Research Article A New Approach and Solution Technique to Solve Time Fractional Nonlinear Reaction-Diffusion Equations Inci Cilingir Sungu 1 and Huseyin Demir 2 1 Department of Elementary School Mathematics Education, Education Faculty, Ondokuz Mayıs University, 55139 Samsun, Turkey 2 Department of Mathematics, Arts and Science Faculty, Ondokuz Mayıs University, 55139 Samsun, Turkey Correspondence should be addressed to Huseyin Demir; hdemir@omu.edu.tr Received 19 August 2014; Accepted 20 November 2014 Academic Editor: Samir B. Belhaouari Copyright © 2015 I. Cilingir Sungu and H. Demir. Tis 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. A new application of the hybrid generalized diferential transform and fnite diference method is proposed by solving time fractional nonlinear reaction-difusion equations. Tis method is a combination of the multi-time-stepping temporal generalized diferential transform and the spatial fnite diference methods. Te procedure frst converts the time-evolutionary equations into Poisson equations which are then solved using the central diference method. Te temporal diferential transform method as used in the paper takes care of stability and the fnite diference method on the resulting equation results in a system of diagonally dominant linear algebraic equations. Te Gauss-Seidel iterative procedure then used to solve the linear system thus has assured convergence. To have optimized convergence rate, numerical experiments were done by using a combination of factors involving multi-time-stepping, spatial step size, and degree of the polynomial ft in time. It is shown that the hybrid technique is reliable, accurate, and easy to apply. 1. Introduction Te nonlinear reaction-difusion equations have found numerous applications in pattern formation, in many branches of biology, chemistry, and physics [1–4]. Reaction- difusion (RD) equations have also been applied to other areas of science and can be successfully modelled by the use of fractional order derivatives. [5–18]; for example, the RD equations are employed to describe the CO oxidation on Pt (110) [5], the study of Ca 2 + waves on Xenopus oocytes [11], and the study of reentry in heart tissue [7, 13]. A great deal of efort has been expended over the last 10 years in attempting to fnd robust and stable numerical and analytical methods for solving fractional partial diferential equations of physical interest. Tere has also been a wide variety of numerical methods, for example, fnite diference techniques, fnite element methods, spectral techniques, adaptive and nonadaptive algorithms, and so forth, which have been developed for RD’s numerical solution [19, 20]. Te diferential transform method was used frst by Zhou [21] who solved linear and nonlinear initial value problems in electric circuit analysis. Tis method constructs an analytical solution in the form of a polynomial. It is diferent from the traditional higher order Taylor series method, which requires symbolic computation of the necessary derivatives of the data functions. Te Taylor series method computationally takes long time for large orders. Te diferential transform is an iterative procedure for obtaining analytic Taylor series solution of ordinary or partial diferential equations. Te method is well addressed in [22–26]. Recently, the application of diferential transform method is successfully extended to obtain analytical approximate solutions to ordinary difer- ential equations of fractional order. Fractional diferential transform method (FDTM) is a method that Arikoglu and Ozkol [23] developed for solving linear and nonlinear inte- grodiferential equations of fractional order. Tis method solves problems with high accuracy while constructing semi- analytic solutions in the polynomial forms. FDTM is based on classical diferential transform method, fractional power series, and Caputo fractional derivative [27]. Arikoglu and Ozkol [23] tested their approach on several examples and the results obtained are in good agreement with the existing Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 457013, 13 pages http://dx.doi.org/10.1155/2015/457013