Fractional Collocation Method for Third-Kind Volterra Integral Equations with Nonsmooth Solutions

In this paper, we develop a collocation method for solving third-kind Volterra integral equations. In order to achieve high order convergence for problems with nonsmooth solutions, we construct a collocation scheme on a modified graded mesh using a basis of fractional polynomials, depending on a certain parameter λ . For the proposed method, we derive an error estimate in the L^∞ -norm, which shows that the optimal order of global convergence can be obtained by choosing the appropriate parameter λ and modified mesh, even when the exact solution has low regularity. Numerical experiments confirm the theoretical results and illustrate the performance of the method.