Numerical Methods for Solving Nonlinear Fractional Differential Equations with Non-Uniform Meshes

Abstract

In this dissertation, we consider numerical methods for solving fractional differential equations with non-uniform meshes. We first introduce some basic definitions and theories for fractional differential equations and then we consider the numerical methods fro solving fractional differential equation. In the literature, the popular numerical methods for solving fractional differential equation include the rectangle method, trapezoid method and predictor-corrector methods. We reviewed such methods and the ways to prove the stability and the error estimates for these methods. Since the fractional differential equation is a nonlocal problem, the computation cost is very long compared with the local problem. Therefore it is very important to design some higher order numerical methods for solving fractional differential equation. In this dissertation, we introduce a new higher order numerical method for solving fractional differential equation which is based on the quadratic interpolation polynomial approximation to the fractional integral. To capture the singularity near the origin we also introduce the non-uniform meshes. The numerical results show that the optimal convergence order can be recovered by using non-uniform meshes even if the data are not sufficiently smooth.