A High Order Numerical Method for Solving Nonlinear Fractional Differential Equation with Non-uniform Meshes

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11189)

Abstract

We introduce a high-order numerical method for solving nonlinear fractional differential equation with non-uniform meshes. We first transform the fractional nonlinear differential equation into the equivalent Volterra integral equation. Then we approximate the integral by using the quadratic interpolation polynomials. On the first subinterval $$[t_{0}, t_{1}]$$, we approximate the integral with the quadratic interpolation polynomials defined on the nodes $$t_{0}, t_{1}, t_{2}$$ and in the other subinterval $$[t_{j}, t_{j+1}], j=1, 2, \dots N-1$$, we approximate the integral with the quadratic interpolation polynomials defined on the nodes $$t_{j-1}, t_{j}, t_{j+1}$$. A high-order numerical method is obtained. Then we apply this numerical method with the non-uniform meshes with the step size $$\tau _{j}= t_{j+1}- t_{j}= (j+1) \mu$$ where $$\mu = \frac{2T}{N (N+1)}$$. Numerical results show that this method with the non-uniform meshes has the higher convergence order than the standard numerical methods obtained by using the rectangle and the trapzoid rules with the same non-uniform meshes.

Keywords

Nonlinear fractional differential equation Numerical method Non-uniform meshes

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Authors and Affiliations

• Lili Fan
• 1
• Yubin Yan
• 2
1. 1.Department of MathematicsLvliang UniversityLvliangPeople’s Republic of China
2. 2.Department of MathematicsUniversity of ChesterInceUK

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