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Research ArticleAnalytic Solution for Systems of Two-Dimensional TimeConformable Fractional PDEs by Using CFRDTM
Abir Chaouk and Maher Jneid
Department of Mathematics and Computer Science Faculty of Science Beirut Arab University Beirut Lebanon
Correspondence should be addressed to Maher Jneid mjneidbauedulb
Received 2 January 2019 Accepted 24 March 2019 Published 11 April 2019
Academic Editor Birendra Nath Mandal
Copyright copy 2019 Abir Chaouk andMaher JneidThisis an open access article distributed under theCreativeCommonsAttributionLicensewhichpermits unrestricteduse distribution and reproduction in anymedium provided the original work is properly cited
In this studywe use the conformable fractional reduced differential transform (CFRDTM)method to compute solutions for systemsof nonlinear conformable fractional PDEsThe proposedmethod yields a numerical approximate solution in the form of an infiniteseries that converges to a closed form solution which is in many cases the exact solutionWe inspect its efficiency in solving systemsofCFPDEs byworking on four different nonlinear systemsThe results show that CFRDTMgave similar solutions to exact solutionsconfirming its proficiency as a competent technique for solving CFPDEs systems It required very little computational work andhence consumed much less time compared to other numerical methods
1 Introduction
Fractional partial differential equations (FPDEs) are mathe-matical tools employed to depict a broad variety of phenom-ena that arise in distinct areas of science and engineering likebiology physics fluid mechanics chemistry and so on [1ndash4]Recently investigators have centered their attention on study-ing the solution of partial differential equations of fractionalorder However till now not a single definition of fractionalorder derivative has been found [5] Therefore researchersraced to set definitions for fractional order derivatives andconsequently many definitions mostly complicated wereproposed such as the Riemann-Liouville Caputo Hadamardand others [5] However in 2014 an intriguing perspective tothe subject was presented by Khalil [6] who came up withthe conformable fractional derivative an extension of theclassical simple limit definition of derivative we all knowThis new definition unlocked a whole new chapter in theworld of research of FPDEs in whichmany recent works havebeen done To fully grasp these problems and to be able toanalyze those phenomena it is a necessity to find solutions forthose systems of FPDEs or CFPDEs But then again findinga numerical or analytical solution for those systems is a verytough task to achieve and exact solutions for them are noteasily obtained
To tackle this problem researchers worked on findingadequate methods for those systems of FPDEs One of thegreatest methods found by researchers is the RDTM Whatdistinguishes this method is that it provides us with analyticalapproximations which in many cases are exact solutionsthrough convergent series with terms that are relativelyeasy to compute Not to mention its ability in solving ahuge variety of problems which makes it a very significantmathematical tool For example in 2006 Bildik et al [7]solved different partial differential equations using the RDMand ADM and performed a comparison with approximatesolution and analytic solutions Guptav [8] obtained theapproximate solution of Benney-Lin equation through theRDTM and the HPM Acan and Baleanu [9] handled threelinear and nonlinear equations by CRDTM and CADMFor more applications of the present method see [9ndash12]and some references cited therein In this paper we proposethe conformable fractional reduced differential transformmethod (CFRDTM) as a reliable method to solve systems ofnonlinear CFPDEs
This paper is arranged as follows In the second sectionwedemonstrate the properties and theorems of the CFRDTMin addition to reviewing some of the properties of theconformable derivative which are needed in the followingwork In Section 3 we solve 4 systems by using CFRDTM
HindawiInternational Journal of Mathematics and Mathematical SciencesVolume 2019 Article ID 7869516 7 pageshttpsdoiorg10115520197869516
2 International Journal of Mathematics and Mathematical Sciences
to test its efficiency In Section 4 we conclude this article byanalyzing the results we have obtained briefly
2 CFRDTM
In this section we introduce the CFRDTM which we willbe using to solve the system of conformable FPDEs in thispaper Throughout this section let x isin R119899 and q isin (0 1]Formore details about the definitions and properties given inthis section the reader can refer to [13ndash16] and the referencestherin
Definition 1 Given a function u R119899 times (0infin) 997888rarr R theconformable fractional partial derivative of 119906(119909 119905) of orderq with respect to t is given as
120597119902119905 119906 (119909 119905) = lim120576997888rarr0
119906 (119909 119905 + 1205761199051minus119902) minus 119906 (119909 119905)120576
(1)
provided that this limit exists as a finite number
Definition 2 Let 119906(119909 119905) be an infinitely partially q-differen-tiable function near 0 with respect to 119905 of order 119902 then theCFRDT of 119906(119909 119905) is given as
119880119902119896 (119909) =
1119902119896119896 [(120597
119902119905 )119896 119906 (119909 119905)]
119905=0(2)
where (120597119902119905 )119896119906(119909 119905) resembles applying the conformable par-tial fractional derivative 119896-times and 119880119902
119896(119909) is the CFRDT
function
Definition 3 Let 119880119902119896(119909) be the CFRDT of 119906(119909 119905) Then the
inverse CFRDT of 119880119902119896(119909) is given as
119906 (119909 119905) =infin
sum119896=0
119880119902119896 (119909) 119905119902119896
=infin
sum119896=0
1119902119896119896 [(120597
119902119905 )119896 119906 (119909 119905)]
119905=0119905119902119896
(3)
Definition 4 TheCFRDT of 119906(119909 119905) of the initial conditions isdefined as
119880119902119896 (119909) =
1(119902119896) [(120597
119902119905 )119896 119906]
119905=0if 119896119902 isin Z+
0 if 119896119902 notin Z+(4)
for 119896 = 0 1 2 (119899119902 minus 1) where 119899 is the order of CFPDETheorem5 Let 119906(119909 119905) V(119909 119905) and119908(119909 119905)R119899times[0infin) 997888rarr R
be partially q-differentiable at a point 119905 gt 0 and ab isin R Thenthe following is obtained
(1) If 119906(119909 119905) = V(119909 119905)119908(119909 119905) then 119880119902119896(119909) = sum119896119904=0 =
119881119902119896(119909 119904)119882119902
119896minus119904
(2) If V(119909 119905) = 120597119902119905 119906(119909 119905) then 119881119902119896(119909) = 119902(119896 + 1)119880119902
119896+1(119909)
(3) If 119906(119909 119905) = 119886V(119909 119905) plusmn 119887119908(119909 119905) then 119880119902119896(119909) = 119886119881119902
119896(119909) plusmn
119887119882119902119896(119909)
In general for 119906(119909 119905) = V1(119909 119905)V2(119909 119905) sdot sdot sdot V119899(119909 119905) then wehave
119880119902119896 (119909)
=119896119899
sum119896119899minus1=0
sdot sdot sdot1198962
sum1198961=0
119881119902(1)1198961119881119902
(2)1198962minus1198961times sdot sdot sdot 119881119902(119899minus1)119896119899minus1119881
119902
(119899)119896119899minus119896119899minus1
(5)
(4) If 119906(119909 119905) = 119905119898ℎ(119909) then 119880119902119896(119909) = 120575(119896 minus 119898119902)ℎ(119909)where
120575 (119896) =
1 if 119896 = 00 if 119896 = 0
(6)
3 Applications
To demonstrate the excellent performance of this method weapply it on four different systems in this section
Example 6 Consider the nonlinear system of CFDEs
120597119902119905 119906 = 1199062V minus 2119906 + 14 (12059721199092119906 + 12059721199102119906)
120597119902119905 V = 119906 minus 1199062V + 14 (12059721199092V + 12059721199102V)
(7)
with initial conditions
119906 (119909 119910 0) = 119890minus119909minus119910V (119909 119910 0) = 119890119909+119910
(8)
By applying CFRDTM on (7) we obtain the two recurrencerelations
119902 (119896 + 1)119880119902119896+1 =119896
sum119894=0
119894
sum119895=0
119880119902119894minus119895119880119902119895119881119902119896minus119894 minus 2119880119902119896
+ 14 (12059721199092119880119902119896 + 12059721199102119880119902119896)
119902 (119896 + 1) 119881119902119896+1 = 119880119902119896minus119896
sum119894=0
119894
sum119895=0
119880119902119894minus119895119880119902119895119881119902119896minus119894
+ 14 (12059721199092119881119902119896 + 12059721199102119881119902119896 )
(9)
where the IC (8) can be transformed at 119905 = 0 to1198801199020 = 119890minus119909minus119910
1198811199020 = 119890119909+119910(10)
Now substitute 119896 = 0 into (9) one obtains
1198801199021 =minus12119902 119890minus119909minus119910
1198811199021 =12119902119890119909+119910
(11)
International Journal of Mathematics and Mathematical Sciences 3
For k=1 and k=2 we get
1198801199022 =181199022 119890minus119909minus119910 = 1
2221199022 119890minus119909minus119910
1198811199022 =181199022 119890119909+119910 = 1
2221199022 119890119909+119910
(12)
and
1198801199023 =minus1481199023 119890
minus119909minus119910 = minus12331199023 119890
minus119909minus119910
1198811199023 =1
481199023 119890119909+119910 = 1
2331199023 119890119909+119910
(13)
Carrying on the iterative calculations in a similar way forother values of k we obtain the general terms
119880119902119899 =(minus1)1198992119899119899119902119899 119890
minus119909minus119910
119881119902119899 =1
2119899119899119902119899 119890119909+119910
(14)
Applying the inverse transformation of CFRDTM the solu-tion of (7) is given as
119906 (119909 119910 119905) =infin
sum119896=0
119880119902119896 (119909 119910 119905) 119905119896119902 = 119890minus119909minus119910minus1199051199022119902
V (119909 119910 119905) =infin
sum119896=0
119881119902119896 (119909 119910 119905) 119905119896119902 = 119890119909+119910+1199051199022119902(15)
Example 7 Consider the following system of CFDEs
120597119902119905 119906 = minus120597119909119906120597119909V minus 120597119910119906120597119910V minus 119906120597119902119905 V = minus120597119909V120597119909119908 + 120597119910V120597119910119908 + V
120597119902119905119908 = minus120597119909119908120597119909119906 minus 120597119910119908120597119910119906 minus 119908(16)
with the initial conditions
119906 (119909 119910 0) = 119890119909+119910V (119909 119910 0) = 119890119909minus119910119908 (119909 119910 0) = 119890minus119909+119910
(17)
Applying theorems of CFRDTM on (16) we obtain thefollowing system of recurrence relations
(119896 + 1)119880119902119896+1
= minus119896
sum119903=0
120597119909119880119902119896120597119909119881119902119896 minus119896
sum119903=0
120597119910119880119902119896120597119910119881119902119896 minus 119880119902119896
(119896 + 1) 119881119902119896+1 = minus119896
sum119903=0
120597119909119881119902119896 120597119909119882119902
119896+119896
sum119903=0
120597119910119881119902119896 120597119910119882119902
119896+ 119881119902119896
(119896 + 1)119882119902119896+1 = minus119896
sum119903=0
120597119909119882119902119896 120597119909119880119902119896 minus119896
sum119903=0
120597119910119882119902119896 120597119910119880119902119896 minus119882119902119896
(18)
where the IC can be transformed at 119905 = 0 to1198801199020 = 119890119909+119910
1198811199020 = 119890119909minus119910
1198821199020 = 119890minus119909+119910(19)
Substituting 119896 = 0 into (16) we obtain
1198801199021 =minus1119902 119890119909+119910
1198811199020 =1119902119890119909minus119910
1198821199020 =minus1119902 119890minus119909+119910
(20)
Similarly for k=1 and k=2 we get
1198801199022 =121199022 119890119909+119910
1198811199022 =121199022 119890119909minus119910
1198821199022 =121199022 119890minus119909+119910
(21)
and
1198801199023 = minus 161199023 119890119909+119910
1198811199023 =161199023 119890119909minus119910
1198821199023 = minus 161199023 119890minus119909+119910
(22)
Continuing the iterative calculations in a similar way forother values of k we obtain the general terms
119880119902119899 =(minus1)119899119899119902119899 119890
119909+119910
119881119902119899 =1
119899119902119899 119890119909minus119910
119882119902119899 =(minus1)119899119899119902119899 119890
minus119909+119910
(23)
Applying the inverse transformation of CFRDTM the solu-tion of (16) is given as
119906 (119909 119910 119905) =infin
sum119896=0
119880119902119896(119909 119910 119905) 119905119896119902 = 119890119909+119910minus119905119902119902
V (119909 119910 119905) =infin
sum119896=0
119881119902119896(119909 119910 119905) 119905119896119902 = 119890119909minus119910+119905119902119902
119908 (119909 119910 119905) =infin
sum119896=0
119882119902119896(119909 119910 119905) 119905119896119902 = 119890minus119909+119910minus119905119902119902
(24)
4 International Journal of Mathematics and Mathematical Sciences
Example 8 Consider the following system of CFDEs
120597119902119905 119906 + 119906120597119909119906 + V120597119910119906 = 12059721199092119906 + 12059721199102119906
120597119902119905 V + 119906120597119909V + V120597119910V = 12059721199092V + 12059721199102V(25)
subject to the initial conditions
119906 (119909 119910 0) = 119909 + 119910V (119909 119910 0) = 119909 minus 119910
(26)
Applying theorems of CFRDTM on (25) we obtain thefollowing system of recurrence relations
119902 (119896 + 1)119880119902119896+1 +119896
sum119903=0
119880119902119896minus119903120597119909119880119902119903 +119896
sum119903=0
119881119902119896minus119903120597119910119880119902119903
= 12059721199092119880119902119896 + 12059721199102119880119902119896
119902 (119896 + 1)119881119902119896+1 +119896
sum119903=0
119880119902119896minus119903120597119909119881119902119903 +119896
sum119903=0
119881119902119896minus119903120597119910119881119902119903
= 12059721199102119881119902119896 + +12059721199102119881119902119896
(27)
where the IC can be transformed at 119905 = 0 to1198801199020 = 119909 + 1199101198811199020 = 119909 minus 119910
(28)
Substituting 119896 = 0 into (25) we obtain1198801199021 =
minus2119902 119909
1198811199021 =minus2119902 119910
(29)
Similarly for 119896 = 1 119896 = 2 119896 = 3 and 119896 = 4 we getrespectively
1198801199022 =21199022 (119909 + 119910)
1198811199022 =21199022 (119909 minus 119910)
(30)
1198801199023 = minus 41199023119909
1198811199023 = minus 41199023119910
(31)
1198801199024 =41199023 (119909 + 119910)
1198811199024 = minus 41199023 (119909 minus 119910)
(32)
and
1198801199025 = minus 81199025119909
1198811199025 = minus 81199025119910
(33)
Applying the inverse transformation of CFRDTM the solu-tion of (25) is given as
119906 (119909 119910 119905) =infin
sum119896=0
119880119902119896(119909 119910 119905) 119905119896119902
= 119909 + 119910 minus 2119902119909119905119902 + 2
1199022 (119909 + 119910) 1199052119902 minus 411990231199091199053119902
+ 41199023 (119909 + 119910) 1199054119902 minus 8
11990251199091199055119902 + sdot sdot sdot
V (119909 119910 119905) =infin
sum119896=0
119881119902119896(119909 119910 119905) 119905119896119902
= 119909 minus 119910 minus 2119902119910119905119902 + 2
1199022 (119909 minus 119910) 1199052119902 minus 411990231199101199053119902
+ 41199023 (119909 minus 119910) 1199054119902 minus 8
11990251199101199055119902 + sdot sdot sdot
(34)
And by performing some computations the solutions aregiven as
119906 (119909 119910 119905) = 119909 + 119910 minus 21199091199051199021199021 minus 211990521199021199022
V (119909 119910 119905) = 119909 minus 119910 minus 21199091199051199021199021 minus 211990521199021199022
(35)
Example 9 Consider the following system of CFDEs
120597119902119905 119906 = V120597119909119906 + 120597119902119905 V120597119910119906 + 1 minus 119909 + 119910 + 119905119902119902
120597119902119905 V = 119906120597119909V + 120597119902119905 119906120597119910V + 1 minus 119909 minus 119910 minus 119905119902119902
(36)
subject to the initial conditions
119906 (119909 119910 0) = 119909 + 119910 minus 1V (119909 119910 0) = 119909 minus 119910 + 1
(37)
Applying theorems of CFRDTM on (25) we obtain thefollowing system of recurrence relations
119902 (119896 + 1)119880119902119896+1 =119896
sum119903=0
119881119902119903 120597119909119880119902119896minus119903
+119896
sum119903=0
119902 (119896 + 1) 119881119902119896+1minus119903120597119910119880119902119903 + 120575 (119896)
minus 119909 + 119910 minus 1 + 120575 (119896 minus 1)
119902 (119896 + 1) 119881119902119896+1 =119896
sum119903=0
119880119902119903 120597119909119880119902119896minus119903
+119896
sum119903=0
119902 (119896 + 1)119880119902119896+1minus119903120597119910119881119902119903 + 120575 (119896)
minus 119909 minus 119910 minus 1 + 120575 (119896 minus 1)
(38)
International Journal of Mathematics and Mathematical Sciences 5
00
00
00
00
00
00
minus05
minus05minus05
minus05minus05
minus10
minus15
minus20
05
05
05
15
05
0510
u(xy)v(xy)
Figure 1 The approximate solution of Example 9 for 120572 = 099 and t = 01
00
00
00
00
minus05
minus05minus05
minus05
minus10
minus15
05
05
00
00
minus0505
0515
05
10
u(xy) v(xy)
Figure 2The exact solution of Example 9 for 120572 = 099 and t = 01
where the IC can be transformed at 119905 = 0 to
1198801199020 = 119909 + 119910 minus 11198811199020 = 119909 minus 119910 + 1
(39)
Substituting 119896 = 0 into (36) we obtain
1198801199021 =1119902
1198811199021 = minus1119902
(40)
Similarly for 119896 = 1 and 119896 = 2 we get1198801199022 = minus 119909
2119902
1198811199022 =1119902 minus 119910 minus 1
2119902 (41)
and
1198801199023 =minus2119902119909 minus 119909
61199022 + 1 minus 5119902121199023
1198811199023 =minus1199103119902 minus 119902 + 1
121199023 (42)
Therefore the approximate analytical solution of (36) is givenas
119906 = 119909 + 119910 minus 1 + 119905119902119902 minus 1199091199052119902
2119902 + minus2119902119909 minus 11990961199022 1199053119902
+ 1 minus 5119902121199023 119905
3119902 + sdot sdot sdot
V = 119909 minus 119910 + 1 minus 119905119902119902 + 1119902 minus 119910 minus 1
2119902 1199052119902 + minus1199103119902 1199053119902
minus 119902 + 1121199023 119905
3119902 + sdot sdot sdot
(43)
Since this solution does not equal the exact one we usemathematica program to show its accuracy In Figures 1 and3 we plot the approximate solution for Example 9 for 119902 = 099at 119905 = 01 and at 119905 = 02 respectively The graphs in Figures2 and 4 illustrate the exact solution for Example 9 for 119902 =099 at 119905 = 01 and at 119905 = 02 respectively We can clearlyobserve at the time 119905 = 01 and 119905 = 02 that the values of theapproximate solution byCFRDTMand the values of the exactsolution obtained by other methods are extremely close Inother words solution provided by CFRDTM is highly preciseand accurate Particularly CFRDTM is a successful techniquefor solving systems of conformable FPDEs
6 International Journal of Mathematics and Mathematical Sciences
00
minus05
00
00
minus0505
0515
05
10
00
00
00
minus05
minus05
minus05
minus10
minus15
05
05
u(xy) v(xy)
Figure 3 The approximate solution of Example 9 for 120572 = 099 and t = 02
00
00
00
minus05
minus05
minus05minus10minus15
05
05
00
00
minus05
minus05
05
05
u(xy) v(xy)
00
15
05
10
Figure 4 The exact solution of Example 9 for 120572 = 099 and t = 02
4 Conclusion
In this paper our main purpose was to inspect the com-petence of the CFRDTM as a valid technique to solvesystems of nonlinear CFPDEs We successfully applied itto four distinct systems of nonlinear conformable time andspace FPDEs CFRDTM produced a numerical approximatesolution having the form of an infinite series that convergedto a closed form solution which coincided with the exactsolution in the first 3 applications In the 4119905ℎ applicationhowever we obtained an approximate solution that turnedout to be extremely close to the exact solution as we haveinterpreted from the attached Figures 1ndash4 On this basis weconclude that the CFRDTM is a powerful tool and a facileapproach for obtaining solutions for systems of CFPDEs Itreduces the amount of required computational work whencompared to other techniques used for this purpose and givessolutions that are in outstanding agreement with the exactones This provides a good starting point for further researchvia CFRDTM
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations North-HollandMathematics Studies New York NY USA 2006
[2] A S Elwakil ldquoFractional-order circuits and systems an emerg-ing interdisciplinary research areardquo IEEE Circuits and SystemsMagazine vol 10 no 4 pp 40ndash50 2010
[3] M D Ortigueira Fractional Calculus for Scientists and Engi-neers Heidelberg Springer Germany 2011
[4] T J Freeborn ldquoA survey of fractional-order circuit modelsfor biology and biomedicinerdquo IEEE Journal on Emerging andSelected Topics in Circuits and Systems vol 3 no 3 pp 416ndash4242013
[5] E C de Oliveira and J A Tenreiro Machado ldquoA review ofdefinitions for fractional derivatives and integralrdquoMathematicalProblems in Engineering vol 2014 Article ID 238459 6 pages2014
[6] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014
International Journal of Mathematics and Mathematical Sciences 7
[7] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationbydifferential transformmethodandAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006
[8] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers ampMathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011
[9] O Acan and D Baleanu ldquoA new numerical technique forsolving fractional partial differential equationsrdquo Miskolc Math-ematical Notes vol 19 no 1 pp 3ndash18 2018
[10] M Jneid and A El Chakik ldquoAnalytical solution for somesystems of nonlinear conformable fractional differential equa-tionsrdquo Far East Journal of Mathematical Sciences (FJMS) vol109 no 2 pp 243ndash259 2018
[11] O S Iyiola and G O Ojo ldquoOn the analytical solution ofFornberg-Whitham equation with the new fractional deriva-tiverdquo PramanamdashJournal of Physics vol 85 no 4 pp 567ndash5752015
[12] M Jneid and A Chaouk ldquoThe conformable reduced differ-ential transform method for solving Newell-Whitehead-Segelequation with non-integer orderrdquo Journal of Analysis andApplications 2019
[13] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015
[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 no 1 pp889ndash898 2015
[15] O Acan O Firat Y Keskin and G Oturanc ldquoSolution ofconformable fractional partial differential equations by reduceddifferential transform methodrdquo Selcuk Journal of Applied Math-ematics 2016
[16] H Thabet and S Kendre ldquoAnalytical solutions for conformablespace-time fractional partial differential equations via frac-tional differential transformrdquo Chaos Solitons amp Fractals vol109 pp 238ndash245 2018
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Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
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Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
2 International Journal of Mathematics and Mathematical Sciences
to test its efficiency In Section 4 we conclude this article byanalyzing the results we have obtained briefly
2 CFRDTM
In this section we introduce the CFRDTM which we willbe using to solve the system of conformable FPDEs in thispaper Throughout this section let x isin R119899 and q isin (0 1]Formore details about the definitions and properties given inthis section the reader can refer to [13ndash16] and the referencestherin
Definition 1 Given a function u R119899 times (0infin) 997888rarr R theconformable fractional partial derivative of 119906(119909 119905) of orderq with respect to t is given as
120597119902119905 119906 (119909 119905) = lim120576997888rarr0
119906 (119909 119905 + 1205761199051minus119902) minus 119906 (119909 119905)120576
(1)
provided that this limit exists as a finite number
Definition 2 Let 119906(119909 119905) be an infinitely partially q-differen-tiable function near 0 with respect to 119905 of order 119902 then theCFRDT of 119906(119909 119905) is given as
119880119902119896 (119909) =
1119902119896119896 [(120597
119902119905 )119896 119906 (119909 119905)]
119905=0(2)
where (120597119902119905 )119896119906(119909 119905) resembles applying the conformable par-tial fractional derivative 119896-times and 119880119902
119896(119909) is the CFRDT
function
Definition 3 Let 119880119902119896(119909) be the CFRDT of 119906(119909 119905) Then the
inverse CFRDT of 119880119902119896(119909) is given as
119906 (119909 119905) =infin
sum119896=0
119880119902119896 (119909) 119905119902119896
=infin
sum119896=0
1119902119896119896 [(120597
119902119905 )119896 119906 (119909 119905)]
119905=0119905119902119896
(3)
Definition 4 TheCFRDT of 119906(119909 119905) of the initial conditions isdefined as
119880119902119896 (119909) =
1(119902119896) [(120597
119902119905 )119896 119906]
119905=0if 119896119902 isin Z+
0 if 119896119902 notin Z+(4)
for 119896 = 0 1 2 (119899119902 minus 1) where 119899 is the order of CFPDETheorem5 Let 119906(119909 119905) V(119909 119905) and119908(119909 119905)R119899times[0infin) 997888rarr R
be partially q-differentiable at a point 119905 gt 0 and ab isin R Thenthe following is obtained
(1) If 119906(119909 119905) = V(119909 119905)119908(119909 119905) then 119880119902119896(119909) = sum119896119904=0 =
119881119902119896(119909 119904)119882119902
119896minus119904
(2) If V(119909 119905) = 120597119902119905 119906(119909 119905) then 119881119902119896(119909) = 119902(119896 + 1)119880119902
119896+1(119909)
(3) If 119906(119909 119905) = 119886V(119909 119905) plusmn 119887119908(119909 119905) then 119880119902119896(119909) = 119886119881119902
119896(119909) plusmn
119887119882119902119896(119909)
In general for 119906(119909 119905) = V1(119909 119905)V2(119909 119905) sdot sdot sdot V119899(119909 119905) then wehave
119880119902119896 (119909)
=119896119899
sum119896119899minus1=0
sdot sdot sdot1198962
sum1198961=0
119881119902(1)1198961119881119902
(2)1198962minus1198961times sdot sdot sdot 119881119902(119899minus1)119896119899minus1119881
119902
(119899)119896119899minus119896119899minus1
(5)
(4) If 119906(119909 119905) = 119905119898ℎ(119909) then 119880119902119896(119909) = 120575(119896 minus 119898119902)ℎ(119909)where
120575 (119896) =
1 if 119896 = 00 if 119896 = 0
(6)
3 Applications
To demonstrate the excellent performance of this method weapply it on four different systems in this section
Example 6 Consider the nonlinear system of CFDEs
120597119902119905 119906 = 1199062V minus 2119906 + 14 (12059721199092119906 + 12059721199102119906)
120597119902119905 V = 119906 minus 1199062V + 14 (12059721199092V + 12059721199102V)
(7)
with initial conditions
119906 (119909 119910 0) = 119890minus119909minus119910V (119909 119910 0) = 119890119909+119910
(8)
By applying CFRDTM on (7) we obtain the two recurrencerelations
119902 (119896 + 1)119880119902119896+1 =119896
sum119894=0
119894
sum119895=0
119880119902119894minus119895119880119902119895119881119902119896minus119894 minus 2119880119902119896
+ 14 (12059721199092119880119902119896 + 12059721199102119880119902119896)
119902 (119896 + 1) 119881119902119896+1 = 119880119902119896minus119896
sum119894=0
119894
sum119895=0
119880119902119894minus119895119880119902119895119881119902119896minus119894
+ 14 (12059721199092119881119902119896 + 12059721199102119881119902119896 )
(9)
where the IC (8) can be transformed at 119905 = 0 to1198801199020 = 119890minus119909minus119910
1198811199020 = 119890119909+119910(10)
Now substitute 119896 = 0 into (9) one obtains
1198801199021 =minus12119902 119890minus119909minus119910
1198811199021 =12119902119890119909+119910
(11)
International Journal of Mathematics and Mathematical Sciences 3
For k=1 and k=2 we get
1198801199022 =181199022 119890minus119909minus119910 = 1
2221199022 119890minus119909minus119910
1198811199022 =181199022 119890119909+119910 = 1
2221199022 119890119909+119910
(12)
and
1198801199023 =minus1481199023 119890
minus119909minus119910 = minus12331199023 119890
minus119909minus119910
1198811199023 =1
481199023 119890119909+119910 = 1
2331199023 119890119909+119910
(13)
Carrying on the iterative calculations in a similar way forother values of k we obtain the general terms
119880119902119899 =(minus1)1198992119899119899119902119899 119890
minus119909minus119910
119881119902119899 =1
2119899119899119902119899 119890119909+119910
(14)
Applying the inverse transformation of CFRDTM the solu-tion of (7) is given as
119906 (119909 119910 119905) =infin
sum119896=0
119880119902119896 (119909 119910 119905) 119905119896119902 = 119890minus119909minus119910minus1199051199022119902
V (119909 119910 119905) =infin
sum119896=0
119881119902119896 (119909 119910 119905) 119905119896119902 = 119890119909+119910+1199051199022119902(15)
Example 7 Consider the following system of CFDEs
120597119902119905 119906 = minus120597119909119906120597119909V minus 120597119910119906120597119910V minus 119906120597119902119905 V = minus120597119909V120597119909119908 + 120597119910V120597119910119908 + V
120597119902119905119908 = minus120597119909119908120597119909119906 minus 120597119910119908120597119910119906 minus 119908(16)
with the initial conditions
119906 (119909 119910 0) = 119890119909+119910V (119909 119910 0) = 119890119909minus119910119908 (119909 119910 0) = 119890minus119909+119910
(17)
Applying theorems of CFRDTM on (16) we obtain thefollowing system of recurrence relations
(119896 + 1)119880119902119896+1
= minus119896
sum119903=0
120597119909119880119902119896120597119909119881119902119896 minus119896
sum119903=0
120597119910119880119902119896120597119910119881119902119896 minus 119880119902119896
(119896 + 1) 119881119902119896+1 = minus119896
sum119903=0
120597119909119881119902119896 120597119909119882119902
119896+119896
sum119903=0
120597119910119881119902119896 120597119910119882119902
119896+ 119881119902119896
(119896 + 1)119882119902119896+1 = minus119896
sum119903=0
120597119909119882119902119896 120597119909119880119902119896 minus119896
sum119903=0
120597119910119882119902119896 120597119910119880119902119896 minus119882119902119896
(18)
where the IC can be transformed at 119905 = 0 to1198801199020 = 119890119909+119910
1198811199020 = 119890119909minus119910
1198821199020 = 119890minus119909+119910(19)
Substituting 119896 = 0 into (16) we obtain
1198801199021 =minus1119902 119890119909+119910
1198811199020 =1119902119890119909minus119910
1198821199020 =minus1119902 119890minus119909+119910
(20)
Similarly for k=1 and k=2 we get
1198801199022 =121199022 119890119909+119910
1198811199022 =121199022 119890119909minus119910
1198821199022 =121199022 119890minus119909+119910
(21)
and
1198801199023 = minus 161199023 119890119909+119910
1198811199023 =161199023 119890119909minus119910
1198821199023 = minus 161199023 119890minus119909+119910
(22)
Continuing the iterative calculations in a similar way forother values of k we obtain the general terms
119880119902119899 =(minus1)119899119899119902119899 119890
119909+119910
119881119902119899 =1
119899119902119899 119890119909minus119910
119882119902119899 =(minus1)119899119899119902119899 119890
minus119909+119910
(23)
Applying the inverse transformation of CFRDTM the solu-tion of (16) is given as
119906 (119909 119910 119905) =infin
sum119896=0
119880119902119896(119909 119910 119905) 119905119896119902 = 119890119909+119910minus119905119902119902
V (119909 119910 119905) =infin
sum119896=0
119881119902119896(119909 119910 119905) 119905119896119902 = 119890119909minus119910+119905119902119902
119908 (119909 119910 119905) =infin
sum119896=0
119882119902119896(119909 119910 119905) 119905119896119902 = 119890minus119909+119910minus119905119902119902
(24)
4 International Journal of Mathematics and Mathematical Sciences
Example 8 Consider the following system of CFDEs
120597119902119905 119906 + 119906120597119909119906 + V120597119910119906 = 12059721199092119906 + 12059721199102119906
120597119902119905 V + 119906120597119909V + V120597119910V = 12059721199092V + 12059721199102V(25)
subject to the initial conditions
119906 (119909 119910 0) = 119909 + 119910V (119909 119910 0) = 119909 minus 119910
(26)
Applying theorems of CFRDTM on (25) we obtain thefollowing system of recurrence relations
119902 (119896 + 1)119880119902119896+1 +119896
sum119903=0
119880119902119896minus119903120597119909119880119902119903 +119896
sum119903=0
119881119902119896minus119903120597119910119880119902119903
= 12059721199092119880119902119896 + 12059721199102119880119902119896
119902 (119896 + 1)119881119902119896+1 +119896
sum119903=0
119880119902119896minus119903120597119909119881119902119903 +119896
sum119903=0
119881119902119896minus119903120597119910119881119902119903
= 12059721199102119881119902119896 + +12059721199102119881119902119896
(27)
where the IC can be transformed at 119905 = 0 to1198801199020 = 119909 + 1199101198811199020 = 119909 minus 119910
(28)
Substituting 119896 = 0 into (25) we obtain1198801199021 =
minus2119902 119909
1198811199021 =minus2119902 119910
(29)
Similarly for 119896 = 1 119896 = 2 119896 = 3 and 119896 = 4 we getrespectively
1198801199022 =21199022 (119909 + 119910)
1198811199022 =21199022 (119909 minus 119910)
(30)
1198801199023 = minus 41199023119909
1198811199023 = minus 41199023119910
(31)
1198801199024 =41199023 (119909 + 119910)
1198811199024 = minus 41199023 (119909 minus 119910)
(32)
and
1198801199025 = minus 81199025119909
1198811199025 = minus 81199025119910
(33)
Applying the inverse transformation of CFRDTM the solu-tion of (25) is given as
119906 (119909 119910 119905) =infin
sum119896=0
119880119902119896(119909 119910 119905) 119905119896119902
= 119909 + 119910 minus 2119902119909119905119902 + 2
1199022 (119909 + 119910) 1199052119902 minus 411990231199091199053119902
+ 41199023 (119909 + 119910) 1199054119902 minus 8
11990251199091199055119902 + sdot sdot sdot
V (119909 119910 119905) =infin
sum119896=0
119881119902119896(119909 119910 119905) 119905119896119902
= 119909 minus 119910 minus 2119902119910119905119902 + 2
1199022 (119909 minus 119910) 1199052119902 minus 411990231199101199053119902
+ 41199023 (119909 minus 119910) 1199054119902 minus 8
11990251199101199055119902 + sdot sdot sdot
(34)
And by performing some computations the solutions aregiven as
119906 (119909 119910 119905) = 119909 + 119910 minus 21199091199051199021199021 minus 211990521199021199022
V (119909 119910 119905) = 119909 minus 119910 minus 21199091199051199021199021 minus 211990521199021199022
(35)
Example 9 Consider the following system of CFDEs
120597119902119905 119906 = V120597119909119906 + 120597119902119905 V120597119910119906 + 1 minus 119909 + 119910 + 119905119902119902
120597119902119905 V = 119906120597119909V + 120597119902119905 119906120597119910V + 1 minus 119909 minus 119910 minus 119905119902119902
(36)
subject to the initial conditions
119906 (119909 119910 0) = 119909 + 119910 minus 1V (119909 119910 0) = 119909 minus 119910 + 1
(37)
Applying theorems of CFRDTM on (25) we obtain thefollowing system of recurrence relations
119902 (119896 + 1)119880119902119896+1 =119896
sum119903=0
119881119902119903 120597119909119880119902119896minus119903
+119896
sum119903=0
119902 (119896 + 1) 119881119902119896+1minus119903120597119910119880119902119903 + 120575 (119896)
minus 119909 + 119910 minus 1 + 120575 (119896 minus 1)
119902 (119896 + 1) 119881119902119896+1 =119896
sum119903=0
119880119902119903 120597119909119880119902119896minus119903
+119896
sum119903=0
119902 (119896 + 1)119880119902119896+1minus119903120597119910119881119902119903 + 120575 (119896)
minus 119909 minus 119910 minus 1 + 120575 (119896 minus 1)
(38)
International Journal of Mathematics and Mathematical Sciences 5
00
00
00
00
00
00
minus05
minus05minus05
minus05minus05
minus10
minus15
minus20
05
05
05
15
05
0510
u(xy)v(xy)
Figure 1 The approximate solution of Example 9 for 120572 = 099 and t = 01
00
00
00
00
minus05
minus05minus05
minus05
minus10
minus15
05
05
00
00
minus0505
0515
05
10
u(xy) v(xy)
Figure 2The exact solution of Example 9 for 120572 = 099 and t = 01
where the IC can be transformed at 119905 = 0 to
1198801199020 = 119909 + 119910 minus 11198811199020 = 119909 minus 119910 + 1
(39)
Substituting 119896 = 0 into (36) we obtain
1198801199021 =1119902
1198811199021 = minus1119902
(40)
Similarly for 119896 = 1 and 119896 = 2 we get1198801199022 = minus 119909
2119902
1198811199022 =1119902 minus 119910 minus 1
2119902 (41)
and
1198801199023 =minus2119902119909 minus 119909
61199022 + 1 minus 5119902121199023
1198811199023 =minus1199103119902 minus 119902 + 1
121199023 (42)
Therefore the approximate analytical solution of (36) is givenas
119906 = 119909 + 119910 minus 1 + 119905119902119902 minus 1199091199052119902
2119902 + minus2119902119909 minus 11990961199022 1199053119902
+ 1 minus 5119902121199023 119905
3119902 + sdot sdot sdot
V = 119909 minus 119910 + 1 minus 119905119902119902 + 1119902 minus 119910 minus 1
2119902 1199052119902 + minus1199103119902 1199053119902
minus 119902 + 1121199023 119905
3119902 + sdot sdot sdot
(43)
Since this solution does not equal the exact one we usemathematica program to show its accuracy In Figures 1 and3 we plot the approximate solution for Example 9 for 119902 = 099at 119905 = 01 and at 119905 = 02 respectively The graphs in Figures2 and 4 illustrate the exact solution for Example 9 for 119902 =099 at 119905 = 01 and at 119905 = 02 respectively We can clearlyobserve at the time 119905 = 01 and 119905 = 02 that the values of theapproximate solution byCFRDTMand the values of the exactsolution obtained by other methods are extremely close Inother words solution provided by CFRDTM is highly preciseand accurate Particularly CFRDTM is a successful techniquefor solving systems of conformable FPDEs
6 International Journal of Mathematics and Mathematical Sciences
00
minus05
00
00
minus0505
0515
05
10
00
00
00
minus05
minus05
minus05
minus10
minus15
05
05
u(xy) v(xy)
Figure 3 The approximate solution of Example 9 for 120572 = 099 and t = 02
00
00
00
minus05
minus05
minus05minus10minus15
05
05
00
00
minus05
minus05
05
05
u(xy) v(xy)
00
15
05
10
Figure 4 The exact solution of Example 9 for 120572 = 099 and t = 02
4 Conclusion
In this paper our main purpose was to inspect the com-petence of the CFRDTM as a valid technique to solvesystems of nonlinear CFPDEs We successfully applied itto four distinct systems of nonlinear conformable time andspace FPDEs CFRDTM produced a numerical approximatesolution having the form of an infinite series that convergedto a closed form solution which coincided with the exactsolution in the first 3 applications In the 4119905ℎ applicationhowever we obtained an approximate solution that turnedout to be extremely close to the exact solution as we haveinterpreted from the attached Figures 1ndash4 On this basis weconclude that the CFRDTM is a powerful tool and a facileapproach for obtaining solutions for systems of CFPDEs Itreduces the amount of required computational work whencompared to other techniques used for this purpose and givessolutions that are in outstanding agreement with the exactones This provides a good starting point for further researchvia CFRDTM
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations North-HollandMathematics Studies New York NY USA 2006
[2] A S Elwakil ldquoFractional-order circuits and systems an emerg-ing interdisciplinary research areardquo IEEE Circuits and SystemsMagazine vol 10 no 4 pp 40ndash50 2010
[3] M D Ortigueira Fractional Calculus for Scientists and Engi-neers Heidelberg Springer Germany 2011
[4] T J Freeborn ldquoA survey of fractional-order circuit modelsfor biology and biomedicinerdquo IEEE Journal on Emerging andSelected Topics in Circuits and Systems vol 3 no 3 pp 416ndash4242013
[5] E C de Oliveira and J A Tenreiro Machado ldquoA review ofdefinitions for fractional derivatives and integralrdquoMathematicalProblems in Engineering vol 2014 Article ID 238459 6 pages2014
[6] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014
International Journal of Mathematics and Mathematical Sciences 7
[7] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationbydifferential transformmethodandAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006
[8] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers ampMathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011
[9] O Acan and D Baleanu ldquoA new numerical technique forsolving fractional partial differential equationsrdquo Miskolc Math-ematical Notes vol 19 no 1 pp 3ndash18 2018
[10] M Jneid and A El Chakik ldquoAnalytical solution for somesystems of nonlinear conformable fractional differential equa-tionsrdquo Far East Journal of Mathematical Sciences (FJMS) vol109 no 2 pp 243ndash259 2018
[11] O S Iyiola and G O Ojo ldquoOn the analytical solution ofFornberg-Whitham equation with the new fractional deriva-tiverdquo PramanamdashJournal of Physics vol 85 no 4 pp 567ndash5752015
[12] M Jneid and A Chaouk ldquoThe conformable reduced differ-ential transform method for solving Newell-Whitehead-Segelequation with non-integer orderrdquo Journal of Analysis andApplications 2019
[13] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015
[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 no 1 pp889ndash898 2015
[15] O Acan O Firat Y Keskin and G Oturanc ldquoSolution ofconformable fractional partial differential equations by reduceddifferential transform methodrdquo Selcuk Journal of Applied Math-ematics 2016
[16] H Thabet and S Kendre ldquoAnalytical solutions for conformablespace-time fractional partial differential equations via frac-tional differential transformrdquo Chaos Solitons amp Fractals vol109 pp 238ndash245 2018
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The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
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AnalysisInternational Journal of
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Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
International Journal of Mathematics and Mathematical Sciences 3
For k=1 and k=2 we get
1198801199022 =181199022 119890minus119909minus119910 = 1
2221199022 119890minus119909minus119910
1198811199022 =181199022 119890119909+119910 = 1
2221199022 119890119909+119910
(12)
and
1198801199023 =minus1481199023 119890
minus119909minus119910 = minus12331199023 119890
minus119909minus119910
1198811199023 =1
481199023 119890119909+119910 = 1
2331199023 119890119909+119910
(13)
Carrying on the iterative calculations in a similar way forother values of k we obtain the general terms
119880119902119899 =(minus1)1198992119899119899119902119899 119890
minus119909minus119910
119881119902119899 =1
2119899119899119902119899 119890119909+119910
(14)
Applying the inverse transformation of CFRDTM the solu-tion of (7) is given as
119906 (119909 119910 119905) =infin
sum119896=0
119880119902119896 (119909 119910 119905) 119905119896119902 = 119890minus119909minus119910minus1199051199022119902
V (119909 119910 119905) =infin
sum119896=0
119881119902119896 (119909 119910 119905) 119905119896119902 = 119890119909+119910+1199051199022119902(15)
Example 7 Consider the following system of CFDEs
120597119902119905 119906 = minus120597119909119906120597119909V minus 120597119910119906120597119910V minus 119906120597119902119905 V = minus120597119909V120597119909119908 + 120597119910V120597119910119908 + V
120597119902119905119908 = minus120597119909119908120597119909119906 minus 120597119910119908120597119910119906 minus 119908(16)
with the initial conditions
119906 (119909 119910 0) = 119890119909+119910V (119909 119910 0) = 119890119909minus119910119908 (119909 119910 0) = 119890minus119909+119910
(17)
Applying theorems of CFRDTM on (16) we obtain thefollowing system of recurrence relations
(119896 + 1)119880119902119896+1
= minus119896
sum119903=0
120597119909119880119902119896120597119909119881119902119896 minus119896
sum119903=0
120597119910119880119902119896120597119910119881119902119896 minus 119880119902119896
(119896 + 1) 119881119902119896+1 = minus119896
sum119903=0
120597119909119881119902119896 120597119909119882119902
119896+119896
sum119903=0
120597119910119881119902119896 120597119910119882119902
119896+ 119881119902119896
(119896 + 1)119882119902119896+1 = minus119896
sum119903=0
120597119909119882119902119896 120597119909119880119902119896 minus119896
sum119903=0
120597119910119882119902119896 120597119910119880119902119896 minus119882119902119896
(18)
where the IC can be transformed at 119905 = 0 to1198801199020 = 119890119909+119910
1198811199020 = 119890119909minus119910
1198821199020 = 119890minus119909+119910(19)
Substituting 119896 = 0 into (16) we obtain
1198801199021 =minus1119902 119890119909+119910
1198811199020 =1119902119890119909minus119910
1198821199020 =minus1119902 119890minus119909+119910
(20)
Similarly for k=1 and k=2 we get
1198801199022 =121199022 119890119909+119910
1198811199022 =121199022 119890119909minus119910
1198821199022 =121199022 119890minus119909+119910
(21)
and
1198801199023 = minus 161199023 119890119909+119910
1198811199023 =161199023 119890119909minus119910
1198821199023 = minus 161199023 119890minus119909+119910
(22)
Continuing the iterative calculations in a similar way forother values of k we obtain the general terms
119880119902119899 =(minus1)119899119899119902119899 119890
119909+119910
119881119902119899 =1
119899119902119899 119890119909minus119910
119882119902119899 =(minus1)119899119899119902119899 119890
minus119909+119910
(23)
Applying the inverse transformation of CFRDTM the solu-tion of (16) is given as
119906 (119909 119910 119905) =infin
sum119896=0
119880119902119896(119909 119910 119905) 119905119896119902 = 119890119909+119910minus119905119902119902
V (119909 119910 119905) =infin
sum119896=0
119881119902119896(119909 119910 119905) 119905119896119902 = 119890119909minus119910+119905119902119902
119908 (119909 119910 119905) =infin
sum119896=0
119882119902119896(119909 119910 119905) 119905119896119902 = 119890minus119909+119910minus119905119902119902
(24)
4 International Journal of Mathematics and Mathematical Sciences
Example 8 Consider the following system of CFDEs
120597119902119905 119906 + 119906120597119909119906 + V120597119910119906 = 12059721199092119906 + 12059721199102119906
120597119902119905 V + 119906120597119909V + V120597119910V = 12059721199092V + 12059721199102V(25)
subject to the initial conditions
119906 (119909 119910 0) = 119909 + 119910V (119909 119910 0) = 119909 minus 119910
(26)
Applying theorems of CFRDTM on (25) we obtain thefollowing system of recurrence relations
119902 (119896 + 1)119880119902119896+1 +119896
sum119903=0
119880119902119896minus119903120597119909119880119902119903 +119896
sum119903=0
119881119902119896minus119903120597119910119880119902119903
= 12059721199092119880119902119896 + 12059721199102119880119902119896
119902 (119896 + 1)119881119902119896+1 +119896
sum119903=0
119880119902119896minus119903120597119909119881119902119903 +119896
sum119903=0
119881119902119896minus119903120597119910119881119902119903
= 12059721199102119881119902119896 + +12059721199102119881119902119896
(27)
where the IC can be transformed at 119905 = 0 to1198801199020 = 119909 + 1199101198811199020 = 119909 minus 119910
(28)
Substituting 119896 = 0 into (25) we obtain1198801199021 =
minus2119902 119909
1198811199021 =minus2119902 119910
(29)
Similarly for 119896 = 1 119896 = 2 119896 = 3 and 119896 = 4 we getrespectively
1198801199022 =21199022 (119909 + 119910)
1198811199022 =21199022 (119909 minus 119910)
(30)
1198801199023 = minus 41199023119909
1198811199023 = minus 41199023119910
(31)
1198801199024 =41199023 (119909 + 119910)
1198811199024 = minus 41199023 (119909 minus 119910)
(32)
and
1198801199025 = minus 81199025119909
1198811199025 = minus 81199025119910
(33)
Applying the inverse transformation of CFRDTM the solu-tion of (25) is given as
119906 (119909 119910 119905) =infin
sum119896=0
119880119902119896(119909 119910 119905) 119905119896119902
= 119909 + 119910 minus 2119902119909119905119902 + 2
1199022 (119909 + 119910) 1199052119902 minus 411990231199091199053119902
+ 41199023 (119909 + 119910) 1199054119902 minus 8
11990251199091199055119902 + sdot sdot sdot
V (119909 119910 119905) =infin
sum119896=0
119881119902119896(119909 119910 119905) 119905119896119902
= 119909 minus 119910 minus 2119902119910119905119902 + 2
1199022 (119909 minus 119910) 1199052119902 minus 411990231199101199053119902
+ 41199023 (119909 minus 119910) 1199054119902 minus 8
11990251199101199055119902 + sdot sdot sdot
(34)
And by performing some computations the solutions aregiven as
119906 (119909 119910 119905) = 119909 + 119910 minus 21199091199051199021199021 minus 211990521199021199022
V (119909 119910 119905) = 119909 minus 119910 minus 21199091199051199021199021 minus 211990521199021199022
(35)
Example 9 Consider the following system of CFDEs
120597119902119905 119906 = V120597119909119906 + 120597119902119905 V120597119910119906 + 1 minus 119909 + 119910 + 119905119902119902
120597119902119905 V = 119906120597119909V + 120597119902119905 119906120597119910V + 1 minus 119909 minus 119910 minus 119905119902119902
(36)
subject to the initial conditions
119906 (119909 119910 0) = 119909 + 119910 minus 1V (119909 119910 0) = 119909 minus 119910 + 1
(37)
Applying theorems of CFRDTM on (25) we obtain thefollowing system of recurrence relations
119902 (119896 + 1)119880119902119896+1 =119896
sum119903=0
119881119902119903 120597119909119880119902119896minus119903
+119896
sum119903=0
119902 (119896 + 1) 119881119902119896+1minus119903120597119910119880119902119903 + 120575 (119896)
minus 119909 + 119910 minus 1 + 120575 (119896 minus 1)
119902 (119896 + 1) 119881119902119896+1 =119896
sum119903=0
119880119902119903 120597119909119880119902119896minus119903
+119896
sum119903=0
119902 (119896 + 1)119880119902119896+1minus119903120597119910119881119902119903 + 120575 (119896)
minus 119909 minus 119910 minus 1 + 120575 (119896 minus 1)
(38)
International Journal of Mathematics and Mathematical Sciences 5
00
00
00
00
00
00
minus05
minus05minus05
minus05minus05
minus10
minus15
minus20
05
05
05
15
05
0510
u(xy)v(xy)
Figure 1 The approximate solution of Example 9 for 120572 = 099 and t = 01
00
00
00
00
minus05
minus05minus05
minus05
minus10
minus15
05
05
00
00
minus0505
0515
05
10
u(xy) v(xy)
Figure 2The exact solution of Example 9 for 120572 = 099 and t = 01
where the IC can be transformed at 119905 = 0 to
1198801199020 = 119909 + 119910 minus 11198811199020 = 119909 minus 119910 + 1
(39)
Substituting 119896 = 0 into (36) we obtain
1198801199021 =1119902
1198811199021 = minus1119902
(40)
Similarly for 119896 = 1 and 119896 = 2 we get1198801199022 = minus 119909
2119902
1198811199022 =1119902 minus 119910 minus 1
2119902 (41)
and
1198801199023 =minus2119902119909 minus 119909
61199022 + 1 minus 5119902121199023
1198811199023 =minus1199103119902 minus 119902 + 1
121199023 (42)
Therefore the approximate analytical solution of (36) is givenas
119906 = 119909 + 119910 minus 1 + 119905119902119902 minus 1199091199052119902
2119902 + minus2119902119909 minus 11990961199022 1199053119902
+ 1 minus 5119902121199023 119905
3119902 + sdot sdot sdot
V = 119909 minus 119910 + 1 minus 119905119902119902 + 1119902 minus 119910 minus 1
2119902 1199052119902 + minus1199103119902 1199053119902
minus 119902 + 1121199023 119905
3119902 + sdot sdot sdot
(43)
Since this solution does not equal the exact one we usemathematica program to show its accuracy In Figures 1 and3 we plot the approximate solution for Example 9 for 119902 = 099at 119905 = 01 and at 119905 = 02 respectively The graphs in Figures2 and 4 illustrate the exact solution for Example 9 for 119902 =099 at 119905 = 01 and at 119905 = 02 respectively We can clearlyobserve at the time 119905 = 01 and 119905 = 02 that the values of theapproximate solution byCFRDTMand the values of the exactsolution obtained by other methods are extremely close Inother words solution provided by CFRDTM is highly preciseand accurate Particularly CFRDTM is a successful techniquefor solving systems of conformable FPDEs
6 International Journal of Mathematics and Mathematical Sciences
00
minus05
00
00
minus0505
0515
05
10
00
00
00
minus05
minus05
minus05
minus10
minus15
05
05
u(xy) v(xy)
Figure 3 The approximate solution of Example 9 for 120572 = 099 and t = 02
00
00
00
minus05
minus05
minus05minus10minus15
05
05
00
00
minus05
minus05
05
05
u(xy) v(xy)
00
15
05
10
Figure 4 The exact solution of Example 9 for 120572 = 099 and t = 02
4 Conclusion
In this paper our main purpose was to inspect the com-petence of the CFRDTM as a valid technique to solvesystems of nonlinear CFPDEs We successfully applied itto four distinct systems of nonlinear conformable time andspace FPDEs CFRDTM produced a numerical approximatesolution having the form of an infinite series that convergedto a closed form solution which coincided with the exactsolution in the first 3 applications In the 4119905ℎ applicationhowever we obtained an approximate solution that turnedout to be extremely close to the exact solution as we haveinterpreted from the attached Figures 1ndash4 On this basis weconclude that the CFRDTM is a powerful tool and a facileapproach for obtaining solutions for systems of CFPDEs Itreduces the amount of required computational work whencompared to other techniques used for this purpose and givessolutions that are in outstanding agreement with the exactones This provides a good starting point for further researchvia CFRDTM
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations North-HollandMathematics Studies New York NY USA 2006
[2] A S Elwakil ldquoFractional-order circuits and systems an emerg-ing interdisciplinary research areardquo IEEE Circuits and SystemsMagazine vol 10 no 4 pp 40ndash50 2010
[3] M D Ortigueira Fractional Calculus for Scientists and Engi-neers Heidelberg Springer Germany 2011
[4] T J Freeborn ldquoA survey of fractional-order circuit modelsfor biology and biomedicinerdquo IEEE Journal on Emerging andSelected Topics in Circuits and Systems vol 3 no 3 pp 416ndash4242013
[5] E C de Oliveira and J A Tenreiro Machado ldquoA review ofdefinitions for fractional derivatives and integralrdquoMathematicalProblems in Engineering vol 2014 Article ID 238459 6 pages2014
[6] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014
International Journal of Mathematics and Mathematical Sciences 7
[7] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationbydifferential transformmethodandAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006
[8] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers ampMathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011
[9] O Acan and D Baleanu ldquoA new numerical technique forsolving fractional partial differential equationsrdquo Miskolc Math-ematical Notes vol 19 no 1 pp 3ndash18 2018
[10] M Jneid and A El Chakik ldquoAnalytical solution for somesystems of nonlinear conformable fractional differential equa-tionsrdquo Far East Journal of Mathematical Sciences (FJMS) vol109 no 2 pp 243ndash259 2018
[11] O S Iyiola and G O Ojo ldquoOn the analytical solution ofFornberg-Whitham equation with the new fractional deriva-tiverdquo PramanamdashJournal of Physics vol 85 no 4 pp 567ndash5752015
[12] M Jneid and A Chaouk ldquoThe conformable reduced differ-ential transform method for solving Newell-Whitehead-Segelequation with non-integer orderrdquo Journal of Analysis andApplications 2019
[13] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015
[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 no 1 pp889ndash898 2015
[15] O Acan O Firat Y Keskin and G Oturanc ldquoSolution ofconformable fractional partial differential equations by reduceddifferential transform methodrdquo Selcuk Journal of Applied Math-ematics 2016
[16] H Thabet and S Kendre ldquoAnalytical solutions for conformablespace-time fractional partial differential equations via frac-tional differential transformrdquo Chaos Solitons amp Fractals vol109 pp 238ndash245 2018
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4 International Journal of Mathematics and Mathematical Sciences
Example 8 Consider the following system of CFDEs
120597119902119905 119906 + 119906120597119909119906 + V120597119910119906 = 12059721199092119906 + 12059721199102119906
120597119902119905 V + 119906120597119909V + V120597119910V = 12059721199092V + 12059721199102V(25)
subject to the initial conditions
119906 (119909 119910 0) = 119909 + 119910V (119909 119910 0) = 119909 minus 119910
(26)
Applying theorems of CFRDTM on (25) we obtain thefollowing system of recurrence relations
119902 (119896 + 1)119880119902119896+1 +119896
sum119903=0
119880119902119896minus119903120597119909119880119902119903 +119896
sum119903=0
119881119902119896minus119903120597119910119880119902119903
= 12059721199092119880119902119896 + 12059721199102119880119902119896
119902 (119896 + 1)119881119902119896+1 +119896
sum119903=0
119880119902119896minus119903120597119909119881119902119903 +119896
sum119903=0
119881119902119896minus119903120597119910119881119902119903
= 12059721199102119881119902119896 + +12059721199102119881119902119896
(27)
where the IC can be transformed at 119905 = 0 to1198801199020 = 119909 + 1199101198811199020 = 119909 minus 119910
(28)
Substituting 119896 = 0 into (25) we obtain1198801199021 =
minus2119902 119909
1198811199021 =minus2119902 119910
(29)
Similarly for 119896 = 1 119896 = 2 119896 = 3 and 119896 = 4 we getrespectively
1198801199022 =21199022 (119909 + 119910)
1198811199022 =21199022 (119909 minus 119910)
(30)
1198801199023 = minus 41199023119909
1198811199023 = minus 41199023119910
(31)
1198801199024 =41199023 (119909 + 119910)
1198811199024 = minus 41199023 (119909 minus 119910)
(32)
and
1198801199025 = minus 81199025119909
1198811199025 = minus 81199025119910
(33)
Applying the inverse transformation of CFRDTM the solu-tion of (25) is given as
119906 (119909 119910 119905) =infin
sum119896=0
119880119902119896(119909 119910 119905) 119905119896119902
= 119909 + 119910 minus 2119902119909119905119902 + 2
1199022 (119909 + 119910) 1199052119902 minus 411990231199091199053119902
+ 41199023 (119909 + 119910) 1199054119902 minus 8
11990251199091199055119902 + sdot sdot sdot
V (119909 119910 119905) =infin
sum119896=0
119881119902119896(119909 119910 119905) 119905119896119902
= 119909 minus 119910 minus 2119902119910119905119902 + 2
1199022 (119909 minus 119910) 1199052119902 minus 411990231199101199053119902
+ 41199023 (119909 minus 119910) 1199054119902 minus 8
11990251199101199055119902 + sdot sdot sdot
(34)
And by performing some computations the solutions aregiven as
119906 (119909 119910 119905) = 119909 + 119910 minus 21199091199051199021199021 minus 211990521199021199022
V (119909 119910 119905) = 119909 minus 119910 minus 21199091199051199021199021 minus 211990521199021199022
(35)
Example 9 Consider the following system of CFDEs
120597119902119905 119906 = V120597119909119906 + 120597119902119905 V120597119910119906 + 1 minus 119909 + 119910 + 119905119902119902
120597119902119905 V = 119906120597119909V + 120597119902119905 119906120597119910V + 1 minus 119909 minus 119910 minus 119905119902119902
(36)
subject to the initial conditions
119906 (119909 119910 0) = 119909 + 119910 minus 1V (119909 119910 0) = 119909 minus 119910 + 1
(37)
Applying theorems of CFRDTM on (25) we obtain thefollowing system of recurrence relations
119902 (119896 + 1)119880119902119896+1 =119896
sum119903=0
119881119902119903 120597119909119880119902119896minus119903
+119896
sum119903=0
119902 (119896 + 1) 119881119902119896+1minus119903120597119910119880119902119903 + 120575 (119896)
minus 119909 + 119910 minus 1 + 120575 (119896 minus 1)
119902 (119896 + 1) 119881119902119896+1 =119896
sum119903=0
119880119902119903 120597119909119880119902119896minus119903
+119896
sum119903=0
119902 (119896 + 1)119880119902119896+1minus119903120597119910119881119902119903 + 120575 (119896)
minus 119909 minus 119910 minus 1 + 120575 (119896 minus 1)
(38)
International Journal of Mathematics and Mathematical Sciences 5
00
00
00
00
00
00
minus05
minus05minus05
minus05minus05
minus10
minus15
minus20
05
05
05
15
05
0510
u(xy)v(xy)
Figure 1 The approximate solution of Example 9 for 120572 = 099 and t = 01
00
00
00
00
minus05
minus05minus05
minus05
minus10
minus15
05
05
00
00
minus0505
0515
05
10
u(xy) v(xy)
Figure 2The exact solution of Example 9 for 120572 = 099 and t = 01
where the IC can be transformed at 119905 = 0 to
1198801199020 = 119909 + 119910 minus 11198811199020 = 119909 minus 119910 + 1
(39)
Substituting 119896 = 0 into (36) we obtain
1198801199021 =1119902
1198811199021 = minus1119902
(40)
Similarly for 119896 = 1 and 119896 = 2 we get1198801199022 = minus 119909
2119902
1198811199022 =1119902 minus 119910 minus 1
2119902 (41)
and
1198801199023 =minus2119902119909 minus 119909
61199022 + 1 minus 5119902121199023
1198811199023 =minus1199103119902 minus 119902 + 1
121199023 (42)
Therefore the approximate analytical solution of (36) is givenas
119906 = 119909 + 119910 minus 1 + 119905119902119902 minus 1199091199052119902
2119902 + minus2119902119909 minus 11990961199022 1199053119902
+ 1 minus 5119902121199023 119905
3119902 + sdot sdot sdot
V = 119909 minus 119910 + 1 minus 119905119902119902 + 1119902 minus 119910 minus 1
2119902 1199052119902 + minus1199103119902 1199053119902
minus 119902 + 1121199023 119905
3119902 + sdot sdot sdot
(43)
Since this solution does not equal the exact one we usemathematica program to show its accuracy In Figures 1 and3 we plot the approximate solution for Example 9 for 119902 = 099at 119905 = 01 and at 119905 = 02 respectively The graphs in Figures2 and 4 illustrate the exact solution for Example 9 for 119902 =099 at 119905 = 01 and at 119905 = 02 respectively We can clearlyobserve at the time 119905 = 01 and 119905 = 02 that the values of theapproximate solution byCFRDTMand the values of the exactsolution obtained by other methods are extremely close Inother words solution provided by CFRDTM is highly preciseand accurate Particularly CFRDTM is a successful techniquefor solving systems of conformable FPDEs
6 International Journal of Mathematics and Mathematical Sciences
00
minus05
00
00
minus0505
0515
05
10
00
00
00
minus05
minus05
minus05
minus10
minus15
05
05
u(xy) v(xy)
Figure 3 The approximate solution of Example 9 for 120572 = 099 and t = 02
00
00
00
minus05
minus05
minus05minus10minus15
05
05
00
00
minus05
minus05
05
05
u(xy) v(xy)
00
15
05
10
Figure 4 The exact solution of Example 9 for 120572 = 099 and t = 02
4 Conclusion
In this paper our main purpose was to inspect the com-petence of the CFRDTM as a valid technique to solvesystems of nonlinear CFPDEs We successfully applied itto four distinct systems of nonlinear conformable time andspace FPDEs CFRDTM produced a numerical approximatesolution having the form of an infinite series that convergedto a closed form solution which coincided with the exactsolution in the first 3 applications In the 4119905ℎ applicationhowever we obtained an approximate solution that turnedout to be extremely close to the exact solution as we haveinterpreted from the attached Figures 1ndash4 On this basis weconclude that the CFRDTM is a powerful tool and a facileapproach for obtaining solutions for systems of CFPDEs Itreduces the amount of required computational work whencompared to other techniques used for this purpose and givessolutions that are in outstanding agreement with the exactones This provides a good starting point for further researchvia CFRDTM
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations North-HollandMathematics Studies New York NY USA 2006
[2] A S Elwakil ldquoFractional-order circuits and systems an emerg-ing interdisciplinary research areardquo IEEE Circuits and SystemsMagazine vol 10 no 4 pp 40ndash50 2010
[3] M D Ortigueira Fractional Calculus for Scientists and Engi-neers Heidelberg Springer Germany 2011
[4] T J Freeborn ldquoA survey of fractional-order circuit modelsfor biology and biomedicinerdquo IEEE Journal on Emerging andSelected Topics in Circuits and Systems vol 3 no 3 pp 416ndash4242013
[5] E C de Oliveira and J A Tenreiro Machado ldquoA review ofdefinitions for fractional derivatives and integralrdquoMathematicalProblems in Engineering vol 2014 Article ID 238459 6 pages2014
[6] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014
International Journal of Mathematics and Mathematical Sciences 7
[7] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationbydifferential transformmethodandAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006
[8] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers ampMathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011
[9] O Acan and D Baleanu ldquoA new numerical technique forsolving fractional partial differential equationsrdquo Miskolc Math-ematical Notes vol 19 no 1 pp 3ndash18 2018
[10] M Jneid and A El Chakik ldquoAnalytical solution for somesystems of nonlinear conformable fractional differential equa-tionsrdquo Far East Journal of Mathematical Sciences (FJMS) vol109 no 2 pp 243ndash259 2018
[11] O S Iyiola and G O Ojo ldquoOn the analytical solution ofFornberg-Whitham equation with the new fractional deriva-tiverdquo PramanamdashJournal of Physics vol 85 no 4 pp 567ndash5752015
[12] M Jneid and A Chaouk ldquoThe conformable reduced differ-ential transform method for solving Newell-Whitehead-Segelequation with non-integer orderrdquo Journal of Analysis andApplications 2019
[13] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015
[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 no 1 pp889ndash898 2015
[15] O Acan O Firat Y Keskin and G Oturanc ldquoSolution ofconformable fractional partial differential equations by reduceddifferential transform methodrdquo Selcuk Journal of Applied Math-ematics 2016
[16] H Thabet and S Kendre ldquoAnalytical solutions for conformablespace-time fractional partial differential equations via frac-tional differential transformrdquo Chaos Solitons amp Fractals vol109 pp 238ndash245 2018
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
International Journal of Mathematics and Mathematical Sciences 5
00
00
00
00
00
00
minus05
minus05minus05
minus05minus05
minus10
minus15
minus20
05
05
05
15
05
0510
u(xy)v(xy)
Figure 1 The approximate solution of Example 9 for 120572 = 099 and t = 01
00
00
00
00
minus05
minus05minus05
minus05
minus10
minus15
05
05
00
00
minus0505
0515
05
10
u(xy) v(xy)
Figure 2The exact solution of Example 9 for 120572 = 099 and t = 01
where the IC can be transformed at 119905 = 0 to
1198801199020 = 119909 + 119910 minus 11198811199020 = 119909 minus 119910 + 1
(39)
Substituting 119896 = 0 into (36) we obtain
1198801199021 =1119902
1198811199021 = minus1119902
(40)
Similarly for 119896 = 1 and 119896 = 2 we get1198801199022 = minus 119909
2119902
1198811199022 =1119902 minus 119910 minus 1
2119902 (41)
and
1198801199023 =minus2119902119909 minus 119909
61199022 + 1 minus 5119902121199023
1198811199023 =minus1199103119902 minus 119902 + 1
121199023 (42)
Therefore the approximate analytical solution of (36) is givenas
119906 = 119909 + 119910 minus 1 + 119905119902119902 minus 1199091199052119902
2119902 + minus2119902119909 minus 11990961199022 1199053119902
+ 1 minus 5119902121199023 119905
3119902 + sdot sdot sdot
V = 119909 minus 119910 + 1 minus 119905119902119902 + 1119902 minus 119910 minus 1
2119902 1199052119902 + minus1199103119902 1199053119902
minus 119902 + 1121199023 119905
3119902 + sdot sdot sdot
(43)
Since this solution does not equal the exact one we usemathematica program to show its accuracy In Figures 1 and3 we plot the approximate solution for Example 9 for 119902 = 099at 119905 = 01 and at 119905 = 02 respectively The graphs in Figures2 and 4 illustrate the exact solution for Example 9 for 119902 =099 at 119905 = 01 and at 119905 = 02 respectively We can clearlyobserve at the time 119905 = 01 and 119905 = 02 that the values of theapproximate solution byCFRDTMand the values of the exactsolution obtained by other methods are extremely close Inother words solution provided by CFRDTM is highly preciseand accurate Particularly CFRDTM is a successful techniquefor solving systems of conformable FPDEs
6 International Journal of Mathematics and Mathematical Sciences
00
minus05
00
00
minus0505
0515
05
10
00
00
00
minus05
minus05
minus05
minus10
minus15
05
05
u(xy) v(xy)
Figure 3 The approximate solution of Example 9 for 120572 = 099 and t = 02
00
00
00
minus05
minus05
minus05minus10minus15
05
05
00
00
minus05
minus05
05
05
u(xy) v(xy)
00
15
05
10
Figure 4 The exact solution of Example 9 for 120572 = 099 and t = 02
4 Conclusion
In this paper our main purpose was to inspect the com-petence of the CFRDTM as a valid technique to solvesystems of nonlinear CFPDEs We successfully applied itto four distinct systems of nonlinear conformable time andspace FPDEs CFRDTM produced a numerical approximatesolution having the form of an infinite series that convergedto a closed form solution which coincided with the exactsolution in the first 3 applications In the 4119905ℎ applicationhowever we obtained an approximate solution that turnedout to be extremely close to the exact solution as we haveinterpreted from the attached Figures 1ndash4 On this basis weconclude that the CFRDTM is a powerful tool and a facileapproach for obtaining solutions for systems of CFPDEs Itreduces the amount of required computational work whencompared to other techniques used for this purpose and givessolutions that are in outstanding agreement with the exactones This provides a good starting point for further researchvia CFRDTM
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations North-HollandMathematics Studies New York NY USA 2006
[2] A S Elwakil ldquoFractional-order circuits and systems an emerg-ing interdisciplinary research areardquo IEEE Circuits and SystemsMagazine vol 10 no 4 pp 40ndash50 2010
[3] M D Ortigueira Fractional Calculus for Scientists and Engi-neers Heidelberg Springer Germany 2011
[4] T J Freeborn ldquoA survey of fractional-order circuit modelsfor biology and biomedicinerdquo IEEE Journal on Emerging andSelected Topics in Circuits and Systems vol 3 no 3 pp 416ndash4242013
[5] E C de Oliveira and J A Tenreiro Machado ldquoA review ofdefinitions for fractional derivatives and integralrdquoMathematicalProblems in Engineering vol 2014 Article ID 238459 6 pages2014
[6] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014
International Journal of Mathematics and Mathematical Sciences 7
[7] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationbydifferential transformmethodandAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006
[8] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers ampMathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011
[9] O Acan and D Baleanu ldquoA new numerical technique forsolving fractional partial differential equationsrdquo Miskolc Math-ematical Notes vol 19 no 1 pp 3ndash18 2018
[10] M Jneid and A El Chakik ldquoAnalytical solution for somesystems of nonlinear conformable fractional differential equa-tionsrdquo Far East Journal of Mathematical Sciences (FJMS) vol109 no 2 pp 243ndash259 2018
[11] O S Iyiola and G O Ojo ldquoOn the analytical solution ofFornberg-Whitham equation with the new fractional deriva-tiverdquo PramanamdashJournal of Physics vol 85 no 4 pp 567ndash5752015
[12] M Jneid and A Chaouk ldquoThe conformable reduced differ-ential transform method for solving Newell-Whitehead-Segelequation with non-integer orderrdquo Journal of Analysis andApplications 2019
[13] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015
[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 no 1 pp889ndash898 2015
[15] O Acan O Firat Y Keskin and G Oturanc ldquoSolution ofconformable fractional partial differential equations by reduceddifferential transform methodrdquo Selcuk Journal of Applied Math-ematics 2016
[16] H Thabet and S Kendre ldquoAnalytical solutions for conformablespace-time fractional partial differential equations via frac-tional differential transformrdquo Chaos Solitons amp Fractals vol109 pp 238ndash245 2018
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
6 International Journal of Mathematics and Mathematical Sciences
00
minus05
00
00
minus0505
0515
05
10
00
00
00
minus05
minus05
minus05
minus10
minus15
05
05
u(xy) v(xy)
Figure 3 The approximate solution of Example 9 for 120572 = 099 and t = 02
00
00
00
minus05
minus05
minus05minus10minus15
05
05
00
00
minus05
minus05
05
05
u(xy) v(xy)
00
15
05
10
Figure 4 The exact solution of Example 9 for 120572 = 099 and t = 02
4 Conclusion
In this paper our main purpose was to inspect the com-petence of the CFRDTM as a valid technique to solvesystems of nonlinear CFPDEs We successfully applied itto four distinct systems of nonlinear conformable time andspace FPDEs CFRDTM produced a numerical approximatesolution having the form of an infinite series that convergedto a closed form solution which coincided with the exactsolution in the first 3 applications In the 4119905ℎ applicationhowever we obtained an approximate solution that turnedout to be extremely close to the exact solution as we haveinterpreted from the attached Figures 1ndash4 On this basis weconclude that the CFRDTM is a powerful tool and a facileapproach for obtaining solutions for systems of CFPDEs Itreduces the amount of required computational work whencompared to other techniques used for this purpose and givessolutions that are in outstanding agreement with the exactones This provides a good starting point for further researchvia CFRDTM
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
References
[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations North-HollandMathematics Studies New York NY USA 2006
[2] A S Elwakil ldquoFractional-order circuits and systems an emerg-ing interdisciplinary research areardquo IEEE Circuits and SystemsMagazine vol 10 no 4 pp 40ndash50 2010
[3] M D Ortigueira Fractional Calculus for Scientists and Engi-neers Heidelberg Springer Germany 2011
[4] T J Freeborn ldquoA survey of fractional-order circuit modelsfor biology and biomedicinerdquo IEEE Journal on Emerging andSelected Topics in Circuits and Systems vol 3 no 3 pp 416ndash4242013
[5] E C de Oliveira and J A Tenreiro Machado ldquoA review ofdefinitions for fractional derivatives and integralrdquoMathematicalProblems in Engineering vol 2014 Article ID 238459 6 pages2014
[6] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014
International Journal of Mathematics and Mathematical Sciences 7
[7] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationbydifferential transformmethodandAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006
[8] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers ampMathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011
[9] O Acan and D Baleanu ldquoA new numerical technique forsolving fractional partial differential equationsrdquo Miskolc Math-ematical Notes vol 19 no 1 pp 3ndash18 2018
[10] M Jneid and A El Chakik ldquoAnalytical solution for somesystems of nonlinear conformable fractional differential equa-tionsrdquo Far East Journal of Mathematical Sciences (FJMS) vol109 no 2 pp 243ndash259 2018
[11] O S Iyiola and G O Ojo ldquoOn the analytical solution ofFornberg-Whitham equation with the new fractional deriva-tiverdquo PramanamdashJournal of Physics vol 85 no 4 pp 567ndash5752015
[12] M Jneid and A Chaouk ldquoThe conformable reduced differ-ential transform method for solving Newell-Whitehead-Segelequation with non-integer orderrdquo Journal of Analysis andApplications 2019
[13] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015
[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 no 1 pp889ndash898 2015
[15] O Acan O Firat Y Keskin and G Oturanc ldquoSolution ofconformable fractional partial differential equations by reduceddifferential transform methodrdquo Selcuk Journal of Applied Math-ematics 2016
[16] H Thabet and S Kendre ldquoAnalytical solutions for conformablespace-time fractional partial differential equations via frac-tional differential transformrdquo Chaos Solitons amp Fractals vol109 pp 238ndash245 2018
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
International Journal of Mathematics and Mathematical Sciences 7
[7] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationbydifferential transformmethodandAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006
[8] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers ampMathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011
[9] O Acan and D Baleanu ldquoA new numerical technique forsolving fractional partial differential equationsrdquo Miskolc Math-ematical Notes vol 19 no 1 pp 3ndash18 2018
[10] M Jneid and A El Chakik ldquoAnalytical solution for somesystems of nonlinear conformable fractional differential equa-tionsrdquo Far East Journal of Mathematical Sciences (FJMS) vol109 no 2 pp 243ndash259 2018
[11] O S Iyiola and G O Ojo ldquoOn the analytical solution ofFornberg-Whitham equation with the new fractional deriva-tiverdquo PramanamdashJournal of Physics vol 85 no 4 pp 567ndash5752015
[12] M Jneid and A Chaouk ldquoThe conformable reduced differ-ential transform method for solving Newell-Whitehead-Segelequation with non-integer orderrdquo Journal of Analysis andApplications 2019
[13] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015
[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 no 1 pp889ndash898 2015
[15] O Acan O Firat Y Keskin and G Oturanc ldquoSolution ofconformable fractional partial differential equations by reduceddifferential transform methodrdquo Selcuk Journal of Applied Math-ematics 2016
[16] H Thabet and S Kendre ldquoAnalytical solutions for conformablespace-time fractional partial differential equations via frac-tional differential transformrdquo Chaos Solitons amp Fractals vol109 pp 238ndash245 2018
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom