Collocation method for numerical solution of two-dimensional Linear Volterra integral equations and prove its convergence

In this paper, we extend the collocation method for the numerical solution of two-dimensional Volterra integral equations. For this purpose, we first prove the existence and uniqueness of the solution of these types of equations and present a resolvent kernel representation for their solution. Then, we extend the collocation method using piecewise polynomials to solve the mentioned equations and obtain the corresponding algebraic system of equations and show that the system has a unique solution. We also prove the convergence of the method and obtain the order of convergence of the method by proving a theorem. Finally, we present some numerical examples to show the efficiency of the method and confirm the obtained theoretical results.