Finite difference scheme on non-uniform meshes for third-kind Volterra integral equations with nonsmooth solutions and its error analysis

This paper presents a high-order numerical method for nonlinear Volterra integral equations of the third kind (VIE3) with weakly singular kernels. To overcome the accuracy loss caused by the unbounded derivatives of the solution near the origin, we propose a fractional Adams-Simpson-type method on a graded mesh. The stability and convergence of the proposed scheme, along with detailed error estimates, are rigorously established. It is shown that the method achieves an optimal convergence order of $4-\alpha -\lambda \beta$, provided the mesh grading exponent $\lambda$ is chosen appropriately. Numerical experiments are presented to validate the theoretical convergence results and to demonstrate the effectiveness of the method in resolving initial singularities.