Solving nonlinear mixed Volterra–Fredholm integral equations with two dimensional block-pulse functions using direct method

Few numerical methods such as projection methods, time collocation method, trapezoidal Nystrom method, Adomian decomposition method and some else are used for mixed Volterra–Fredholm integral equations. The main purpose of this paper is to use the piecewise constant two-dimensional block-pulse functions (2D-BPFs) and their operational matrices for solving mixed nonlinear Volterra–Fredholm integral equations of the first kind (VFIE). This method leads to a linear system of equations by expanding unknown function as 2D-BPFs with unknown coefficients. The properties of 2D-BPFs are then utilized to evaluate the unknown coefficients. The error analysis and rate of convergence are given. Finally, some numerical examples show the implementation and accuracy of this method.

Research highlights
► The block-pulse functions can lead more easily to recursive computations of concrete problems.
► Block-pulse functions transform a mixed nonlinear (V-FIE) to a linear equations system.
► The coefficients matrix is a lower triangular block matrix, so the computational cost is very low.
► Representation error is obtained when a differentiable function is represented in a series of BPFs.