Singularity separation Chebyshev collocation method for weakly singular Volterra integral equations of the second kind

Volterra integral equation of the second kind with weakly singular kernel usually exhibits singular behavior at the origin, which deteriorates the accuracy of standard numerical methods. This paper develops a singularity separation Chebyshev collocation method to solve this kind of Volterra integral equation by splitting the interval into a singular subinterval and a regular one. In the singular subinterval, the general psi-series expansion for the solution about the origin or its Padé approximation is used to approximate the solution. In the regular subinterval, the Chebyshev collocation method is used to discretize the equation. The details of the implementation are also discussed. Specifically, a stable and fast recurrence procedure is derived to evaluate the singular weight integrals involving Chebyshev polynomials analytically. The convergence of the method is proved. We further extend the method to the nonlinear Volterra integral equation by using the Newton method. Three numerical examples are provided to show that the singularity separation Chebyshev collocation method in this paper can effectively solve linear and nonlinear weakly singular Volterra integral equations with high precision.