### A New Approach for the Solution of Space-Time Fractional Order Heat-Like Partial Differential Equations by Residual Power Series Method

#### Abstract

The main concern of this article has been to apply the *Residual Power Series Method* (RPSM) effectively to find the exact solutions of fractional-order space-time dependent nonhomogeneous partial differential equations in the Caputo sense. Our first step is to reduce fractional-order space-time dependent non-homogeneous partial differential equations to fractional-order space-time dependent homogeneous partial differential equations before applying the proposed method. Obtaining fractional power series solutions of the problem and reproducing the exact solution is the main step. The illustrative examples reveal that RPSM is a very significant and powerful method for obtaining the solution of any-order time-space fractional non-homogeneous partial differential equations in the form of fractional power series.

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S. Abbasbandy and E. Shivanian, Predictor homotopy analysis method and its application to some nonlinear problems, Commun. Non. Sci. & Num. Simul. 16 (2011), 2456 – 2468, DOI: 10.1016/j.cnsns.2010.09.027.

M. Alquran, Analytical solutions of fractional foam drainage equation by residual power series method, Math. Sci. 8 (2014), 153 – 160, DOI: 10.1007/s40096-015-0141-1.

O. A. Arqub, A. El-Ajou and S. Momani, Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations, J. Compu. Phy. 293 (2015), 385 – 399, DOI: 10.1016/j.jcp.2014.09.034.

O. A. Arqub, A. El-Ajou, A. Bataineh and I. Hashim, A representation of the exact solution of generalized Lane–Emden equations using a new analytical method, Abstr. Appl. Anal. 2013 (2013), Article ID 378593, 10 pages, DOI: 10.1155/2013/378593.

O. A. Arqub, Series solution of Fuzzy differential equations under strongly generalized differentiability, J. Adv. Res. Appl. Math. 5 (2013), 31 – 52, DOI: 10.5373/jaram.1447.051912.

O. A. Arqub, Z. Abo-Hammour, R. Al-Badarneh and S. Momani, A reliable analytical method for solving higher-order initial value problems, Dis. Dyn. Nat. Soc. 2013 (2013), Article ID 673829, 12 pages, DOI: 10.1155/2013/673829.

A. El-Ajou, O. A. Arqub and S. Momani, Approximate analytical solution of the nonlinear fractional KdV–Burgers equation: A new iterative algorithm, J. Compu. Phy. 293 (2015), 81 – 95, DOI: 10.1016/j.jcp.2014.08.004.

A. El-Ajou, O. A. Arqub, S. Momani, D. Baleanu and A. Alsaedi, A novel expansion iterative method for solving linear partial differential equations of fractional order, Appl. Math. Modell. 257 (2015), 119 – 133, DOI: 10.1016/j.amc.2014.12.121.

A. El-Ajou, O. A. Arqub, Z. A. Zhour and S. Momani, New results on fractional power series: theories and applications, Entropy 15 (2013), 5305 – 5323, DOI: 10.3390/e15125305.

J. H. He, Some applications of nonlinear fractional differential equations and their approximations, Sci. Technol. Soc. 15 (1999), 86 – 90.

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam, Elsevier (2006).

S. Kumar and M. M. Rashidi, New analytical method for gas dynamic equation arising in shock fronts, Comput. Phy. Commun. 185 (2014), 1947 – 1954, DOI: 10.1016/j.cpc.2014.03.025.

S. Kumar and O. P. Singh, Numerical inversion of Abel integral equation using homotopy perturbation method, Z. Naturfors 65 (2010), 677 – 682, DOI: 10.1515/zna-2010-8-908.

S. Kumar, A new analytical modeling for telegraph equation via Laplace transforms, Appl. Math. Modell. 38 (2014), 3154 – 3163, DOI: 10.1016/j.apm.2013.11.035.

S. Kumar, A. Kumar and D. Baleanau, Two analytical methods for time-fractional nonlinear coupled Boussinesq Burgers equations arise in propogation of shallow water waves, Nonlinear Dyn. 85 (2016), 699 – 715, DOI: 10.1007/s11071-016-2716-2.

A. Kumar, S. Kumar and M. Singh, Residual power series method for fractional Sharma-Tasso-Olever equation, Comm. Num. Anal. 1 (2016), 1 – 10, DOI: 10.5899/2016/cna-00235.

K. S. Miller and B. Ross, An Introduction to the Fractional Integrals and Derivatives – theory and Application, Wiley, New York (1993).

Z. M. Odibat, A reliable modification of the rectangular decomposition method, Appl. Math. Comput. 183 (2006), 1226 – 1234, DOI: 10.1016/j.amc.2006.06.048.

K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, USA (1974).

I. Podlubny, Fractional Differential Equations, Academic, New York (1999).

J. Sabatier, O. P. Agarwal and J. A. T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Dordrecht, Springer (2007).

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives Theory and Applications, Amsterdam, Gordon and Breach (1993).

V. K. Srivastava, M. K. Awasthi and S. Kumar, Analytical approximations of two and three dimensional time-fractional telegraphic equation by reduced differential transform method, Egyp. J. Basic Appl. Sci. 1 (2014), 60 – 66, DOI: 10.1016/j.ejbas.2014.01.002.

F. Tchier, M. Inc, Z. S. Korpinar and D. Baleanau, Solutions of the time fractional reactiondiffusion equations with residual power series method, Adv. Mech. Eng. 8 (2016), 1 – 10, DOI: 10.1177/1687814016670867.

K. Vishal, S. Kumar and S. Das, Application of homotopy analysis method for fractional swift Hohenberg equation-revisited, Appl. Math. Modell. 36 (2012), 3630 – 3637, DOI: 10.1016/j.apm.2011.10.001.

A.-M. Wazwaz, The combined Laplace transform-Adomain decomposition method for handling nonlinear Voltraintegro differential equations, Appl. Math. Comput. 216 (2010), 1304 – 1309, DOI: 10.1016/j.amc.2010.02.023.

DOI: http://dx.doi.org/10.26713%2Fcma.v10i3.626

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