Comparative study on solving fractional differential equations via shifted Jacobi collocation method

In this paper, operational matrices of Riemann-Liouville fractional integration and Caputo fractional differentiation for shifted Jacobi polynomials are considered. Using the given initial conditions, we transform the fractional differential equation (FDE) into a modified fractional differential equation with zero initial conditions. Next, all the existing functions in modified differential equation are approximated by shifted Jacobi polynomials. Then, operational matrices and spectral collocation method are applied to obtain a linear or nonlinear system of algebraic equations. System of algebraic equations can be simultaneously solved (e.g. using Mathematica^{TM}). Main characteristic behind of the this technique is that only a small number of shifted Jacobi polynomials is needed to obtain a satisfactory result which demonstrates the validity and efficiency of the method. Comparison between this method and some other methods confirm the good performance of the presented method. Also, this method is generalized for the multi-point fractional differential equation.