Vandermonde-Interpolation Method with Chebyshev Nodes for Solving Volterra Integral Equations of the Second Kind with Weakly Singular Kernels

in this work, we present an advanced interpolation method via the Vandermonde matrix for solving weakly singular Volterra integral equations of the second kind. The optimal rules for the node distributions of the two kernel variables were created to guarantee that the kernel's singularity was isolated. The unknown function is interpolated using three matrices: one of which is the monomial matrix, based on the Vandermonde matrix and Chebyshev nodes; the second is the known square Vandermyde matrix, and the third is the unknown coefficient matrix. The singular kernel is interpolated twice and transformed into a double-interpolated non-singular function through five matrices, two of which are monomials. A linear algebraic system can be obtained without using the collocation points by inserting the interpolated unknown function on the left and right sides of the integral equation. The solution of the obtained system yields the unknown coefficients matrix and thereby finds the interpolated solution. The obtained results from solving six examples are faster to converge to the exact ones using the lowest degree of interpolants and are better than those achieved by the other indicated method, which confirms the novelty and efficiency of the presented method.