On Numerical Treatments to Solve a Volterra - Hammerstein Integral Equation

In this paper a Volterra – Hammerstein integral equation (V-HIE), with two continuous kernels of position k (x, y) and of time F(t,T) , is considered in the Banach space C ([0,1] x [0,T]), T < 1. The existence of a unique solution of the V-HIE, is discussed and proved. A quadratic numerical method is used to obtain a system of Hammerstein integral equations (SHIEs) in position and the existence of a unique solution of the SHIEs, under certain conditions, is proved. Moreover, we use two different methods, quadratic method (QM) and Simpson's rule (SR), to transform, in each method, the SHIEs into a nonlinear algebraic system (NAS). In addition, the existence of a unique solution of each algebraic system is guaranteed and proved. The Adomian decomposition method (ADM) is used to solve SHIEs without having to convert the system to a linearity. Finally, some applications contain numerical results, in some different time, are calculated and the error estimate, in each case, is computed.