Acta Universitatis Apulensis ISSN: 1582-5329 http://www.uab.ro/auajournal/ No. 40/2014 pp. 103-112 doi: 10.17114/j.aua.2014.40.09 THE APPROXIMATE SOLUTIONS OF TIME-FRACTIONAL DIFFUSION EQUATION BY USING CRANK NICHOLSON METHOD H. Bulut, S. Tuluce Demiray, M. Kayhan Abstract. In this study, we consider approximate solution of Time-Fractional Diffusion Equation (TFDE) by using Crank-Nicholson Method. Besides, we uti- lize property of Riemann-Liouville derivative to obtain this solution. Then, we draw three dimensional graphics of this solution by means of programming language Map- ple. Finally, we show table including error analysis for some values of α, x, t, M and N. Numerical results ensure to illustrate the effectiveness and reliability of this method. 2000 Mathematics Subject Classification : 65XX02, 65Axx, 65Gxx. Keywords: Crank-Nicholson Method, Time-fractional Diffusion Equation, Riemann- Liouville. 1. Introduction The exploration of solutions of nonlinear fractional differantial equations has a very important role in several sciences such as biology, system identification, physics, viscoelasticity, signal processing, probability and statistics, mechanical engineering, hydrodynamics, chemistry, solid state physics, finance, optical fibers, fluid mechan- ics, electric control theory, thermodynamics, heat transfer and fractional dynamics [1, 2]. In recent years, most authors have improved a lot of methods to find solutions of fractional differantial equations such as variational iteration method [3], homo- topy decomposition method (HDM) [4], generalized Kudryashov method [5, 6], the modified Gauss elimination method [7], the Sinc-Legendre collocation method [9]. Time-fractional diffusion equation recently takes attention because it is a highly beneficial tool to identify problems involving non-Markovian random walks. This type of equation is procured from standard diffusion equation by substituting the first-order time derivative with a fractional derivative of α. The diffusion equation 103