Solving Volterra-Fredholm integral equations by non-polynomial spline function based on weighted residual methods

In this paper, a method that utilizes a non-polynomial spline function based on the weighted residual technique to approximate solutions for linear Volterra-Fredholm integral equations is presented. The approach begins with the selection of a series of knots along the integration interval. We then create a set of basis functions, defined as non-polynomial spline functions, between each pair of adjacent knots. The unknown function is expressed as a linear combination of these basis functions to approximate the solution of integral equations. The coefficients of the spline function are calculated by solving a system of linear equations derived from substituting the spline approximation into the integral equation while maintaining continuity and smoothness at the knots. Non-polynomial splines are beneficial for approximating functions with complex shapes and for solving integral equations with non-smooth kernels. However, the solution’s accuracy significantly relies on the selection of knots, and the method may require extensive computational resources for large systems. To illustrate the effectiveness of the method, three examples are presented, implemented using Python version 3.9. The paper also addresses the error analysis theorem relevant to the proposed non-polynomial spline function.