Analysis of a finite difference scheme for a nonlinear Caputo fractional differential equation on an adaptive grid

Abstract

A nonlinear initial value problem whose differential operator is a Caputo derivative of order α with 0<α<1 is studied. By using the Riemann-Liouville fractional integral transformation, this problem is reformulated as a Volterra integral equation, which is discretized by using the right rectangle formula. Both a priori and an a posteriori error analysis are conducted. Based on the a priori error bound and mesh equidistribution principle, we show that there exists a nonuniform grid that gives first-order convergent result, which is robust with respect to α. Then an a posteriori error estimation is derived and used to design an adaptive grid generation algorithm. Numerical results complement the theoretical findings.

Keywords

• caputo fractional derivative
• initial value problem
• adaptive grid
• mesh equidistribution