On The Stability of Volterra Runge-Kutta Method for Nonlinear Volterra- Hammerstein Integral Equation
Akter, Mosammat Arifa
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Abstract: Some Runge-Kutta methods for the numerical solution of Volterra integral equations of the second kind have been considered for the last 30 years, and these methods can be generalized in a natural 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 insight is readily obtained. Such insight may provide confidence limits when constructing practical algorithms . Here I investigate the behavior of the analytical and numerical solution of Volterra-Hammerstein equation where the linear part of the kernel has a constant sign and I provide conditions for the boundedness or decay of solutions and approximate solutions obtained by Volterra Runge-Kutta method. I also devoted to establishing bounds on the analytical solution of Volterra-Hammerstein equation, on ( or on under certain conditions on kernel k(.,.) and the functions f(.) and , where I also discussed the numerical method which will be needed to obtain the numerical solution of nonlinear Volterra-Hammerstein Integral equation of 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.