Abstract
In this work, a modern and novel approach method called the residual power series technique has been applied to find an analytical solution for an important equation in optical fibers called the Hirota–Satsuma coupled KdV equation with time as a series solution. Comparison of the analytical approximate solution with the exact solution concluded that the present method is an important addition for analyzing a system of partial differential equations that have a strong nonlinear term. We also represented graphically and discussed the effect of initial condition parameters and reaction of time on the model.
Highlights
Nonlinear coupled partial differential equations arise in many fields, such as mathematics, biology, chemistry, engineering, and physics such as plasma physics, water waves, fluid mechanics, thermodynamics, and other physical phenomena
A modern and novel approach method called the residual power series technique has been applied to find an analytical solution for an important equation in optical fibers called the Hirota–Satsuma coupled KdV equation with time as a series solution
The residual power series method is effectively applied to find a new solution in solitary wave type for the time Hirota–Satsuma–KdV equation
Summary
Nonlinear coupled partial differential equations arise in many fields, such as mathematics, biology, chemistry, engineering, and physics such as plasma physics, water waves, fluid mechanics, thermodynamics, and other physical phenomena. Many researchers in mathematical physics have given importance to these topics, and many analytical or numerical methods have been proposed, for example, homotopy perturbation method, variational iteration method, and a combination of both; homotopy analysis and differential transformation methods; DTM-Padé method; and wavelet homotopy analysis method.. We develop additional new approaches for the analytical solution of the combined Hirota–Satsuma and KdV equations. The Hirota–Satsuma system was proposed by Hirota and Satsuma.. Hirota and Satsuma and KdV equations have been reviewed by many authors and solved via different methods The method was first proposed by Abu Arqub.. The residual power series (RPS) technique has been applied successfully to get numerical solutions for many problems We used the RPSM to obtain approximate solutions for the Hirota–Satsuma–KdV equations, and comparisons with exact solutions show that the RPSM is an efficient method for solving the generalized Hirota–Satsuma–KdV equations
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