Efficient high-order algorithm for nonlinear functional mixed integral equations

Despite the wide applications of nonlinear functional mixed Volterra-Fredholm equations (VFIEs), not much attention has been paid to their numerical analysis. Further, a well-known challenge in solving nonlinear integral equations is the problem of simultaneously achieving high-order accuracy, computational efficiency and avoiding to solve nonlinear algebraic systems. For contraction maps in Banach spaces, fixed-point iterative methods can address the problem of solving systems. However, the issues of computational efficiency, high-order accuracy, and approximation of functional VFIEs remain largely unaddressed. In this article, a new cubature rule is proposed and used to develop a high (fourth) order method for nonlinear functional mixed VFIEs. To ensure computational efficiency and avoid solving systems, a Gauss–Seidel-type algorithm (GSTA) is formulated. In this case of GSTA, the convergence proof becomes quite challenging (no wonder the inefficient Jacobi-type idea is very popular in the literature). We use the Banach contraction principle and mathematical induction to rigorously prove the fourth-order convergence of the method. Several numerical examples are used to verify the theoretical convergence results. It is our belief that both the numerical scheme and convergence proof presented in this paper will serve researchers in devising and analyzing efficient, high-order schemes for other integral equations, even in higher dimensions.