Tau numerical solution of the Volterra-Fredholm Hammerstein integro-differential equations with the Bernstein multi-scaling functions

Abstract

This paper involves the development of the Tau method with Bernstein multi-scaling (BMS) functions basis for the numerical solution of the Volterra-Fredholm Hammerstein integro-differential equations (VFHIDEs). For this purpose at the beginning we define BMS functions and express briefly some properties of BMS functions and after function approximation by using BMS functions, will be presented. Then, the operator matrix representation for the differential and integral parts seeming in the equation using the operational Tau method base on BMS functions basis, will be displaced. The operational Tau method transforms the differential and integration parts of the desired VFHIDEs to some operational matrices. In fact, this method reduces VFHIDEs to a system of algebraic equations. Numerical examples demonstrate the validity and applicability of the proposed method with BMS functions basis.