A New Predictor-Corrector Method for Solving Nonlinear Fractional Differential Equations with Graded Meshes

Abstract

In this dissertation we consider the numerical methods for solving non-linear fractional differential equations. We first review the predictor-corrector methods for solving the nonlinear fractional differential equation with uniform meshes and discussed in detail how to prove the error estimates. The convergence orders of the predictorcorrector methods for solving nonlinear fractional differential equations available in the literature are only O(h1+α ), where α ∈ (0, 1) denotes the fractional order and h is the step size. It will take a long time to obtain the good approximate solutions by using such method. Therefore it is necessary to construct some higher order numerical methods to solve the nonlinear fractional differential equations. We construct a higher order numerical method with the convergence order O(h1+2α) by approximating the Riemann-Liouville fractional integral with the quadratic interpolation polynomials. The graded meshes can be used in the numerical methods to capture the singularity of the problem. Numerical examples are given to show that the numerical results are consistent with the theoretical results.