Solve Riemann–Liouville boundary value problems using collocation boundary value methods with the graded mesh

Abstract

The paper explores numerical approaches to a class of fractional boundary value problems that can be converted into Volterra integral equations with multiple singular kernels. We propose a collocation boundary value method with a graded mesh for the resulting weakly singular integral equations. Moreover, we demonstrate the local convergence property of the presented method using the Gronwall’s inequality. In addition, we assess the stability of the algorithm by examining the L2-norm of the approximate solution. It is found that the proposed collocation method exhibits both fast convergence order and good stability. The numerical experiments provide rigorous validation of the theoretical results.