematical Analysis and Applications vol350 no 1 pp 256ndash261 2009

[6] S LieTheorie der Transoformationsgruppen vol 3 BG TubnerEd Liepzig Germany 1893

[7] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2002

[8] G W Bluman A F Cheviakov and S C Anco Applications ofSymmetry Methods to Partial Differential Equations SpringerNew York NY USA 2010

[9] G W Bluman and S Kumei Symmetries and DifferentialEquations Springer New York NY USA 1989

[10] B J Cantwell Introduction to Symmetry Analysis CambridgeUniversity Press Cambridge UK 2002

[11] A Ahmad A H Bokhari A H Kara and F D ZamanldquoSymmetry classifications and reductions of some classes of (2+1)-nonlinear heat equationrdquo Journal of Mathematical Analysisand Applications vol 339 no 1 pp 175ndash181 2008

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

Table 1 Commutator table of the tumor equation

[XiXj] X1 X2

X1 0 minus2X2

X2 2X2 0

In this subcase we have two generators which are

X1 =120597

120597119909

X2 =120597

120597119905

(24)

312 Subcase (b) (119870(119906)=0) Using these conditions arising inthis case into (13) it becomes

minus2119863120585 + 2119909119863120585119909+ 1199092120585119905+ 3119863119909

2120585119909119909= 0 (25)

Differentiating (25) with respect to 119906 we get

31199092120585119909119909+ 2119909120585

119909minus 2120585 = 0 (26)

The above equation is a Cauchy Euler differential equationits solution given by

120585 = 119860 (119905) 119909 + 119861 (119905) 119909minus(23)

(27)

To require consistency of 120585 found above we use (27) into(25) This suggests that (25) is satisfied when the followingdifferential constraint is met

1198601199051199093+ 11986111990511990943

= 0 (28)

Differentiating (28) four times with respect to 119909 we get119861(119905) =1198882and hence 119860(119905) = 119888

1 Therefore

120585 = 1198881119909 + 1198882119909minus(23)

(29)

Using (29) into (17) we conclude that

120591 = 21198881119905 minus

4

3

1198882119909minus(53)

+ 1198883 (30)

From (16) and (30) we infer that 1198882= 0 and consequently the

general expressions of 120585 120591 120601119863 and119870 are

120585 = 1198881119909 120591 = 2119888

1119905 + 1198883 120601 = 0

119870 (119906) = 0 119863 (119906) is arbitrary(31)

The two symmetry generators associatedwith above infinites-imals are given by

1198831= 119909

120597

120597119909

+ 2119905

120597

120597119905

1198832=

120597

120597119905

(32)

The commutation relation for each of the above symmetrygenerators is listed in Table 1

313 Subcase (c) (120591119905=0=119870(119906)) Following the procedure

adopted in earlier cases we obtain the same symmetry gen-erators in cases (a) and (b)

32 Case II ((119863119863119906)119906119906=0) Solving equation (119863119863

119906)119906119906= 0

for119863(119906) instantly yields

119863 (119906) = (119887119906)1119887 (33)

where 119887 is a constant Using (33) into (11) we obtain

120601 = (119887119906) (2120585119909minus 120591119905) (34)

Using (34) and (33) into (13) we obtain a differential relationin the 120585 given by

minus 2(119887119906)1119887120585 + 2119909(119887119906)

1119887120585119909+ 1199092120585119905

+ 1199092(4119887 minus 1) (119887119906)

1119887120585119909119909= 0

(35)

Differentiating (35) with respect to 119906 we obtain

minus2120585 + 2119909120585119909+ 1199092(4119887 minus 1) 120585

119909119909= 0 (36)

From (36) two cases arise

(a) 119887 = 14(b) 119887 = 14

321 Subcase (a) (119887 = 14) This subcase gives that

120585 = 119860 (119905) 119909 + 119861 (119905) 119909minus2119898

1 where 119898

1= 4119887 minus 1 (37)

Using (37) into (35) yields

1198601199051199093+ 119861119905119909minus(2119898

1)+2

= 0 (38)

Differentiating (38) with respect to 119909 4 times we concludethat 119861(119905) = 119888

2and hence 119860(119905) = 119888

1 Therefore

120585 = 1198881119909 + 1198882119909minus2119898

1 where 119898

1= 4119887 minus 1 (39)

As a result of (39) we obtain

120601 = 119887119906(21198881minus

4

1198981

1198882119909minus(2119898

1)minus120591119905) (40)

Using (39) and (40) into (12) we obtain

minus 1198871199062119870119906(21198881minus

4

1198981

1198882119909minus(2119898

1)minus1

minus 120591119905) + 119887119906120591

119905119905minus 119906119870120591

119905

minus

8

1198981

1198882(119887119906)(1119887)+1

(minus

2

1198981

minus 1)119909minus(2119898

1)minus3

minus

4

1198981

1198882(119887119906)(1119887)+1

(minus

2

1198981minus 1

)(minus

2

1198981minus 2

)119909minus(2119898

1)minus3

= 0

(41)

Differentiating (41) with respect to 119909 yields

minus 1198871199062119870119906

4

1198981

(

minus2

1198981

minus 1) 1198882119909(minus2119898

1)minus2

minus

8

1198981

1198882(119887119906)(1119887)+1

times (minus

2

1198981

minus 1)(minus

2

1198981

minus 3)119909minus(2119898

1)minus4

minus

4

1198981

1198882(119887119906)(1119887)+1

times (minus

2

1198981

minus 1)(minus

2

1198981

minus 1)(minus

2

1198981

minus 3)119909minus(2119898

1)minus4

= 0

(42)

4 Journal of Applied Mathematics

The above equation holds if 1198882= 0 and substituting this value

into (41) gives

minus119887119906119870119906(21198881minus 120591119905) + 119887120591

119905119905minus 119870120591119905= 0 (43)

Differentiating (43) with respect to 119905

minus119887119906119870119906minus 119870 =

minus119887120591119905119905119905

120591119905119905

(44)

The equality in (44) holds if

minus119887119906119870119906minus 119870 =

minus119887120591119905119905119905

120591119905119905

= 120574 (45)

where 120574 is 119886 constantIn accordance with (45) we conclude that

120591 (119905) =

11988831198872

1205742

exp(120574

119887

119905) + 1198884119905 + 1198885

119870 (119906) = 120574 + 1205821199061119887

(46)

Using (46) into (43) yields

minus211988811205821199061119887minus 1205741198884= 0 (47)

Differentiating (47) with respect to 119906 leads to

minus21198881

1

119887

120582119906(1119887)minus1

= 0 (48)

From (48) three cases arise namely

(a1) 1198881= 0 and 120582 = 0

(a2) 1198881= 0 and 120582 = 0

(a3) 1198881= 0 and 120582 = 0

Subsubcase (a1) (119863(119906) = (119887119906)1119887 and119870(119906) = 120574+1205821199061119887)Usingthese conditions arising in this case into (47) we obtain 119888

4= 0

and hence the infinitesimals are determined

120585 = 0 120591 =

1198883119887

120574

exp(120574

119887

119905) + 1198885

120601 = minus1198883119887119906 exp(

120574

119887

119905)

(49)

The two symmetry generators associated with the above in-finitesimals are given by

X1 =119887

120574

exp (120574

119887

119905)

120597

120597119905

minus 119887119906 exp(120574

119887

119905)

120597

120597119906

X2 =120597

120597119905

(50)

The commutation relation for these generators is given inTable 2

Table 2 Commutator table of the tumor equation

[XiXj] X1 X2

X1 0 minus

120574

119887

X1

X2120574

119887

X1 0

Table 3 Commutator table of the tumor equation

[XiXj] X1 X2 X3

X1 0 0 minus

120574

119887

X1

1198832

0 0 0

1198833

120574

119887

X1 0 0

Subsubcase (a2) (119863(119906) = (119887119906)1119887 and 119870(119906) = 120574) Similarly

using these conditions arising in this case into (47) we obtain1198884= 0 and hence the infinitesimals are determined

120585 = 1198881119909 120591 =

1198883119887

120574

exp (120574

119887

119905) + 1198885

120601 = 119887119906 (21198881minus 1198883exp(

120574

119887

119905))

(51)

The three symmetry generators associated with the aboveinfinitesimals are given by

X1 =119887

120574

exp(120574

119887

119905)

120597

120597119905

minus 119887119906 exp (120574

119887

119905)

120597

120597119906

X2 = 119909120597

120597119909

+ 2119887119906

120597

120597119906

X3 =120597

120597119905

(52)

The commutation relation for these generators is given inTable 3

Subsubcase (a3) (119863(119906) = (119887119906)1119887 and 119870(119906) = 120574) using theseconditions arising in this case into (47) we obtain 119888

4= 0 and

hence the infinitesimals are

120585 = 0 120591 =

1198883119887

120574

exp(120574

119887

119905) + 1198885

120601 = minus1198883119887119906 exp(

120574

119887

119905)

(53)

The two symmetry generators associated with the above in-finitesimals are given by

X1 =119887

120574

exp(120574

119887

119905)

120597

120597119905

minus 119887119906 exp (120574

119887

119905)

120597

120597119906

X2 =120597

120597119905

(54)

The commutation relation for these generators is given inTable 4

322 Subcase (b) (119887=14) In accordance with this subcase(35) becomes

2119909120585119909minus 2120585 = 0 then 120585 = 119909120573 (119905) (55)

Journal of Applied Mathematics 5

Table 4 Commutator table of the tumor equation

[XiXj] X1 X2

X1 0 minus

120574

119887

X1

X2120574

119887

X1 0

Using (33) into (35) gives 120573 = 1198881and hence

120585 = 1198881119909 120601 =

1

4

119906 (21198881minus 120591119905) (56)

Using (56) into (12) yields

minus

1

4

119906119870119906(21198881minus 120591119905) +

1

4

120591119905119905minus 119870120591119905= 0 (57)

Differentiating (57) with respect to 119905

minus

1

4

119906119870119906minus 119870 =

minus (14) 120591119905119905119905

120591119905119905

(58)

The equality in (58) holds if

minus119887119906119870119906minus 119870 =

minus (14) 120591119905119905119905

120591119905119905

= 120574 where 120574 is a constant

(59)

In accordance with (59) we have

120591 (119905) =

1198883

161205742exp (4120574119905) + 119888

4119905 + 1198885 119870 (119906) = 120574 + 120582119906

4 (60)

Using (60) into (57) yields

minus211988811205821199064minus 1205741198884= 0 (61)

Differentiating (61) with respect to 119906 leads to

minus811988811205821199063= 0 (62)

From (62) three cases arise

(b1) 1198881= 0 and 120582 = 0

(b2) 1198881= 0 and 120582 = 0

(b3) 1198881= 0 and 120582 = 0

Subsubcase (b1) (119863(119906) = ((14)119906)4 and 119870(119906) = 120574 + 1205821199064) Inaccordance with this subsubcase and (61) we obtain 119888

4= 0

and hence the infinitesimals are determined as

120585 = 0 120591 =

1198883

4120574

exp (4120574119905) + 1198885

120601 = minus

1

4

1198883119906 exp (4120574119905)

(63)

The generators associated with this subsubcase are

X1=

1

4120574

exp (4120574119905) 120597120597119905

minus

1

4

119906 exp (4120574119905) 120597120597119906

X2 =120597

120597119905

(64)

Table 5 Commutator table of the tumor equation

[XiXj] X1 X2

X1 0 minus4120574X1

X2 4120574X1 0

Table 6 Commutator table of the tumor equation

[XiXj] X1 X2 X3

X1 0 0 minus4120574X1

1198832

0 0 0

1198833

4120574X1 0 0

The commutation relation for these generators is given inTable 5

Subsubcase (b2) (119863(119906) = ((14)119906)4 and 119870(119906) = 120574) This

subsubcase with (61) yields to 1198884= 0 and hence the infin-

itesimals are

120585 = 1198881119909 120591 =

1198883

4120574

exp (4120574119905) + 1198885

120601 =

1

4

119906 (21198881minus 1198883exp (4120574119905))

(65)

The generators associated with this subsubcase are

X1 =1

4120574

exp (4120574119905) 120597120597119905

minus

1

4

119906 exp (4120574119905) 120597120597119906

X2 = 119909120597

120597119909

+

1

2

119906

120597

120597119906

X3 =120597

120597119905

(66)

The commutation relation for these generators is given inTable 6

Subsubcase (b3) (119863(119906) = ((14)119906)4 and 119870(119906) = 120574) In ac-

cordance with this subsubcase and (61) we obtain 1198884= 0 and

hence the infinitesimals are

120585 = 0 120591 =

1198883

4120574

exp (4120574119905) + 1198885

120601 = minus

1

4

1198883119906 exp (4120574119905)

(67)

The generators corresponding to this subsubcase are

X1 =1

4120574

exp (4120574119905) 120597120597119905

minus

1

4

119906 exp (4120574119905) 120597120597119906

X2 =120597

120597119905

(68)

Remark 1 Case (III) represents the particular case of (I)and (II) So by similar manipulation as in the previous caseswe obtain the same symmetry generators

4 Some Reduction

In this section we present solutions of (4) via reductionsThese reductions are obtained by the similarity variablesobtained through symmetry generators

6 Journal of Applied Mathematics

Figure 1 Plot of solution (74) with 1198881= 1 1198882= 0

Case 1 (119870(119906) and 119863(119906) are arbitrary) In this case we haveonly one generator that is X = 120597120597119905 Thus the characteristicequation corresponding to this generator is

119889119909

0

=

119889119905

1

=

119889119906

0

(69)

Solving the above equation it is straightforward [11] to findthat it yields the similarity variables 119903 = 119909 and 119908 = 119906Replacing 119906 in (4) in terms of new variables it becomes

119903119863 (119908)119908119903119903+ 1199031198631199081199082

119903+ 2119863 (119908)119908

119903minus 119903119870 (119908)119908 = 0 (70)

Case 2 (119870(119906) = 0 and 119863(119906) are arbitrary) We consider thegenerator X

1= 119909(120597120597119909) + 2119905(120597120597119905) Thus the characteristic

equation associated with this generator is119889119909

119909

=

119889119905

2119905

=

119889119906

0

(71)

The similarity variables corresponding to the above equationbecome 119903 = 119909119905

minus(12) and 119908 = 119906 These new variables reduce(4) to a ODE of the form

1199032119863(119908)119908

119903119903+ 11990321198631199081199082

119903+ 2119903119863 (119908)119908

119903+

1

2

1199033119908119903= 0 (72)

Choosing 119863(119908) = 1 in the above equation then the solutionof the resulting equation is

119908 (119903) = 1198882+ 1198881(minus

1

119903

exp(minus1199032

4

) minus

1

2

radicΠErf( 1199032

)) (73)

Recasting the above equation in its original coordinates theexact solution of (4) becomes

119906 (119909 119905) = 1198882+ 1198881(minus

radic119905

119909

exp(minus1199092

4119905

) minus

1

2

radicΠErf( 119909

2radic119905

))

(74)

The graph of this solution is plotted in Figure 1

Case 3 (119870(119906) = 120574 + 1205821199061119887 and 119863(119906) = 119886(119887119906)1119887) We take thegenerator

X1 =119887

120574

exp(120574

119887

119905)

120597

120597119905

minus 119887119906 exp(120574

119887

119905)

120597

120597119906

(75)

Figure 2 Plot of solution (79) with 1198881= 0 120574 = 119888

2= 1

and its characteristic equation

119889119909

0

=

120574

119887

exp (minus120574

119887

119905) 119889119905 = exp(minus120574

119887

119905)

119889119906

minus119887119906

(76)

gives the similarity variables 119903 = 119909 and 119908 = exp(120574119905)119906 Inaccordance of these similarities (4) is transformed to theODE in the form

119903119908119903119903+

119903

119887

1

119908

1199082

119903+ 2119908119903minus 119903120582119887minus(1119887)

119908 = 0 (77)

Choosing 119887 = 120582 = 120574 = 1 in the above equation then itssolution becomes

119908 (119903) =

1198882

radic119903

radiccosh (radic2119903 + 1198941198881) (78)

Recasting the above equation in its original coordinates theexact solution of (4) becomes

119906 (119909 119905) = exp (minus119905)1198882

radic119909

radiccosh (radic2119909 + 1198941198881) (79)

The graph of this solution is plotted in Figure 2

Case 4 (119870(119906) = 120574 and 119863(119906) = 119886(119887119906)1119887) In this case we

consider the generator

X1 =119887

120574

exp(120574

119887

119905)

120597

120597119905

minus 119887119906 exp (120574

119887

119905)

120597

120597119906

(80)

The characteristic equation corresponding to this generatoris

119889119909

0

=

120574

119887

exp(minus120574

119887

119905) 119889119905 = exp(minus120574

119887

119905)

119889119906

minus119887119906

(81)

solving the above characteristic equation gives 119903 = 119909 with119908 = exp(120574119905)119906 These variables can be used to recast (4) to anODE

119903119908119903119903+

119903

119887

1

119908

1199082

119903+ 2119908119903= 0 (82)

Journal of Applied Mathematics 7

Figure 3 Plot of solution (84) with 119887 = 14 120574 = 1198881= 1198882= 1

Choosing 119887 = (14) in (82) then its solution becomes

119908 (119903) = (

minus119903

5(1198881minus 1198882119903)

)

minus15

(83)

Recasting the above equation in its original coordinates theexact solution of (4) becomes

119906 (119909 119905) = exp (minus120574119905) ( minus119909

5(1198881minus 1198882119909)

)

minus15

(84)

The graph of this solution is plotted in Figure 3

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to acknowledge the support providedby the Deanship of Research at King Fahd University ofPetroleum and Minerals (KFUPM) for funding this workthrough Project IN111008

References

[1] D S Jones M Plank and B D Saleem Differential Equationsand Mathematical Biology Mathematical amp ComputationalBiology Chapman and HallCRC New York NY USA 2011

[2] G C Cruywagen D E Woodward P Tracqui G T BartooJ D Murray and C A Ellsworth ldquoThe modelling of diffusivetumoursrdquo Journal of Biological Systems vol 3 no 4 pp 937ndash9451995

[3] S Moyo and P G L Leach ldquoSymmetry methods applied to amathematical model of a tumour of the brainrdquo Proceedings ofInstitute of Mathematics of NAS of Ukraine vol 50 pp 204ndash2102004

[4] P Tracqui G C Cruywagen D E Woodward G T Bartoo JD Murray and E C Alvord ldquoA mathematical model of gliomagrowth the effect of chemotherapy on spatio-temporal growthrdquoCell Proliferation vol 28 no 1 pp 17ndash31 1995

[5] A H Bokhari A H Kara and F D Zaman ldquoOn the solutionsand conservation laws of the model for tumor growth in thebrainrdquo Journal of Mathematical Analysis and Applications vol350 no 1 pp 256ndash261 2009

[6] S LieTheorie der Transoformationsgruppen vol 3 BG TubnerEd Liepzig Germany 1893

[7] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2002

[8] G W Bluman A F Cheviakov and S C Anco Applications ofSymmetry Methods to Partial Differential Equations SpringerNew York NY USA 2010

[9] G W Bluman and S Kumei Symmetries and DifferentialEquations Springer New York NY USA 1989

[10] B J Cantwell Introduction to Symmetry Analysis CambridgeUniversity Press Cambridge UK 2002

[11] A Ahmad A H Bokhari A H Kara and F D ZamanldquoSymmetry classifications and reductions of some classes of (2+1)-nonlinear heat equationrdquo Journal of Mathematical Analysisand Applications vol 339 no 1 pp 175ndash181 2008

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

4 Journal of Applied Mathematics

The above equation holds if 1198882= 0 and substituting this value

into (41) gives

minus119887119906119870119906(21198881minus 120591119905) + 119887120591

119905119905minus 119870120591119905= 0 (43)

Differentiating (43) with respect to 119905

minus119887119906119870119906minus 119870 =

minus119887120591119905119905119905

120591119905119905

(44)

The equality in (44) holds if

minus119887119906119870119906minus 119870 =

minus119887120591119905119905119905

120591119905119905

= 120574 (45)

where 120574 is 119886 constantIn accordance with (45) we conclude that

120591 (119905) =

11988831198872

1205742

exp(120574

119887

119905) + 1198884119905 + 1198885

119870 (119906) = 120574 + 1205821199061119887

(46)

Using (46) into (43) yields

minus211988811205821199061119887minus 1205741198884= 0 (47)

Differentiating (47) with respect to 119906 leads to

minus21198881

1

119887

120582119906(1119887)minus1

= 0 (48)

From (48) three cases arise namely

(a1) 1198881= 0 and 120582 = 0

(a2) 1198881= 0 and 120582 = 0

(a3) 1198881= 0 and 120582 = 0

Subsubcase (a1) (119863(119906) = (119887119906)1119887 and119870(119906) = 120574+1205821199061119887)Usingthese conditions arising in this case into (47) we obtain 119888

4= 0

and hence the infinitesimals are determined

120585 = 0 120591 =

1198883119887

120574

exp(120574

119887

119905) + 1198885

120601 = minus1198883119887119906 exp(

120574

119887

119905)

(49)

The two symmetry generators associated with the above in-finitesimals are given by

X1 =119887

120574

exp (120574

119887

119905)

120597

120597119905

minus 119887119906 exp(120574

119887

119905)

120597

120597119906

X2 =120597

120597119905

(50)

The commutation relation for these generators is given inTable 2

Table 2 Commutator table of the tumor equation

[XiXj] X1 X2

X1 0 minus

120574

119887

X1

X2120574

119887

X1 0

Table 3 Commutator table of the tumor equation

[XiXj] X1 X2 X3

X1 0 0 minus

120574

119887

X1

1198832

0 0 0

1198833

120574

119887

X1 0 0

Subsubcase (a2) (119863(119906) = (119887119906)1119887 and 119870(119906) = 120574) Similarly

using these conditions arising in this case into (47) we obtain1198884= 0 and hence the infinitesimals are determined

120585 = 1198881119909 120591 =

1198883119887

120574

exp (120574

119887

119905) + 1198885

120601 = 119887119906 (21198881minus 1198883exp(

120574

119887

119905))

(51)

The three symmetry generators associated with the aboveinfinitesimals are given by

X1 =119887

120574

exp(120574

119887

119905)

120597

120597119905

minus 119887119906 exp (120574

119887

119905)

120597

120597119906

X2 = 119909120597

120597119909

+ 2119887119906

120597

120597119906

X3 =120597

120597119905

(52)

The commutation relation for these generators is given inTable 3

Subsubcase (a3) (119863(119906) = (119887119906)1119887 and 119870(119906) = 120574) using theseconditions arising in this case into (47) we obtain 119888

4= 0 and

hence the infinitesimals are

120585 = 0 120591 =

1198883119887

120574

exp(120574

119887

119905) + 1198885

120601 = minus1198883119887119906 exp(

120574

119887

119905)

(53)

The two symmetry generators associated with the above in-finitesimals are given by

X1 =119887

120574

exp(120574

119887

119905)

120597

120597119905

minus 119887119906 exp (120574

119887

119905)

120597

120597119906

X2 =120597

120597119905

(54)

The commutation relation for these generators is given inTable 4

322 Subcase (b) (119887=14) In accordance with this subcase(35) becomes

2119909120585119909minus 2120585 = 0 then 120585 = 119909120573 (119905) (55)

Journal of Applied Mathematics 5

Table 4 Commutator table of the tumor equation

[XiXj] X1 X2

X1 0 minus

120574

119887

X1

X2120574

119887

X1 0

Using (33) into (35) gives 120573 = 1198881and hence

120585 = 1198881119909 120601 =

1

4

119906 (21198881minus 120591119905) (56)

Using (56) into (12) yields

minus

1

4

119906119870119906(21198881minus 120591119905) +

1

4

120591119905119905minus 119870120591119905= 0 (57)

Differentiating (57) with respect to 119905

minus

1

4

119906119870119906minus 119870 =

minus (14) 120591119905119905119905

120591119905119905

(58)

The equality in (58) holds if

minus119887119906119870119906minus 119870 =

minus (14) 120591119905119905119905

120591119905119905

= 120574 where 120574 is a constant

(59)

In accordance with (59) we have

120591 (119905) =

1198883

161205742exp (4120574119905) + 119888

4119905 + 1198885 119870 (119906) = 120574 + 120582119906

4 (60)

Using (60) into (57) yields

minus211988811205821199064minus 1205741198884= 0 (61)

Differentiating (61) with respect to 119906 leads to

minus811988811205821199063= 0 (62)

From (62) three cases arise

(b1) 1198881= 0 and 120582 = 0

(b2) 1198881= 0 and 120582 = 0

(b3) 1198881= 0 and 120582 = 0

Subsubcase (b1) (119863(119906) = ((14)119906)4 and 119870(119906) = 120574 + 1205821199064) Inaccordance with this subsubcase and (61) we obtain 119888

4= 0

and hence the infinitesimals are determined as

120585 = 0 120591 =

1198883

4120574

exp (4120574119905) + 1198885

120601 = minus

1

4

1198883119906 exp (4120574119905)

(63)

The generators associated with this subsubcase are

X1=

1

4120574

exp (4120574119905) 120597120597119905

minus

1

4

119906 exp (4120574119905) 120597120597119906

X2 =120597

120597119905

(64)

Table 5 Commutator table of the tumor equation

[XiXj] X1 X2

X1 0 minus4120574X1

X2 4120574X1 0

Table 6 Commutator table of the tumor equation

[XiXj] X1 X2 X3

X1 0 0 minus4120574X1

1198832

0 0 0

1198833

4120574X1 0 0

The commutation relation for these generators is given inTable 5

Subsubcase (b2) (119863(119906) = ((14)119906)4 and 119870(119906) = 120574) This

subsubcase with (61) yields to 1198884= 0 and hence the infin-

itesimals are

120585 = 1198881119909 120591 =

1198883

4120574

exp (4120574119905) + 1198885

120601 =

1

4

119906 (21198881minus 1198883exp (4120574119905))

(65)

The generators associated with this subsubcase are

X1 =1

4120574

exp (4120574119905) 120597120597119905

minus

1

4

119906 exp (4120574119905) 120597120597119906

X2 = 119909120597

120597119909

+

1

2

119906

120597

120597119906

X3 =120597

120597119905

(66)

The commutation relation for these generators is given inTable 6

Subsubcase (b3) (119863(119906) = ((14)119906)4 and 119870(119906) = 120574) In ac-

cordance with this subsubcase and (61) we obtain 1198884= 0 and

hence the infinitesimals are

120585 = 0 120591 =

1198883

4120574

exp (4120574119905) + 1198885

120601 = minus

1

4

1198883119906 exp (4120574119905)

(67)

The generators corresponding to this subsubcase are

X1 =1

4120574

exp (4120574119905) 120597120597119905

minus

1

4

119906 exp (4120574119905) 120597120597119906

X2 =120597

120597119905

(68)

Remark 1 Case (III) represents the particular case of (I)and (II) So by similar manipulation as in the previous caseswe obtain the same symmetry generators

4 Some Reduction

In this section we present solutions of (4) via reductionsThese reductions are obtained by the similarity variablesobtained through symmetry generators

6 Journal of Applied Mathematics

Figure 1 Plot of solution (74) with 1198881= 1 1198882= 0

Case 1 (119870(119906) and 119863(119906) are arbitrary) In this case we haveonly one generator that is X = 120597120597119905 Thus the characteristicequation corresponding to this generator is

119889119909

0

=

119889119905

1

=

119889119906

0

(69)

Solving the above equation it is straightforward [11] to findthat it yields the similarity variables 119903 = 119909 and 119908 = 119906Replacing 119906 in (4) in terms of new variables it becomes

119903119863 (119908)119908119903119903+ 1199031198631199081199082

119903+ 2119863 (119908)119908

119903minus 119903119870 (119908)119908 = 0 (70)

Case 2 (119870(119906) = 0 and 119863(119906) are arbitrary) We consider thegenerator X

1= 119909(120597120597119909) + 2119905(120597120597119905) Thus the characteristic

equation associated with this generator is119889119909

119909

=

119889119905

2119905

=

119889119906

0

(71)

The similarity variables corresponding to the above equationbecome 119903 = 119909119905

minus(12) and 119908 = 119906 These new variables reduce(4) to a ODE of the form

1199032119863(119908)119908

119903119903+ 11990321198631199081199082

119903+ 2119903119863 (119908)119908

119903+

1

2

1199033119908119903= 0 (72)

Choosing 119863(119908) = 1 in the above equation then the solutionof the resulting equation is

119908 (119903) = 1198882+ 1198881(minus

1

119903

exp(minus1199032

4

) minus

1

2

radicΠErf( 1199032

)) (73)

Recasting the above equation in its original coordinates theexact solution of (4) becomes

119906 (119909 119905) = 1198882+ 1198881(minus

radic119905

119909

exp(minus1199092

4119905

) minus

1

2

radicΠErf( 119909

2radic119905

))

(74)

The graph of this solution is plotted in Figure 1

Case 3 (119870(119906) = 120574 + 1205821199061119887 and 119863(119906) = 119886(119887119906)1119887) We take thegenerator

X1 =119887

120574

exp(120574

119887

119905)

120597

120597119905

minus 119887119906 exp(120574

119887

119905)

120597

120597119906

(75)

Figure 2 Plot of solution (79) with 1198881= 0 120574 = 119888

2= 1

and its characteristic equation

119889119909

0

=

120574

119887

exp (minus120574

119887

119905) 119889119905 = exp(minus120574

119887

119905)

119889119906

minus119887119906

(76)

gives the similarity variables 119903 = 119909 and 119908 = exp(120574119905)119906 Inaccordance of these similarities (4) is transformed to theODE in the form

119903119908119903119903+

119903

119887

1

119908

1199082

119903+ 2119908119903minus 119903120582119887minus(1119887)

119908 = 0 (77)

Choosing 119887 = 120582 = 120574 = 1 in the above equation then itssolution becomes

119908 (119903) =

1198882

radic119903

radiccosh (radic2119903 + 1198941198881) (78)

Recasting the above equation in its original coordinates theexact solution of (4) becomes

119906 (119909 119905) = exp (minus119905)1198882

radic119909

radiccosh (radic2119909 + 1198941198881) (79)

The graph of this solution is plotted in Figure 2

Case 4 (119870(119906) = 120574 and 119863(119906) = 119886(119887119906)1119887) In this case we

consider the generator

X1 =119887

120574

exp(120574

119887

119905)

120597

120597119905

minus 119887119906 exp (120574

119887

119905)

120597

120597119906

(80)

The characteristic equation corresponding to this generatoris

119889119909

0

=

120574

119887

exp(minus120574

119887

119905) 119889119905 = exp(minus120574

119887

119905)

119889119906

minus119887119906

(81)

solving the above characteristic equation gives 119903 = 119909 with119908 = exp(120574119905)119906 These variables can be used to recast (4) to anODE

119903119908119903119903+

119903

119887

1

119908

1199082

119903+ 2119908119903= 0 (82)

Journal of Applied Mathematics 7

Figure 3 Plot of solution (84) with 119887 = 14 120574 = 1198881= 1198882= 1

Choosing 119887 = (14) in (82) then its solution becomes

119908 (119903) = (

minus119903

5(1198881minus 1198882119903)

)

minus15

(83)

Recasting the above equation in its original coordinates theexact solution of (4) becomes

119906 (119909 119905) = exp (minus120574119905) ( minus119909

5(1198881minus 1198882119909)

)

minus15

(84)

The graph of this solution is plotted in Figure 3

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to acknowledge the support providedby the Deanship of Research at King Fahd University ofPetroleum and Minerals (KFUPM) for funding this workthrough Project IN111008

References

[1] D S Jones M Plank and B D Saleem Differential Equationsand Mathematical Biology Mathematical amp ComputationalBiology Chapman and HallCRC New York NY USA 2011

[2] G C Cruywagen D E Woodward P Tracqui G T BartooJ D Murray and C A Ellsworth ldquoThe modelling of diffusivetumoursrdquo Journal of Biological Systems vol 3 no 4 pp 937ndash9451995

[3] S Moyo and P G L Leach ldquoSymmetry methods applied to amathematical model of a tumour of the brainrdquo Proceedings ofInstitute of Mathematics of NAS of Ukraine vol 50 pp 204ndash2102004

[4] P Tracqui G C Cruywagen D E Woodward G T Bartoo JD Murray and E C Alvord ldquoA mathematical model of gliomagrowth the effect of chemotherapy on spatio-temporal growthrdquoCell Proliferation vol 28 no 1 pp 17ndash31 1995

[5] A H Bokhari A H Kara and F D Zaman ldquoOn the solutionsand conservation laws of the model for tumor growth in thebrainrdquo Journal of Mathematical Analysis and Applications vol350 no 1 pp 256ndash261 2009

[6] S LieTheorie der Transoformationsgruppen vol 3 BG TubnerEd Liepzig Germany 1893

[7] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2002

[8] G W Bluman A F Cheviakov and S C Anco Applications ofSymmetry Methods to Partial Differential Equations SpringerNew York NY USA 2010

[9] G W Bluman and S Kumei Symmetries and DifferentialEquations Springer New York NY USA 1989

[10] B J Cantwell Introduction to Symmetry Analysis CambridgeUniversity Press Cambridge UK 2002

[11] A Ahmad A H Bokhari A H Kara and F D ZamanldquoSymmetry classifications and reductions of some classes of (2+1)-nonlinear heat equationrdquo Journal of Mathematical Analysisand Applications vol 339 no 1 pp 175ndash181 2008

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

Journal of Applied Mathematics 5

Table 4 Commutator table of the tumor equation

[XiXj] X1 X2

X1 0 minus

120574

119887

X1

X2120574

119887

X1 0

Using (33) into (35) gives 120573 = 1198881and hence

120585 = 1198881119909 120601 =

1

4

119906 (21198881minus 120591119905) (56)

Using (56) into (12) yields

minus

1

4

119906119870119906(21198881minus 120591119905) +

1

4

120591119905119905minus 119870120591119905= 0 (57)

Differentiating (57) with respect to 119905

minus

1

4

119906119870119906minus 119870 =

minus (14) 120591119905119905119905

120591119905119905

(58)

The equality in (58) holds if

minus119887119906119870119906minus 119870 =

minus (14) 120591119905119905119905

120591119905119905

= 120574 where 120574 is a constant

(59)

In accordance with (59) we have

120591 (119905) =

1198883

161205742exp (4120574119905) + 119888

4119905 + 1198885 119870 (119906) = 120574 + 120582119906

4 (60)

Using (60) into (57) yields

minus211988811205821199064minus 1205741198884= 0 (61)

Differentiating (61) with respect to 119906 leads to

minus811988811205821199063= 0 (62)

From (62) three cases arise

(b1) 1198881= 0 and 120582 = 0

(b2) 1198881= 0 and 120582 = 0

(b3) 1198881= 0 and 120582 = 0

Subsubcase (b1) (119863(119906) = ((14)119906)4 and 119870(119906) = 120574 + 1205821199064) Inaccordance with this subsubcase and (61) we obtain 119888

4= 0

and hence the infinitesimals are determined as

120585 = 0 120591 =

1198883

4120574

exp (4120574119905) + 1198885

120601 = minus

1

4

1198883119906 exp (4120574119905)

(63)

The generators associated with this subsubcase are

X1=

1

4120574

exp (4120574119905) 120597120597119905

minus

1

4

119906 exp (4120574119905) 120597120597119906

X2 =120597

120597119905

(64)

Table 5 Commutator table of the tumor equation

[XiXj] X1 X2

X1 0 minus4120574X1

X2 4120574X1 0

Table 6 Commutator table of the tumor equation

[XiXj] X1 X2 X3

X1 0 0 minus4120574X1

1198832

0 0 0

1198833

4120574X1 0 0

The commutation relation for these generators is given inTable 5

Subsubcase (b2) (119863(119906) = ((14)119906)4 and 119870(119906) = 120574) This

subsubcase with (61) yields to 1198884= 0 and hence the infin-

itesimals are

120585 = 1198881119909 120591 =

1198883

4120574

exp (4120574119905) + 1198885

120601 =

1

4

119906 (21198881minus 1198883exp (4120574119905))

(65)

The generators associated with this subsubcase are

X1 =1

4120574

exp (4120574119905) 120597120597119905

minus

1

4

119906 exp (4120574119905) 120597120597119906

X2 = 119909120597

120597119909

+

1

2

119906

120597

120597119906

X3 =120597

120597119905

(66)

The commutation relation for these generators is given inTable 6

Subsubcase (b3) (119863(119906) = ((14)119906)4 and 119870(119906) = 120574) In ac-

cordance with this subsubcase and (61) we obtain 1198884= 0 and

hence the infinitesimals are

120585 = 0 120591 =

1198883

4120574

exp (4120574119905) + 1198885

120601 = minus

1

4

1198883119906 exp (4120574119905)

(67)

The generators corresponding to this subsubcase are

X1 =1

4120574

exp (4120574119905) 120597120597119905

minus

1

4

119906 exp (4120574119905) 120597120597119906

X2 =120597

120597119905

(68)

Remark 1 Case (III) represents the particular case of (I)and (II) So by similar manipulation as in the previous caseswe obtain the same symmetry generators

4 Some Reduction

In this section we present solutions of (4) via reductionsThese reductions are obtained by the similarity variablesobtained through symmetry generators

6 Journal of Applied Mathematics

Figure 1 Plot of solution (74) with 1198881= 1 1198882= 0

Case 1 (119870(119906) and 119863(119906) are arbitrary) In this case we haveonly one generator that is X = 120597120597119905 Thus the characteristicequation corresponding to this generator is

119889119909

0

=

119889119905

1

=

119889119906

0

(69)

Solving the above equation it is straightforward [11] to findthat it yields the similarity variables 119903 = 119909 and 119908 = 119906Replacing 119906 in (4) in terms of new variables it becomes

119903119863 (119908)119908119903119903+ 1199031198631199081199082

119903+ 2119863 (119908)119908

119903minus 119903119870 (119908)119908 = 0 (70)

Case 2 (119870(119906) = 0 and 119863(119906) are arbitrary) We consider thegenerator X

1= 119909(120597120597119909) + 2119905(120597120597119905) Thus the characteristic

equation associated with this generator is119889119909

119909

=

119889119905

2119905

=

119889119906

0

(71)

The similarity variables corresponding to the above equationbecome 119903 = 119909119905

minus(12) and 119908 = 119906 These new variables reduce(4) to a ODE of the form

1199032119863(119908)119908

119903119903+ 11990321198631199081199082

119903+ 2119903119863 (119908)119908

119903+

1

2

1199033119908119903= 0 (72)

Choosing 119863(119908) = 1 in the above equation then the solutionof the resulting equation is

119908 (119903) = 1198882+ 1198881(minus

1

119903

exp(minus1199032

4

) minus

1

2

radicΠErf( 1199032

)) (73)

Recasting the above equation in its original coordinates theexact solution of (4) becomes

119906 (119909 119905) = 1198882+ 1198881(minus

radic119905

119909

exp(minus1199092

4119905

) minus

1

2

radicΠErf( 119909

2radic119905

))

(74)

The graph of this solution is plotted in Figure 1

Case 3 (119870(119906) = 120574 + 1205821199061119887 and 119863(119906) = 119886(119887119906)1119887) We take thegenerator

X1 =119887

120574

exp(120574

119887

119905)

120597

120597119905

minus 119887119906 exp(120574

119887

119905)

120597

120597119906

(75)

Figure 2 Plot of solution (79) with 1198881= 0 120574 = 119888

2= 1

and its characteristic equation

119889119909

0

=

120574

119887

exp (minus120574

119887

119905) 119889119905 = exp(minus120574

119887

119905)

119889119906

minus119887119906

(76)

gives the similarity variables 119903 = 119909 and 119908 = exp(120574119905)119906 Inaccordance of these similarities (4) is transformed to theODE in the form

119903119908119903119903+

119903

119887

1

119908

1199082

119903+ 2119908119903minus 119903120582119887minus(1119887)

119908 = 0 (77)

Choosing 119887 = 120582 = 120574 = 1 in the above equation then itssolution becomes

119908 (119903) =

1198882

radic119903

radiccosh (radic2119903 + 1198941198881) (78)

Recasting the above equation in its original coordinates theexact solution of (4) becomes

119906 (119909 119905) = exp (minus119905)1198882

radic119909

radiccosh (radic2119909 + 1198941198881) (79)

The graph of this solution is plotted in Figure 2

Case 4 (119870(119906) = 120574 and 119863(119906) = 119886(119887119906)1119887) In this case we

consider the generator

X1 =119887

120574

exp(120574

119887

119905)

120597

120597119905

minus 119887119906 exp (120574

119887

119905)

120597

120597119906

(80)

The characteristic equation corresponding to this generatoris

119889119909

0

=

120574

119887

exp(minus120574

119887

119905) 119889119905 = exp(minus120574

119887

119905)

119889119906

minus119887119906

(81)

solving the above characteristic equation gives 119903 = 119909 with119908 = exp(120574119905)119906 These variables can be used to recast (4) to anODE

119903119908119903119903+

119903

119887

1

119908

1199082

119903+ 2119908119903= 0 (82)

Journal of Applied Mathematics 7

Figure 3 Plot of solution (84) with 119887 = 14 120574 = 1198881= 1198882= 1

Choosing 119887 = (14) in (82) then its solution becomes

119908 (119903) = (

minus119903

5(1198881minus 1198882119903)

)

minus15

(83)

Recasting the above equation in its original coordinates theexact solution of (4) becomes

119906 (119909 119905) = exp (minus120574119905) ( minus119909

5(1198881minus 1198882119909)

)

minus15

(84)

The graph of this solution is plotted in Figure 3

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to acknowledge the support providedby the Deanship of Research at King Fahd University ofPetroleum and Minerals (KFUPM) for funding this workthrough Project IN111008

References

[1] D S Jones M Plank and B D Saleem Differential Equationsand Mathematical Biology Mathematical amp ComputationalBiology Chapman and HallCRC New York NY USA 2011

[2] G C Cruywagen D E Woodward P Tracqui G T BartooJ D Murray and C A Ellsworth ldquoThe modelling of diffusivetumoursrdquo Journal of Biological Systems vol 3 no 4 pp 937ndash9451995

[3] S Moyo and P G L Leach ldquoSymmetry methods applied to amathematical model of a tumour of the brainrdquo Proceedings ofInstitute of Mathematics of NAS of Ukraine vol 50 pp 204ndash2102004

[4] P Tracqui G C Cruywagen D E Woodward G T Bartoo JD Murray and E C Alvord ldquoA mathematical model of gliomagrowth the effect of chemotherapy on spatio-temporal growthrdquoCell Proliferation vol 28 no 1 pp 17ndash31 1995

[5] A H Bokhari A H Kara and F D Zaman ldquoOn the solutionsand conservation laws of the model for tumor growth in thebrainrdquo Journal of Mathematical Analysis and Applications vol350 no 1 pp 256ndash261 2009

[6] S LieTheorie der Transoformationsgruppen vol 3 BG TubnerEd Liepzig Germany 1893

[7] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2002

[8] G W Bluman A F Cheviakov and S C Anco Applications ofSymmetry Methods to Partial Differential Equations SpringerNew York NY USA 2010

[9] G W Bluman and S Kumei Symmetries and DifferentialEquations Springer New York NY USA 1989

[10] B J Cantwell Introduction to Symmetry Analysis CambridgeUniversity Press Cambridge UK 2002

[11] A Ahmad A H Bokhari A H Kara and F D ZamanldquoSymmetry classifications and reductions of some classes of (2+1)-nonlinear heat equationrdquo Journal of Mathematical Analysisand Applications vol 339 no 1 pp 175ndash181 2008

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 Journal of Applied Mathematics

Figure 1 Plot of solution (74) with 1198881= 1 1198882= 0

Case 1 (119870(119906) and 119863(119906) are arbitrary) In this case we haveonly one generator that is X = 120597120597119905 Thus the characteristicequation corresponding to this generator is

119889119909

0

=

119889119905

1

=

119889119906

0

(69)

Solving the above equation it is straightforward [11] to findthat it yields the similarity variables 119903 = 119909 and 119908 = 119906Replacing 119906 in (4) in terms of new variables it becomes

119903119863 (119908)119908119903119903+ 1199031198631199081199082

119903+ 2119863 (119908)119908

119903minus 119903119870 (119908)119908 = 0 (70)

Case 2 (119870(119906) = 0 and 119863(119906) are arbitrary) We consider thegenerator X

1= 119909(120597120597119909) + 2119905(120597120597119905) Thus the characteristic

equation associated with this generator is119889119909

119909

=

119889119905

2119905

=

119889119906

0

(71)

The similarity variables corresponding to the above equationbecome 119903 = 119909119905

minus(12) and 119908 = 119906 These new variables reduce(4) to a ODE of the form

1199032119863(119908)119908

119903119903+ 11990321198631199081199082

119903+ 2119903119863 (119908)119908

119903+

1

2

1199033119908119903= 0 (72)

Choosing 119863(119908) = 1 in the above equation then the solutionof the resulting equation is

119908 (119903) = 1198882+ 1198881(minus

1

119903

exp(minus1199032

4

) minus

1

2

radicΠErf( 1199032

)) (73)

Recasting the above equation in its original coordinates theexact solution of (4) becomes

119906 (119909 119905) = 1198882+ 1198881(minus

radic119905

119909

exp(minus1199092

4119905

) minus

1

2

radicΠErf( 119909

2radic119905

))

(74)

The graph of this solution is plotted in Figure 1

Case 3 (119870(119906) = 120574 + 1205821199061119887 and 119863(119906) = 119886(119887119906)1119887) We take thegenerator

X1 =119887

120574

exp(120574

119887

119905)

120597

120597119905

minus 119887119906 exp(120574

119887

119905)

120597

120597119906

(75)

Figure 2 Plot of solution (79) with 1198881= 0 120574 = 119888

2= 1

and its characteristic equation

119889119909

0

=

120574

119887

exp (minus120574

119887

119905) 119889119905 = exp(minus120574

119887

119905)

119889119906

minus119887119906

(76)

gives the similarity variables 119903 = 119909 and 119908 = exp(120574119905)119906 Inaccordance of these similarities (4) is transformed to theODE in the form

119903119908119903119903+

119903

119887

1

119908

1199082

119903+ 2119908119903minus 119903120582119887minus(1119887)

119908 = 0 (77)

Choosing 119887 = 120582 = 120574 = 1 in the above equation then itssolution becomes

119908 (119903) =

1198882

radic119903

radiccosh (radic2119903 + 1198941198881) (78)

Recasting the above equation in its original coordinates theexact solution of (4) becomes

119906 (119909 119905) = exp (minus119905)1198882

radic119909

radiccosh (radic2119909 + 1198941198881) (79)

The graph of this solution is plotted in Figure 2

Case 4 (119870(119906) = 120574 and 119863(119906) = 119886(119887119906)1119887) In this case we

consider the generator

X1 =119887

120574

exp(120574

119887

119905)

120597

120597119905

minus 119887119906 exp (120574

119887

119905)

120597

120597119906

(80)

The characteristic equation corresponding to this generatoris

119889119909

0

=

120574

119887

exp(minus120574

119887

119905) 119889119905 = exp(minus120574

119887

119905)

119889119906

minus119887119906

(81)

solving the above characteristic equation gives 119903 = 119909 with119908 = exp(120574119905)119906 These variables can be used to recast (4) to anODE

119903119908119903119903+

119903

119887

1

119908

1199082

119903+ 2119908119903= 0 (82)

Journal of Applied Mathematics 7

Figure 3 Plot of solution (84) with 119887 = 14 120574 = 1198881= 1198882= 1

Choosing 119887 = (14) in (82) then its solution becomes

119908 (119903) = (

minus119903

5(1198881minus 1198882119903)

)

minus15

(83)

Recasting the above equation in its original coordinates theexact solution of (4) becomes

119906 (119909 119905) = exp (minus120574119905) ( minus119909

5(1198881minus 1198882119909)

)

minus15

(84)

The graph of this solution is plotted in Figure 3

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to acknowledge the support providedby the Deanship of Research at King Fahd University ofPetroleum and Minerals (KFUPM) for funding this workthrough Project IN111008

References

[1] D S Jones M Plank and B D Saleem Differential Equationsand Mathematical Biology Mathematical amp ComputationalBiology Chapman and HallCRC New York NY USA 2011

[2] G C Cruywagen D E Woodward P Tracqui G T BartooJ D Murray and C A Ellsworth ldquoThe modelling of diffusivetumoursrdquo Journal of Biological Systems vol 3 no 4 pp 937ndash9451995

[3] S Moyo and P G L Leach ldquoSymmetry methods applied to amathematical model of a tumour of the brainrdquo Proceedings ofInstitute of Mathematics of NAS of Ukraine vol 50 pp 204ndash2102004

[4] P Tracqui G C Cruywagen D E Woodward G T Bartoo JD Murray and E C Alvord ldquoA mathematical model of gliomagrowth the effect of chemotherapy on spatio-temporal growthrdquoCell Proliferation vol 28 no 1 pp 17ndash31 1995

[5] A H Bokhari A H Kara and F D Zaman ldquoOn the solutionsand conservation laws of the model for tumor growth in thebrainrdquo Journal of Mathematical Analysis and Applications vol350 no 1 pp 256ndash261 2009

[6] S LieTheorie der Transoformationsgruppen vol 3 BG TubnerEd Liepzig Germany 1893

[7] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2002

[8] G W Bluman A F Cheviakov and S C Anco Applications ofSymmetry Methods to Partial Differential Equations SpringerNew York NY USA 2010

[9] G W Bluman and S Kumei Symmetries and DifferentialEquations Springer New York NY USA 1989

[10] B J Cantwell Introduction to Symmetry Analysis CambridgeUniversity Press Cambridge UK 2002

[11] A Ahmad A H Bokhari A H Kara and F D ZamanldquoSymmetry classifications and reductions of some classes of (2+1)-nonlinear heat equationrdquo Journal of Mathematical Analysisand Applications vol 339 no 1 pp 175ndash181 2008

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

Journal of Applied Mathematics 7

Figure 3 Plot of solution (84) with 119887 = 14 120574 = 1198881= 1198882= 1

Choosing 119887 = (14) in (82) then its solution becomes

119908 (119903) = (

minus119903

5(1198881minus 1198882119903)

)

minus15

(83)

Recasting the above equation in its original coordinates theexact solution of (4) becomes

119906 (119909 119905) = exp (minus120574119905) ( minus119909

5(1198881minus 1198882119909)

)

minus15

(84)

The graph of this solution is plotted in Figure 3

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to acknowledge the support providedby the Deanship of Research at King Fahd University ofPetroleum and Minerals (KFUPM) for funding this workthrough Project IN111008

References

[1] D S Jones M Plank and B D Saleem Differential Equationsand Mathematical Biology Mathematical amp ComputationalBiology Chapman and HallCRC New York NY USA 2011

[2] G C Cruywagen D E Woodward P Tracqui G T BartooJ D Murray and C A Ellsworth ldquoThe modelling of diffusivetumoursrdquo Journal of Biological Systems vol 3 no 4 pp 937ndash9451995

[3] S Moyo and P G L Leach ldquoSymmetry methods applied to amathematical model of a tumour of the brainrdquo Proceedings ofInstitute of Mathematics of NAS of Ukraine vol 50 pp 204ndash2102004

[4] P Tracqui G C Cruywagen D E Woodward G T Bartoo JD Murray and E C Alvord ldquoA mathematical model of gliomagrowth the effect of chemotherapy on spatio-temporal growthrdquoCell Proliferation vol 28 no 1 pp 17ndash31 1995

[5] A H Bokhari A H Kara and F D Zaman ldquoOn the solutionsand conservation laws of the model for tumor growth in thebrainrdquo Journal of Mathematical Analysis and Applications vol350 no 1 pp 256ndash261 2009

[6] S LieTheorie der Transoformationsgruppen vol 3 BG TubnerEd Liepzig Germany 1893

[7] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2002

[8] G W Bluman A F Cheviakov and S C Anco Applications ofSymmetry Methods to Partial Differential Equations SpringerNew York NY USA 2010

[9] G W Bluman and S Kumei Symmetries and DifferentialEquations Springer New York NY USA 1989

[10] B J Cantwell Introduction to Symmetry Analysis CambridgeUniversity Press Cambridge UK 2002

[11] A Ahmad A H Bokhari A H Kara and F D ZamanldquoSymmetry classifications and reductions of some classes of (2+1)-nonlinear heat equationrdquo Journal of Mathematical Analysisand Applications vol 339 no 1 pp 175ndash181 2008

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