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Abstract
In this thesis, the explicit finite difference approximation (EFDA) and the nonstandard finite difference method (NSFDM) for solving numerically the twodimensional space fractional diffusion equation (SFDE) are considered. The concept of fractional derivative is considered in the sense of the right-shifted Grünwald. In order to study the stability analysis and the truncation error of the schemes, some theorems with proofs are presented. We are concluded that the NSFDM scheme preserves numerical stability in larger regions than the EFDA. Numerical test examples are given to demonstrate the effectiveness of the method. Moreover, from the comparison between EFDA and NSFDM we can conclude that, for some kind of non-linear fractional differential equations, NSFDM leads to faster convergence and more accurate results than EFDA