Numerical approximation for a nonlinear variable-order fractional differential equation via an integral equation method

TL;DR

This paper introduces a novel numerical method for solving nonlinear variable-order fractional differential equations by transforming them into integral equations, with improved error estimates and convergence analysis.

Contribution

It develops a collocation method based on an integral equation approach that handles non-monotonic discretization coefficients and provides sharper error estimates.

Findings

Error estimates are improved with a new mesh grading parameter.

Convergence rates depend critically on the initial value of the variable order.

The method effectively approximates solutions to complex fractional differential equations.

Abstract

We study a numerical approximation for a nonlinear variable-order fractional differential equation via an integral equation method. Due to the lack of the monotonicity of the discretization coefficients of the variable-order fractional derivative in standard approximation schemes, existing numerical analysis techniques do not apply directly. By an approximate inversion technique, the proposed model is transformed as a second kind Volterra integral equation, based on which a collocation method under uniform or graded mesh is developed and analyzed. In particular, the error estimates improve the existing results by proving a consistent and sharper mesh grading parameter and characterizing the convergence rates in terms of the initial value of the variable order, which demonstrates its critical role in determining the smoothness of the solutions and thus the numerical accuracy.