Dynamical control on the homotopy analysis method for solving nonlinear shallow water wave equation

Abstract

In this paper, the nonlinear shallow water wave equation is illustrated. The famous semi-analytical method, homotopy analysis method (HAM) is applied for solving this equation. The main novelty, of this study is to validate the numerical results using the stochastic arithmetic, the CESTAC method and the CADNA library. Based on this method, we can find the optimal iteration of the HAM, optimal approximation of the shallow water wave equation and optimal error. The main theorem of the CESTAC method is proved. Based on this theorem, we can show that the number of common significant digits for two successive approximations are almost equal to the number of common significant digits for exact and approximate solutions. Thus instead of traditional absolute error to show the accuracyof method we can apply the new termination criterion depends on two successive approximations. In order to find the convergence region of the HAM, several ħ-curves are demonstrated.