A New Wavelet Method for Solving a Class of Nonlinear Volterra-Fredholm Integral Equations

Abstract

A new approach, Coiflet-type wavelet Galerkin method, is proposed for numerically solving the Volterra-Fredholm integral equations. Based on the Coiflet-type wavelet approximation scheme, arbitrary nonlinear term of the unknown function in an equation can be explicitly expressed. By incorporating such a modified wavelet approximation scheme into the conventional Galerkin method, the nonsingular property of the connection coefficients significantly reduces the computational complexity and achieves high precision in a very simple way. Thus, one can obtain a stable, highly accurate, and efficient numerical method without calculating the connection coefficients in traditional Galerkin method for solving the nonlinear algebraic equations. At last, numerical simulations are performed to show the efficiency of the method proposed.

An Algorithm for the Solution of Nonlinear Volterra–Fredholm Integral Equations with a Singular Kernel

“…They are based on general approximate functions, Bernstein polynomials, Chelyshkov polynomials, Fibonacci polynomials, Boubaker polynomials, Bell polynomials, Lucas polynomials, Muntz-Legendre polynomials, Jacobi polynomials, Bernoulli polynomials, and block-pulse functions. Galerkin methods are also one of the methods that attracted the attention of researchers and are widely used with general approximate functions, Bernstein polynomials [12], Legendre polynomials [13], Alpert's multiwavelet bases [14], and conflict-type wavelets [15]. Furthermore, among the numerical methods that have been developed are the Quadrature methods [16], Homotopy analysis methods [17], Modified homotopy perturbation methods [18], and least squares approximation methods [19].…”