An efficient algorithm on timefractional. equations with variable coefficients. Research Article OPEN ACCESS. Jamshad Ahmad*, Syed Tauseef Mohyud-Din

Transcription

1 OPEN ACCESS Research Article An efficient algorithm on timefractional partial differential equations with variable coefficients Jamshad Ahmad*, Syed Tauseef Mohyud-Din Department of Mathematics, Faculty of Sciences, HITEC University Taxila Cantt Pakistan * ABSTRACT In this paper, a fractional complex transform (FCT) is used to convert the given fractional partial differential equations (FPDEs) into corresponding partial differential equations (PDEs), and subsequently reduced differential transform method (RDTM) is applied on the transformed PDEs. The results obtained are re-stated by making use of inverse transformation that yields in terms of original variables. It is observed that the proposed algorithm is highly efficient and appropriate for solving time fractional PDEs arising in mathematical physics, hence can be extended to other diverse problems. Keywords: fractional differential equation, Jumarie s fractional derivative, fractional complex transform, reduced differential transform method /connect.04.7 Submitted: 6 January 04 Accepted: 7 February 04 ª 04 Ahmad, Mohyud-Din, licensee Bloomsbury Qatar Foundation Journals. This is an open access article distributed under the terms of the Creative Commons Attribution license CC BY 4.0, which permits unrestricted use, distribution and reproduction in any medium, provided the original work is properly cited. Cite this article as: Ahmad J, Mohyud-Din ST. An efficient algorithm on time-fractional partial differential equations with variable coefficients, QScience Connect 04:7

2 Page of 0. INTRODUCTION Fractional differential equations arise in almost all areas of physics, applied and engineering sciences. 4 In order to better understand these physical phenomena, as well as further applications in practical scientific research, it is important to find their exact solutions. The investigation of exact solutions to these equations is interesting and important. In the past, many authors had studied the solution of such equations. Recently, several analytical and numerical techniques were successfully applied to deal with differential equations, fractional differential equations, and local fractional differential equations, including variational iteration method (VIM) 9 that has been applied to various nonlinear problems and local fractional Fourier transform. 0 Studies of Adomian s decomposition method (ADM), homotopy perturbation method (HPM), homotopy analysis method (HAM) and variation of parameter method (VPM) are successfully applied to obtain the exact solutions of differential equations. 8 In this paper, our purpose is to apply the reduced differential transform method (RDTM), 9 3 to construct appropriate solutions to time-fractional partial differential equations with variable coefficients. The reduced differential transform technique is an iterative procedure for obtaining Taylor series solution of differential equations. This method reduces the size of computational work and is easily applicable to many physical problems.. JUMARIE S FRACTIONAL DERIVATIVE 5 Some useful formulas and results of Jumarie s fractional derivative are summarized as: Dx a c ¼ 0; a $ 0; c ¼ constant: ðþ D a x cf ðþš x ¼ cdx a f ðþ; x a $ 0; c ¼ constant: ðþ Dx a x b ¼ G þ b x ba ; b $ a $ 0: ð3þ G þ b a D a x fðþgx x ¼ Dx a fðþgx x ðþþfðþda x x gx ðþ : ð4þ Dx a fðxðþ t Þ ¼ f 0 xðþx x a ðþ: t ð5þ 3. FRACTIONAL COMPLEX TRANSFORM 5 Consider the following general fractional differential equation f u; ut a ; ub x ; ug y ; ul z ; ua t ; ux b ; u g y ; ul z ;... ¼ 0; ð6þ where ut a ¼ a uðx;y; z;tþ t denotes the modified Riemann Liouville derivative. 0, a #, 0, b #, a 0, g #, 0, l #. Introducing the following transforms 8 T ¼ >< X ¼ Y ¼ >: Z ¼ pt a GðaþÞ qx b G bþ ð Þ ky g G þg ð Þ lz l GðþlÞ ð7þ where p, q, k, and l are unknown constants.

3 Page 3 of 0 Using the above transforms, we can convert fractional derivatives into classical derivatives 8 a u t ¼ p u a T b u >< b ¼ q u T >: g u t ¼ k u g T l u t ¼ l u l T ð8þ Therefore, we can easily covert the fractional partial differential equations into partial differential corresponding partial differential equations, so that everyone familiar with advanced calculus can deal with fractional calculus without difficulty. 4. REDUCED DIFFERENTIAL TRANSFORM METHOD (RDTM) To illustrate the basic idea of the DTM, differential transform of k th derivative of a function u x; t, that is analytic and differentiated continuously in the domain of interest is defined as " U k ðþ¼ x # k u x; t k! t k ð9þ t¼t 0 ; The differential inverse transform of U k ðþis x defined as follows: X u x; t ¼ k¼0 U k ðþt x ð t 0 Þ k ; ð0þ Eqn. (0) is known as the Taylor series expansion of u x; t, around t ¼ t0. Combining (9) and (0) X u x; t ¼ k¼0 k! " # k u x; t t k ðt t 0 Þ k ; ðþ t¼t 0 when t 0 ¼ 0, above equation reduces to X u x; t ¼ k¼0 " # k u x; t k! t k t¼t 0 t k ; ðþ and Eqn. (0) reduces to X u x; t ¼ k¼0 U k ðþt x k : ð3þ Theorem : If the original function is u x; t ¼ w x; t U k ðþ¼w x k ðþþv x k ðþ: x þ v x; t, then the transformed function is Theorem : If u x; t ¼ aw x; t, then Uk ðþ¼aw x k ðþ. x Theorem 3: If u x; t ¼ m wðx;tþ t, then U m k x ðþ¼ ðkþmþ! k! W k ðþ. x Theorem 4: If u x; t w x;t ¼ ð Þ, then U k ðþ¼ x W kðþ. x

4 Page 4 of 0 w x;y;t Theorem 5: If u x; y; t ¼ ð Þ, then U k x; y ¼ W k x; y. Theorem 6: If u x; y; z; t w x;y;z;t ¼ ð Þ, then U k x; y; z ¼ W k x; y; z. Theorem 7: If u x; t ¼ x m t n w x; t, then Uk ðþ¼x x m W kn ðþ. x Theorem 8: If u x; t ¼ w x; t, then Uk ðþ¼ x P k r¼0 W rðþw x kr ðþ. x 5. NUMERICAL APPLICATIONS OF RDTM In this section, we apply the new approach to find the solutions of the FPDEs in one, two and three dimensions with variable coefficients, and compared them with those obtained by other methods. Example 5. Consider the one-dimensional heat equation with variable coefficients in the form subject to the initial condition a u t ¼ a x u ; x. 0; t. 0; 0, a # ; ð4þ u x; 0 ¼ x : ð5þ u T ¼ x u : ð6þ Applying the DTM to (6) and (5), we obtain the following recursive formula ðk þ ÞU k ðþ¼ x x U k ðþ x : ð7þ U 0 ðþ¼x x ; ð8þ substituting (8) into (7), we obtain the following values successively U k ðþ x U ðþ¼x x ; U ðþ¼ x x ; U 3ðÞ¼ x x 3! ; U 4ðÞ¼ x x 4! ;... The series solution is given by u x; t ¼ x þ x T þ x T þ x 3! T 3 þ x 4! T 4 þ... u x; t ¼ x þ x t a Gða þ Þ þ x t a t 3a t 4a G ða þ Þ þ x 3! G 3 ða þ Þ þ x 4! G 4 þ... ð9þ ða þ Þ

5 Page 5 of 0 Setting a ¼, the closed form solution is u x; t ¼ x e t : Example 5. Consider the two-dimensional heat equation with variable coefficients as subject to the initial condition a u t ¼ a y u þ x u ; x. 0; y. 0; t. 0; 0, a # ; ð0þ y u x; y; 0 ¼ y : ðþ u T ¼ y u þ x u y ; ðþ Taking differential transform of () and (), we obtain the following recursive formula ðk þ ÞU k x; y ¼ y U k x; y þ x U k x; y y : ð3þ U 0 x; y ¼ y : ð4þ Now, substituting (4) into (3), we obtain the following values successively U k x; y U x; y ¼ x y ; U x; y ¼ ; U x 3 x; y ¼ 3! ; U y 4 x; y ¼ 4! ; U x 5 x; y ¼ 5! ; U 6 y x; y ¼ 6! ;... The approximate series solution is given u x; y; T ¼ y þ x T þ y T þ x 3! T 3 þ y 4! T 4 þ x 5! T 5 þ y 6! T 6... u x; y; t ¼ y þ x t a Gða þ Þ þ y þ y 6! t 6a G 6 ða þ Þ þ... Setting a ¼, the closed form solution is u x; t ¼ x sinht þ y cosht: t a t 3a t 4a t 5a G ða þ Þ þ x 3! G 3 ða þ Þ þ y 4! G 4 ða þ Þ þ x 5! G 5 ða þ Þ Example 5.3 Considering three-dimensional heat equation with variable coefficient a u t a x 4 y 4 z 4 36 x u þ y u y þ z u z ¼ 0; x; y; z. 0; t. 0; 0, a # ; ð5þ

6 Page 6 of 0 with the initial condition u x; y; z; 0 ¼ 0: ð6þ u T x 4 y 4 z 4 36 x u þ y u y þ z u z ¼ 0: ð7þ Taking differential transform of (7) and (6), we obtain the following recursive formula ðk þ ÞU k x; y; z ¼ x 4 y 4 z 4 þ 36 x U k x; y; z þ y U k x; y; z þ z U k x; y; z y z ; ð8þ U 0 ðþ¼0: x Now, substituting (9) into (8), we obtain the following values successively U k x; y; z ð9þ U x; y; z ¼ x 4 y 4 z 4 x 4 y 4 z 4 x 4 y 4 z 4 x 4 y 4 z 4 ; U x; y; z ¼ ; U 3 x; y; z ¼ ; U 4 x; y; z ¼ ;... 3! 4! The approximate series solution is given u x; y; z; T ¼ x 4 y 4 z 4 T þ x 4 y 4 z 4 T þ x 4 y 4 z 4 T 3 þ x 4 y 4 z 4 T 4 þ... u x; y; z; T ¼ x 4 y 4 z 4 þ t a Gða þ Þ þ x 4 y 4 z 4 t a G ða þ Þ þ x 4 y 4 z 4 t 3a 3! G 3 ða þ Þ þ x 4 y 4 z 4 t 4a 4! G 4 ða þ Þ þ... setting a ¼, the closed form solution is u x; y; z; t ¼ x 4 y 4 z 4 e t : Example 5.4 Considering the two-dimensional wave equation with variable coefficient as subject to the initial condition a u t a x u y u ¼ 0; x; y. 0; t. 0; 0, a # ; ð30þ y u x; y; 0 ¼ x 4 ; u t x; y; 0 ¼ y 4 ð3þ u T x u y u ¼ 0; ð3þ y Taking differential transform of (3) and (3), we obtain the following recursive formula ðk þ Þðk þ ÞU k x; y ¼ x U k x; y þ x U k x; y ; ð33þ U 0 ðþ¼x x 4 ; U ðþ¼y x 4 ; ð34þ

7 Page 7 of 0 Consequently, U ðþ¼ x x 4 ; The series solution is given U 3 ðþ¼ x y 4 3! ; U 4ðÞ¼ x 4! x 4 ; U 5 ðþ¼ x y 4 5! ;... u x; y; T ¼ x 4 þ y 4 T þ x 4 T þ y 4 3! T 3 þ x 4 4! T 4 þ y 4 5! T 5 þ... u x; y; t ¼ x 4 þ y 4 t a Gða þ Þ þ x 4 t a G ða þ Þ þ y 4 t 3a 3! G 3 ða þ Þ þ x 4 t 4a 4! G 4 ða þ Þ þ y 4 t 5a 5! G 5 ða þ Þ þ : ð35þ setting a ¼, the closed form solution is u x; y; t ¼ x 4 cosht þ y 4 sinht: Example 5.5 Considering the three-dimensional wave equation with variable coefficient a a u t a x þ y þ z with the initial condition x u þ y u y þ z u z ¼ 0; x; y; z. 0; t. 0; 0, a # ; ð36þ u x; y; z; 0 ¼ 0; ut x; y; z; 0 ¼ x þ y z : ð37þ u T x þ y þ z x u þ y u y þ z u z ¼ 0; Taking differential transform of (38) and (37), we obtain the following recursive formula ðk þ Þðk þ ÞU k x; y; z ¼ x þ y þ z þ x U k x; y; z þ y U k x; y; z y þ z U k x; y; z z ð38þ ð39þ U 0 ðþ¼0; x U ðþ¼ x x þ y z ; ð40þ Now, substituting (40) into (39), we obtain the following values successive U k x; y; z U x; y; z ¼ U 4 x; y; z ¼ 4! x þ y þ z ; U3 x; y; z ¼ 3! x þ y þ z ; U5 x; y; z ¼ 5! x þ y z ; x þ y z ;... The approximate series solution is given u x; y; z; T ¼ x þ y z T þ x þ y þ z T þ x þ y z T 3 3! þ x þ y þ z T 4 þ... 4!

8 Page 8 of 0 u x; y; z; T ¼ x þ y z t a GðaþÞ þ x þ y þ z t a G ðaþþ þ 3! x þ y z þ 4! x þ y þ z t 4a G 4 ðaþþ þ 5! x þ y z þ... setting a ¼, the exact solution is given in closed form by u x; y; z; t ¼ x þ y e t þ z e t x þ y þ z : t 3a G 3 ðaþþ Example 5.6 Considering the linear Klein-Gordon equation in the form with the initial condition a u t u u ¼ 0; x. 0; t. 0; 0, a # ; ð4þ a u x; 0 ¼ þ sinx; ut x; y; z; 0 ¼ 0: ð4þ u T u u ¼ 0; ð43þ Taking differential transform of (43) and (4), we obtain the following recursive formula ðk þ Þðk þ ÞU k ðþ¼ x U k ðþ x þ U k ðþ; x ð44þ U 0 ðþ¼ x þ sinx; U ðþ¼0; x ð45þ Consequently, U ðþ¼ x ; U 3ðÞ¼0; x U 4 ðþ¼ x 4! ; U 5ðÞ¼0; x U 6 ðþ¼ x 6! ;... The series solution is given u x; T ¼ þ sinx þ T þ 4! T 4 þ 6! T 6 þ... t a u x; t ¼ þ sinx þ G ða þ Þ þ t 4a 4! G 4 ða þ Þ þ t 6a 6! G 6 þ... ð46þ ða þ Þ setting a ¼, the closed form solution is u x; y; t ¼ sinx þ cosht: Example 5.7 Considering the nonlinear partial differential equation a u t a þ u þ u ¼ 0; x. 0; t. 0; 0, a # ; ð47þ

9 Page 9 of 0 with the initial condition u x; 0 ¼ x : ð48þ u T þ u þ u ¼ 0; ð49þ Taking differential transform of (49) and (48), we obtain the following recursive formula ðk þ ÞU k ðþ¼ x U kðþ x Xk i¼0 U k ðþu x ki ðþ; x ð50þ U 0 ðþ¼ x x ; ð5þ Consequently, U ðþ¼ x 4x ; U ðþ¼ x 8x 3 ; U 3ðÞ¼ x 6x 4 ; U 4ðÞ¼ x 3x 5 ;... The series solution is given u x; T ¼ x þ 4x T þ 8x T þ 3 6x T 3 þ 4 3x T 4 þ... 5 u x; y; t ¼ x þ t a 4x Gða þ Þ þ t a 8x 3 G ða þ Þ þ t 3a 6x 4 G 3 ða þ Þ þ t 4a 3x 5 G 4 ða þ Þ þ... setting a ¼, the closed form solution is u x; y; t ¼ x t : CONCLUSION Applied fractional complex transform (FCT) proved very effective to convert the given fractional partial differential equations (FPDEs) into corresponding partial differential equations (PDEs). The same is true for its subsequent effect in reduced differential transform method (RDTM), which was implemented on the transformed PDEs. The solution obtained by reduced differential transform method (RDTM) is an infinite power series for appropriate initial condition, which can in turn express the exact solutions in a closed form. The results show that the reduced differential transform method (RDTM) is a powerful mathematical tool for solving partial differential equations with variable coefficients. Computational work fully confirms the reliability and efficacy of the proposed algorithm, hence it may be concluded that the presented scheme may be applied to a wide range of physical and engineering problems. REFERENCES [] Noor MA, Mohyud-Din ST. Modified variational iteration method for heat and wave-like equations. Acta Appl Math. 008;04(3): [] Abbasbandy S. A new application of He s variational iteration method for quadratic Riccati differential equation by using Adomian s polynomials. J Comput Appl Math. 007;07(): [3] Abbasbandy S. Numerical solutions of nonlinear Klein-Gordon equation by variational iteration method. Int J Numer Meth Eng. 007;70(7): [4] He JH. Some applications of nonlinear fractional differential equation and their approximations. B Sci Technol Soc. 999;5(): [5] Guo S, Mei L. The fractional variational iteration method using He s polynomials. Phys Lett A. 0;375: [6] Guo S, Mei L, Ye F, Qiu Z. Compacton and solitary pattern solutions for nonlinear dispersive KdV-type equations involving Jumarie s fractional derivative. Phys Lett A. 0;376:58 64.

10 Page 0 of 0 [7] Yang XJ, Baleanu D. Fractal heat conduction problem solved by local fractional variation iteration method. Therm Sci. 03;7(): [8] Baleanu D, Machado JAT, Cattani C, Baleanu MC, Yang XJ. Local fractional variational iteration and decomposition methods for wave equation on Cantor sets within local fractional operators. Abstr Appl Anal. 03;04:6. Article ID [9] Ahmad J, Hassan QM, Mohyud-Din ST. Solitary solutions of the fractional KdV equation using modified Remann- Liouville derivative. J Fract Calc Appl. 03;4(): [0] Zhao Y, Baleanu D, Baleanu MC, Cheng DF, Yang XJ. Mappings for special functions on Cantor sets and special integral transforms via local fractional operators. Abstr Appl Anal. 03;03:6. Article ID [] Ahmad J, Mohyud-Din ST, Yang XJ. Local fractional decomposition method on wave equation in fractal strings. Mitteilungen Klosterneuburg. 04;64(). [] Yang XJ, Baleanu D, Zhong WP. Approximate solutions for diffusion equations on cantor space-time. Proc Rom Acad A. 03;4():7 33. [3] Momani S, Al-Khaled K. Numerical solution for systems of fractional differential equations by the decomposition method. Appl Math Comput. 005;6(3): [4] Odibat Z, Momani S. Numerical solution of Fokker-Planck equation with space- and time-fractional derivatives. Phys Lett A. 007;369: [5] Ganji Z, Ganji D, Rostamiyan Y. Solitary wave solutions for a time-fraction generalized Hirota-Satsuma coupled KdV equation by an analytical technique. Appl Math Model. 009;33: [6] Yıldırım A, Koçak H. Homotopy perturbation method for solving the space-time fractional advection-dispersion equation. Adv Water Resour. 009;3():7 76. [7] Matinfar M, Saeidy M. Application of homotopy analysis method to fourth order parabolic partial differential equations. Appl Appl Math. 00;5(): [8] Mohyud-Din ST, Noor MA, Waheed A. Variation of parameter method for solving sixth-order boundary value problems. Commun Korean Math Soc. 009;4(4): [9] Jang MJ, Chen CL, Liu YC. Two-dimensional differential transform for partial differential equations. Appl Math Comput. 00;:6 70. [0] Arikoglu A, Ozkol I. Solution of fractional differential equations by using differential transform method. Chaos, Soliton Fract. 007;34(5): [] Zhou JK. Differential transform and its applications for Electrical Circuits. Wuhan, China: Huazhong University Press; 986. (in Chinese). [] Merdan M, Gokdogan A. Solution of nonlinear oscillators with fractional nonlinearities by using the modified differential transformation method. Math Comput Appl. 0;6(3): [3] Kurnaz A, Oturanç G. The differential transforms approximation for the system of ordinary differential equations. Int J Comput Math. 005;8(6): [4] Li ZB, He JH. Fractional complex transform for fractional differential equations. Math Comput Appl. 00;5(5): [5] Jumarie G. Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions further results. Comput Math Appl. 006;5(9-0):