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

Abstract

<abstract> A nonlinear initial value problem whose differential operator is a Caputo derivative of order $ \alpha $ with $ 0 &lt; \alpha &lt; 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 $ \alpha $. Then an a posteriori error estimation is derived and used to design an adaptive grid generation algorithm. Numerical results complement the theoretical findings. </abstract>