An approximate solution of bivariate nonlinear Fredholm integral equations using hybrid block-pulse functions with Chebyshev polynomials

Abstract

In this paper, we consider a bivariate nonlinear Fredholm integral equation of the second kind. Then an approximate solution for some of these problems is investigated. To determine the aimed solution, the hybrid of 2D block-pulse functions with Chebyshev polynomials basis with the operational matrices is applied. In this work, we generalize the operational matrices stated by Behbahani (J Basic Appl 4:131–141, 2015), from one-dimensional to 2D space. Comparing this technique with other works, we can distinguish the advantage of developing the solution with fewer runtime and computations with more satisfactory approximations. In this procedure, using operational matrices, the nonlinear Fredholm integral equations are reduced to a system of nonlinear algebraic equations. Furthermore, the convergence analysis for this numerical method is investigated. Numerical examples are presented to describe the performance and rectitude of this method.