GENERALIZED DIFFERENTIAL TRANSFORM METHOD FOR SOLUTIONS OF NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

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1 December 7 Volume Issue JETIR (ISSN-39-56) GENERALIZED DIFFERENTIAL TRANSFORM METHOD FOR SOLTIONS OF NON-LINEAR PARTIAL DIFFERENTIAL EQATIONS OF FRACTIONAL ORDER Deepanjan Das Department of Matematics Gani Kan Coudury Institute of Engineering and Tecnology NarayanpurMaldaWest Bengal-73India Abstract: In te present paper Generalized Differential Transform Metod (GDTM) is used for obtaining te approximate analytic solutions of non-linear partial differential equations of fractional order. Te fractional derivatives are described in te Caputo sense. Keywords: Fractional differential equations Caputo fractional derivative Generalized Differential transform metod Analytic solution. Matematical Subject Classification (): 6A33 3A8 35A 35R 35C 7H.. Introduction: Differential equations wit fractional order are generalizations of classical differential equations of integer order and ave recently been proved to be valuable tools in te modeling of many pysical penomena in various fields of science and engineering. By using fractional derivatives a lot of wors ave been done for a better description of considered material properties. Based on enanced reological models Matematical modeling naturally leads to differential equations of fractional order and to te necessity of te formulation of te initial conditions to suc equations. Recently various analytical and numerical metods ave been employed to solve linear and nonlinear fractional differential equations. Te differential transform metod (DTM) was proposed by Zou [] to solve linear and nonlinear initial value problems in electric circuit analysis. Tis metod as been used for solving various types of equations by many autors [-5]. DTM constructs an analytical solution in te form of a polynomial and different from te traditional iger order Taylor series metod. For solving two-dimensional linear and nonlinear partial differential equations of fractional order DTM is furter developed as Generalized Differential Transform Metod (GDTM) by Momani Odibat and Ertur in teir papers [6-8].Recently Vedat Suat Ertiira and Saer Momanib applied generalized differential transform metod to solve fractional integro-differential equations [9]. Te GDTM is implemented to derive te solution of space-time fractional telegrap equation by Mridula GargPratiba Manoar and Syam L.Kalla []. Manis Kumar BansalRasmi Jain applied generalized differential transform metod to solve fractional order Riccati differential equation []. Aysegul Cetinaya Onur Kiymaz and Jale Camli applied generalized differential transform metod to solve non linear PDE s of fractional order []. Matematical Preliminaries on Fractional Calculus: Many definitions of fractional calculus ave been developed to solve te problems of fractional differential equations. Te most frequently encountered definitions include Riemann-Liouville Caputo Wely Rize fractional operator. Introducing te following definitions [3 ] in te present analysis:. Definition: Let Te integral operator I α defined on te usual Lebesgue space L (a b) by x d f (x) I α f(x) = x t f (t)dt dx ( ) for x [ ] is called Riemann-Liouville fractional integral operator of order α ( ). It as te following properties: (i) exists for any [ ] (iii) (iv) were [ ]. Definition: Te Riemann-Liouville definition of fractional order derivative is were is an integer tat satisfies JETIR738 Journal of Emerging Tecnologies and Innovative Researc (JETIR) 9

2 December 7 Volume Issue JETIR (ISSN-39-56).3 Definition: A modified fractional differential operator proposed by Caputo is given by were is te order of operation and is an integer tat satisfies. It as te following two basic properties[5]: (i) If or [ ] and ten (ii) If [ ] and if ten. Definition: For m being te smallest integer tat exceeds α te Caputo time-fractional derivative operator of order is defined as in [6] { Relation between Caputo derivative and Riemann-Liouville derivative:. Integrating by parts we get te following formulae as given in [7]: (i) [ ] (ii) For [ ] 3 Generalized two dimensional differential transform metod:- Consider a function of two variables be a product of two single-variable functions i.e wic is analytic and differentiated continuously wit respect to x and y in te domain of interest. Ten te generalized two-dimensional differential transform of te function is given by [6-8] * + were is called te spectrum of and (-times). Te inverse generalized differential transform of is given by (3.) It as te following properties: (i) if ten (ii) if ten (iii) if ten (iv) if ten (v) if ten (vi) if ten (vii) if ten ( ) (viii) if u( x y) f ( x) g( y) and te function f ( x) x ( x) were n and ( x) a x x n (a) (b) (3.) and is arbitrary or is arbitrary and a for n... m were m m. Ten (3.) becomes D D u x y (ix) if vx y f x g y n x as te generalized Taylor series expansion x y x y f x satisfies te conditions given in (viii) and ux y D vx y te function ten x JETIR738 Journal of Emerging Tecnologies and Innovative Researc (JETIR) 9

3 December 7 Volume Issue JETIR (ISSN-39-56) V were and are te differential transformations of te functions and respectively and { Test Problems In tis section we present tree examples to illustrate te applicability of Generalized Differential Transform Metod (GDTM) to solve non linear partial differential equations of fractional order.. Example: Consider te non-linear inomogeneous time-fractional partial differential equation u x t u x t u x t ux t t x t x x u x x subject to initial condition were t t is te fractional differential operator(caputo derivative) of order (..) Applying generalized two-dimensional differential transform (3.) wit on (..) we obtain r s r r s r s (..) and Now utilizing te recurrence relation (..) and te initial condition (..3) we obtain after a little simplification te following values of for and W (..3) JETIR738 Journal of Emerging Tecnologies and Innovative Researc (JETIR) 93

4 December 7 Volume Issue JETIR (ISSN-39-56) and so on Now from (3.) we ave u x t x t 6 8 (..) sing te above values of in (..) te solution of (..) is obtained as 3 u x t t t xt xt xt xt... xt x x t (..5). Example: Consider te non-linear space-fractional telegrap partial differential equation 3 u x t u x t u x t u x t cu x t 3 x t t u t t e x subject to initial condition u t e t were x and x JETIR738 Journal of Emerging Tecnologies and Innovative Researc (JETIR) 9 u t t e x is te fractional differential operator(caputo derivative) of order (..) and c= constant Applying generalized two-dimensional differential transform (3.) wit on (..) we obtain and 3 c r s r 3 s r s!!! (..) (..3)

5 December 7 Volume Issue JETIR (ISSN-39-56) Taing ten (..) and (..3) becomes c r s r 3 s r s and!! (..5) tilizing te recurrence relation (..) and te initial condition (..5) we obtain after a little simplification te following values of for and (..)! 3!! 5! 6! 7! W 3! 3! 5! 5! 7 6! 7! 3 c 5 3 c 5 c 8c 6c 5!! ! 3! 3 5!! 3c 5 5! 5! W 5 c 7 5 c 7 8c 6c 3c 7!! ! 3! 5 7!! 6c 7 5! 5! and so on Now from (3.) and using te above values of te solution of (..) is obtained as u x t x t t t t t t t t t t t t t t x! 3!! 5! 6! 7!! 3!! 5! 6! c 8c 6c c c t t t t 5 5 5!! 5 3! 3! 5!! 3 3c 8c 6c t x 5 c c t t t 5! 5! 7 7 7!! 7 3! 3! JETIR738 Journal of Emerging Tecnologies and Innovative Researc (JETIR) 95

6 December 7 Volume Issue JETIR (ISSN-39-56) 3c 5 6c 5 t t... 7 x!! 7 5! 5! Taing ten (..) and (..3) becomes 3 3 (..6) c r s r 3 s r s and! tilizing te recurrence relation (..7)! (..8) and te initial condition (..8)we obtain after a little simplification te following values of for and! 7 7! W W 3 W !! (..7)! 3!! 5! 6! W 6 W 7 c 7 c 8c 6c 3c 7!! 73 3! 3! 7!! 6c 75 5! 5! 8 W 9 W 5! 6! and so on Now from (3.) and using te above values of te solution of (..) is obtained as u x t x t! 3!! 5! 6! 7!! 3!! 5! 6! 7! 8c 6c 3 c c t t t!! 3! 3! t t t t t t t t t t t t t t x 3c 7 6c 5 t t... x!! 5! 5! (..9) 7! 5 JETIR738 Journal of Emerging Tecnologies and Innovative Researc (JETIR) 96

7 December 7 Volume Issue JETIR (ISSN-39-56).3 Example: Consider te non-linear space time fractional wave partial differential equation u x t u x t t x cu x t x x t n subject to initial condition u x a x n u x b x and were x and t n n t n n ut x cnx are fractional differential operators (Caputo derivative) of order differential transform (3.) wit on (.3.) we obtain JETIR738 Journal of Emerging Tecnologies and Innovative Researc (JETIR) 97 n n (.3.) 3 3 r 6 r s 3 r 5 s 3 r s r and 3 c r s 3 r s r s a (.3.) and (.3.3) becomes b c! Applying generalized two-dimensional (.3.) r r 8 s 3 r 5 s r s 3 r 3 c r s 3 r s r s and a b (.3.3) Taing and (.3.) (.3.5) Now utilizing te recurrence relation (.3.) and te initial conditions (.3.5)we obtain after a little simplification te following values of for and a b 3 ca ca 7 cb c a c 7 b 3 ca a 5 7 c a a c a b 5 ten 5 ca b c a b a 5 ca b b a cb b 3 8 c a ab ab ab 3

8 December 7 Volume Issue JETIR (ISSN-39-56) c a a c 3c 7 9 c b ab ab c aa a 3 3 a a 3 3 b 3 b 5 5 b a 6 6 JETIR738 Journal of Emerging Tecnologies and Innovative Researc (JETIR) ca a a a 3 3 c a a a a a 3 53 c a a a a a a 5 3 b 6 63 c a a a a a a a c a a a a a a a a b a a 6 7 a b a a c a a a a a a a a a and so on Now from (3.) we ave u x t x t 8 (.3.6) sing te above values of in (.3.6) te solution of (.3.) is obtained as u x t a bt cat cab t cat cb t 3c abt c a ct c a b t a b t ca a t ca b b a t c a at cbb t c a ab ab ab t c a a c ct c b a b a b t x a bt c aa a t x a3 b3t c aa3 aa t x a bt caa aa 3 a t x a5 b5t c aa5 aa aa3 t x b 8

9 December 7 Volume Issue JETIR (ISSN-39-56) a b t c a a a a a a a t x a b t ca a a a a a a a t x 3 a8 b8t 6 a5 c aa8 aa 7 aa6 a3a5 a t x... (.3.7) Conclusions In te present study we considered tree examples to exibits te applicability of te generalized differential transform metod (GDTM). It may be concluded tat GDTM is a reliable tecnique to andle linear and nonlinear fractional differential equations. Compared wit oter approximate metods tis tecnique provides more realistic series solutions. References [] J. K. Zou Differential Transformation and Its Applications for Electrical Circuits Huazong niversity PressWuan Cina 986. [] C.K. Cen and S.H. Ho Solving partial differential equations by two dimensional differential transform metod Appl. Mat. Comput. vol.6pp.7 79(999). [3] F. Ayaz Solutions of te systems of differential equations by differential transform Metod Appl. Mat. Comput. vol.7pp (). [] R. Abazari and A. Boranifar Numerical study of te solution of te Burgers and coupled Burgers equations by a differential transformation metod Comput. Mat. Appl.vol.59 pp.7 7(). [5] C.K. Cen Solving partial differantial equations by two dimensional differential transformation metod Appl. Mat. Comput.vol. 6pp. 7 79(999). [6] M.J. Jang and C.K. Cen Two-dimensional differential transformation metod for partial differantial equations Appl. Mat. Comput.vol. pp.6 7(). [7] F. Kangalgil and F. Ayaz Solitary wave solutions for te KDV and mkdv equations by differential transformation metod Coas Solitons Fractals vol. pp.6 7(9). [8] A. Arioglu and I. Ozol Solution of difference equations by using differential transformation metod Appl. Mat. Comput. vol.7 pp.6 8(6). [9] S. Momani Z. Odibat and I. Hasim Algoritms for nonlinear fractional partial differantial equations: A selection of numerical metods Topol. Metod Nonlinear Anal.vol. 3pp. 6(8). [] A. Arioglu and I. Ozol Solution of fractional differential equations by using differential transformation metod Caos Solitons Fractals vol.3 pp.73 8(7). [] B. Soltanalizade and M. Zarebnia Numerical analysis of te linear and nonlinear Kuramoto-Sivasinsy equation by using Differential Transformation metod Inter. J. Appl. Mat. Mecanics vol.7no.pp.63 7(). [] A. Tari M.Y. Raimi S. Samoradb and F. Talati Solving a class of two-dimensional linear and nonlinear Volterra integral equations by te differential transform metod J. Comput. Appl. Mat. vol.8pp.7 76(9). [3] D. Nazari and S. Samorad Application of te fractional differential transform metod to fractional-order integro-differential equations wit nonlocal boundary conditions J. Comput. Appl. Mat. vol.3 pp (). [] A. Boranifar and R. Abazari Exact solutions for non-linear Scr.dinger equations by differential transformation metod J. Appl. Mat. Comput. vol.35pp.37 5(). [5] A. Boranifar and R. Abazari Numerical study of nonlinear Scr.dinger and coupled Scr.dinger equations by differential transformation metod Optics Communicationsvol.83 pp.6 3(). [6] S. Momani Z. Odibat and V. S. Ertur Generalized differential transform metod for solving a space- and time-fractional diffusionwave equation Pysics Letters. A vol. 37no. 5-6 pp (7). [7] Z. Odibat and S. Momani A generalized differential transform metod for linear partial Differential equations of fractional order Applied Matematics Letters vol. no. pp.9 99(8) [8] Z. Odibat S. Momani and V. S. Ertur Generalized differential transform metod:application to differential equations of fractional order Applied Matematics and Computation vol. 97 no. pp.67 77(8). [9] Vedat Suat ErtiiraSaer Momanib On te generalized differential transform metod :application to fractional integro-differential equations Studies in Nonlinear Sciencesvol.no.pp.8-6(). [] Mridula GargPratiba ManoarSyam L.Kalla Generalized differential transform metod to Space-time fractional telegrap equation Int.J.of Differential EquationsHindawi Publising Corporationvol.article id.:58989 pagesdoi.:.55//5898. [] Manis Kumar BansalRasmi Jain Application of generalized differential transform metod to fractional order Riccati differential equation and numerical results Int. J.of Pure and Appl. Mat.vol.99no.3pp (5). [] Aysegul Cetinaya Onur Kiymaz and Jale Camli Solution of non linear PDE s of fractional order wit generalized differential transform metod Int. Matematical Forumvol.6no.pp 39-7(). [3] S.Das Functional Fractional Calculus Springer 8. [] K.S.Miller and B.Ross An Introduction to te Fractional Calculus and Fractional Diff. Equations Jon Wiley and Son 993. [5] M.Caputo Linear models of dissipation wose q is almost frequency independent-ii" Geopys J. R. Astron. Soc vol. 3 pp (967). JETIR738 Journal of Emerging Tecnologies and Innovative Researc (JETIR) 99

10 December 7 Volume Issue JETIR (ISSN-39-56) [6] I. Podlubny Fractional differential equations: An introduction to fractional derivatives fractional differential equations to metods of teir solution and some of teir applications Academic Press 999. [7] R. Almeida and D. F. Torres Necessary and sufficient conditions for te fractional calculus of variations wit caputo derivatives" Communications in Nonlinear Science and Numerical Simulation vol. 6 pp. 9-5(). JETIR738 Journal of Emerging Tecnologies and Innovative Researc (JETIR)