A New Approach Based on Supplementary Newton‐Cotes Rules to Solve Volterra Integral Equations

ABSTRACT The purpose of the study is to propose an innovative approach for solving the Volterra integral equation with high accuracy employing supplementary Newton‐Cotes rules and Picard's sequences, where the kernel is a uniformly Lipschitz function. We provide a simple iterative numerical method based on the Picard sequence and additional Newton‐Cotes rules outside of the integration subinterval as a subset of . Furthermore, we employ two theorems to verify the existence and uniqueness of the solution, and we utilize two equivalent norms on to illustrate the convergence of the approach. To demonstrate the excellent accuracy of the approach, we present numerical examples and error estimation for smooth kernels. A non‐smooth kernel is also addressed, which highlights the various uses of the suggested approach. Finally, we investigate this method for Runge's kernel.