Using 2D and 1D block-pulse functions simultaneously for solving the Barbashin integro-differential equations

Abstract

The goal of this work is to look at how to solve Barbashin equations using a mix of Volterra–Fredholm integro-differential equations and Volterra equations. For the first time, the 1D and 2D block-pulse functions are employed in a hybrid technique to solve this two-dimensional integral problem concurrently. We construct a new matrix called the converter operating matrix for this purpose. The Barbashin equation is reduced to a linear system that can be solved using well-known methods employing direct computing. The proposed method's convergence theorem is demonstrated under appropriate conditions. In addition, some numerical results are presented in the paper, which supports the strong theoretical properties of our approach.