Volterra Direct quadrature methods for the numerical solution of Volterra integral equations of the second kind have been measured for centuries, and these methods are generalized in an ordinary way. By considering a class of variable-step quadrature methods, the Runge-Kutta methods  appear as extensions of the step-by-step quadrature methods, and theoretical understanding is readily obtained. Such understanding may provide confidence limits when creating practical algorithms . We investigate the behavior of the analytical and numerical solution of the Volterra-Hammerstein equation where the linear part of the kernel has a constant sign with conditions for the boundedness or decay of solutions and approximate solutions obtained by Volterra Direct Quadrature Method. Also establishing bounds on the analytical solution of Volterra-Hammerstein equation, on (0,T) or on (0,∞) under certain conditions on kernel k(.,.) and the functions f(.) and g(.) , where we also discussed the numerical method which will be needed to obtain the numerical solution of nonlinear Volterra-Hammerstein Integral equation of the second kind and thus it is committed to obtain similar bounds as analytical solution obtained by the method mentioned above. Some numerical experiments are also reported in this paper.
Volume 12 | 02-Special Issue