Fractional Sumudu Decomposition Method for Solving PDEs of Fractional Order

Document Type : Research Paper


1 Department of Mathematics, Faculty of Education for Pure Sciences, University of Thi-Qar Nasiriyah, Iraq

2 Department of Mathematics, Faculty of Computer Science and Mathematics , University of Thi-Qar, Nasiriyah, Iraq


. In this paper, the fractional Sumudu decomposition method (FSDM) is employed to handle the time-fractional PDEs and system of time-fractional PDEs. The fractional derivative is described in the Caputo sense. The approximate solutions are obtained by using FSDM, which is the coupling method of fractional decomposition method and Sumudu transform. The method, in general, is easy to implement and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the proposed technique.


Main Subjects

[1] Kumar, D., Singh, J., Kumar, S., Numerical computation of nonlinear fractional Zakharov Kuznetsov equation arising in ion-acoustic waves, Journal of the Egyptian Mathematical Society, 22(3), 2014, 373–378.
[2] Guner, O., Aksoy, E., Bekir, A., Cevikel, A.C., Various Methods for Solving Time Fractional KdV-Zakharov Kuznetsov Equation, In. AIP Conference Proceedings, 1738. AIP, New York,2016.
[3] Cenesiz, Y., Tasbozan, O., Kurt, A., Functional variable method for conformable fractional modified KdV-ZK equation and Maccari system, Tbilisi Mathematical Journal, 10(1), 2017, 118–126.
[4] Shah, R., Khan, H., Baleanu, D., Kumam, P., Arif, M., A novel method for the analytical solution of fractional Zakharov–Kuznetsov equations, Advances in Difference Equations, 517, 2019, 1-14.
[5] Liao, S., Homotopy Analysis Method in Nonlinear Differential Equations, Higher education press, Beijing,2012.
[6] Xu, S., Ling, X., Zhao, Y., Jassim, H. K., A Novel Schedule for Solving the Two-Dimensional Diffusion in Fractal Heat Transfer, Thermal Science, 19, 2015, S99-S103.
[7] Jassim, H. K., et al., Fractional variational iteration method to solve one-dimensional second-order hyperbolic telegraph equations, Journal of Physics: Conference Series, 1032(1), 2018, 1-9.
[8] Jafari, H., Jassim, H. K., Tchier, F., Baleanu, D., On the Approximate Solutions of Local Fractional Differential Equations with Local Fractional Operator, Entropy, 18, 2016, 1-12.
[9] Ganji, D. D., Rafei, M., Solitary wave solutions for a generalized Hirota-Satsuma coupled KdV equation by homotopy perturbation method, Physics Letters A, 356( 2), 2006, 131–137.
[10] Yildirim, A., An algorithm for solving the fractional nonlinear Schrodinger equation by means of the homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, 10( 4), 2009, 445–450.
[11] Singh, J., et al. Numerical solution of time- and space-fractional coupled Burger’s equations via homotopy algorithm, Alexandria Engineering Journal, 55 ,2016, 1753–1763.
[12] Jassim, H. K., Homotopy Perturbation  Algorithm Using Laplace Transform for Newell-Whitehead-Segel Equation, International Journal of Advances in Applied Mathematics and Mechanics, 2, 2015, 8-12.
[13] Jassim, H. K., The Approximate Solutions of Three-Dimensional Diffusion and Wave Equations within Local Fractional Derivative Operator, Abstract and Applied Analysis, 2016, 2016, 1-5.
[14] Wang, K., et al., A new Sumudu transform iterative method for time-fractional Cauchy reaction–diffusion equation, Springer Plus, 5, 2016, 1-20.
[15] Baleanu, D., Jassim, H. K., Khan, H., A Modification Fractional Variational Iteration Method for solving Nonlinear Gas Dynamic and Coupled KdV Equations Involving Local Fractional Operators, Thermal Science, 22, 2018, S165-S175.
[16] Jafari, H., Jassim, H. K., Vahidi, J., Reduced Differential Transform and Variational Iteration Methods for 3D Diffusion Model in Fractal Heat Transfer within Local Fractional Operators, Thermal Science, 22, 2018, S301-S307.
[17] Jassim, H. K., Baleanu, D., A novel approach for Korteweg-de Vries equation of fractional order, Journal of  Applied Computational Mechanics, 5(2) , 2019, 192-198.
[18] Zhuo, J. F., Sergiy, R., Hong, G. S., Mushtaq, A. K., A robust kernel-based solver for variable-order time fractional PDEs under 2D/3D irregular domains,  Applied Mathematics Letters, 94, 2019, 105-111.
[19] Zhuo, J. F., Li, W. Y., Hui, Q. Z., Wen, Z. X., A semi-analytical collocation Trefftz scheme for solving multi-term time fractional diffusion-wave equations, Engineering Analysis with Boundary Elements, 98, 2019,137-146.
[20] Jassim, H. K., Analytical Approximate Solutions for Local Fractional Wave Equations, Mathematical Methods in the Applied Sciences, 43(2), 2020, 939-947.
[21] Wang, S. Q., Yang, Y. J., Jassim, H. K., Local Fractional Function Decomposition Method for Solving Inhomogeneous Wave Equations with Local Fractional Derivative, Abstract and Applied Analysis, 2014, 2014, 1-7.
[22] Jafari, H., Jassim, H. K., Tchier, F., Baleanu, D., On the Approximate Solutions of Local Fractional Differential Equations with Local Fractional Operator, Entropy, 18, 2016, 1-12.
[23] Jafari, H., et al., On the Existence and Uniqueness of Solutions for Local differential equations, Entropy, 18, 2016, 1-9.
[24] Yang, Z., Wang, J., Li, Nie, Y., Effective numerical treatment of sub-diffusion equation with non-smooth solution, International Journal of Computer Mathematics, 95, 2018, 1394-1407.