General K-order Franklin wavelet method for numerical solution of integral equations

Abstract

The classical Franklin system is a complete orthonormal set of piecewise linear continuous functions using Haar wavelet collocation points. This paper defines the general K-order Franklin function and introduces the general K-order Franklin wavelet method for solving Fredholm and Volterra integral equations. The method is also applied to solve mixed nonlinear Fredholm–Volterra integral equations. The general K-order Franklin wavelet method, like the higher-order Haar wavelet method, is a collocation method. Its advantage of not requiring constraint equations makes it easier to extend to higher orders, resulting in faster convergence. Several examples are provided to illustrate the reliability and effectiveness of the proposed method. Compared to the fourth-order convergence rate of the higher-order Haar wavelet method, the proposed general K-order Franklin wavelet method achieves a sixth-order convergence rate, improving both the rate of convergence and reducing the absolute error.