Abstract
In this paper, the combined double Sumudu transform with iterative method is successfully implemented to obtain the approximate analytical solution of the one-dimensional coupled nonlinear sine-Gordon equation (NLSGE) subject to the appropriate initial and boundary conditions which cannot be solved by applying double Sumudu transform only. The solution of the nonlinear part of this equation was solved by a successive iterative method, the proposed technique has the advantage of producing an exact solution, and it is easily applied to the given problems analytically. Two test problems from mathematical physics were taken to show the liability, accuracy, convergence, and efficiency of the proposed method. Furthermore, the results indicate that the introduced method is promising for solving other types of systems of NLPDEs.
Highlights
IntroductionIn 2018, the authors of [15] studied the coupled sineGordon equations in nonlinear optics, which describe the propagation of an optical pulse in fiber waveguide, and they derived the new exact solutions of the problem through the use of the well-organized modified Kudryashov method
The system of coupled sine-Gordon equations in the form
In the paper [25], the authors’ applied the double Sumudu transform linked with the Adomian decomposition method or with variational iteration method to find the analytical solution of nonlinear fractional partial differential equations
Summary
In 2018, the authors of [15] studied the coupled sineGordon equations in nonlinear optics, which describe the propagation of an optical pulse in fiber waveguide, and they derived the new exact solutions of the problem through the use of the well-organized modified Kudryashov method. In the paper [25], the authors’ applied the double Sumudu transform linked with the Adomian decomposition method or with variational iteration method to find the analytical solution of nonlinear fractional partial differential equations. The main aim of this work is to apply the double Sumudu transform method coupled with the new iterative method (NIT) proposed by Daftardar-Gejji and Jafari in [33] to find an exact/approximate solution of the nonlinear coupled sine-Gordon equation. Double Sumudu transform for the second partial derivative with respect to x is given by "
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