The HPM provides a reliable technique that requires less work if compared with the traditional techniques and the method does not also require unjustified assumptions, linearization, discretization or perturbation. The numerical results showed that this method is very accurate. Findings - The authors obtained the one soliton solution for the KK equation by HPM. Results derived from this method are shown graphically. The results of numerical examples are presented and only a few terms are required to obtain accurate solutions. ![]() compared with the known analytical solutions. Design/methodology/approach - In this paper, the homotopy perturbation method (HPM) is used for obtaining soliton solution of the KK equation. The most important feature of this method is to obtain the solution without direct transformation. Purpose - The purpose of this paper is to obtain soliton solution of the Kaup-Kupershmidt (KK) equation with initial condition. Several computational examples of advective-diffusive problems are solved to demonstrate the increased stability limits. A stability analysis is performed on a particular version of this method for the diffusion equation. In this paper we develop a methodology for increasing the stability bounds of standard explicit finite differencing methods by combining explicit methods, implicit methods, and iterative methods in a novel way to generate new time-difference schemes, called preconditioned time-difference methods. ![]() The linear systems arising from these implicit methods are generally solved by iterative methods. These bounds are very small and implicit methods are used instead. Stability bounds for explicit time differencing methods on advective-diffusive problems are generally determined by. The disadvantage of explicit methods is the severe restrictions that are placed on stable time-step intervals. Explicit time differencing methods for solving differential equations are advantageous in that they are easy to implement on a computer and are intrinsically very parallel.
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