Numerical Solutions of Volterra Integral Equations Through Galerkin Weighted Residual Method with Charlier Polynomials

Volterra Integral Equations (VIEs) are a significant class of integral equations with broad applications in various fields, such as mathematical physics, engineering, biology, economics, and more. In this paper, we numerically solve the linear VIEs of both the first and second kind, with both regular and singular kernels, using the Galerkin Weighted Residual Method. Actually, we derive a straightforward and efficient matrix formulation by the Galerkin Method for each type of VIE, employing piecewise Charlier polynomials as the basis functions in the trial solution. Several numerical examples are tested to verify the effectiveness of the proposed method. The numerical results obtained by the proposed method converge monotonically to the exact solutions and, in some cases, achieve the exact solution. In addition, the proposed Charlier polynomials-based Galerkin method significantly outperforms other state-of-the-art methods for the numerical solution of VIEs.