Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 746401, 12 pages http://dx.doi.org/10.1155/2013/746401 Research Article Multivariate Padé Approximation for Solving Nonlinear Partial Differential Equations of Fractional Order Veyis Turut 1 and Nuran Güzel 2 1 Department of Mathematics, Faculty of Arts and Sciences, Batman University, Batman, Turkey 2 Department of Mathematics, Faculty of Arts and Sciences, Yıldız Technical University, ˙ Istanbul, Turkey Correspondence should be addressed to Nuran G¨ uzel; nguzel@yildiz.edu.tr Received 27 November 2012; Revised 14 January 2013; Accepted 14 January 2013 Academic Editor: Hassan Eltayeb Copyright © 2013 V. Turut and N. G¨ uzel. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Two tecHniques were implemented, the Adomian decomposition method (ADM) and multivariate Pad´ e approximation (MPA), for solving nonlinear partial differential equations of fractional order. e fractional derivatives are described in Caputo sense. First, the fractional differential equation has been solved and converted to power series by Adomian decomposition method (ADM), then power series solution of fractional differential equation was put into multivariate Pad´ e series. Finally, numerical results were compared and presented in tables and figures. 1. Introduction Recently, differential equations of fractional order have gained much interest in engineering, physics, chemistry, and other sciences. It can be said that the fractional derivative has drawn much attention due to its wide application in engi- neering physics [1–9]. Some approximations and numerical techniques have been used to provide an analytical approx- imation to linear and nonlinear differential equations and fractional differential equations. Among them, the variational iteration method, homotopy perturbation method [10, 11], and the Adomian decomposition method are relatively new approaches [5–9, 12, 13]. e decomposition method has been used to obtain approximate solutions of a large class of linear or nonlinear differential equations [12, 13]. Recently, the application of the method is extended for fractional differential equations [6– 9, 14]. Many definitions and theorems have been developed for multivariate Pad´ e approximations MPA (see [15] for a sur- vey on multivariate Pad´ e approximation). e multivariate Pad´ e Approximation has been used to obtain approximate solutions of linear or nonlinear differential equations [16–19]. Recently, the application of the unvariate Pad´ e approximation is extended for fractional differential equations [20, 21]. e objective of the present paper is to provide approxi- mate solutions for initial value problems of nonlinear partial differential equations of fractional order by using multivariate Pad´ e approximation. 2. Definitions For the concept of fractional derivative, we will adopt Caputo’s definition, which is a modification of the Riemann- Liouville definition and has the advantage of dealing properly with initial value problems in which the initial conditions are given in terms of the field variables and their integer order, which is the case in most physical processes. e definitions can be seen in [3, 4, 22, 23]. 3. Decomposition Method [24] Consider   ∗  (, ) =  (,   ,  ) +  (, ),  − 1 <  ≤ . (1)