Analytical Solution of Two-Dimensional Sine-Gordon Equation

Abstract

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear sine-Gordon equations subject to appropriate initial conditions. Some lemmas which help us to solve the governing problem using the proposed method are proved. This scheme has the advantage of generating an analytical approximate solution or exact solution in a convergent power series form with conveniently determinable components. The method considers the use of the appropriate initial conditions and finds the solution without any discretization, transformation, or restrictive assumptions. The accuracy and efficiency of the proposed method are demonstrated by four of our test problems, and solution behavior of the test problems is presented using tables and graphs. Further, the numerical results are found to be in a good agreement with the exact solutions and the numerical solutions that are available in literature. We have showed the convergence of the proposed method. Also, the obtained results reveal that the introduced method is promising for solving other types of nonlinear partial differential equations (NLPDEs) in the fields of science and engineering.

Analytical Solution of One-Dimensional Nonlinear Conformable Fractional Telegraph Equation by Reduced Differential Transform Method

“…One of the greatest methods found by researchers is the reduced differential transform method (RDTM). What distinguishes this method is that it provides us with analytical approximations, which in many cases are exact solutions, in a rapidly convergent power series form with elegantly computed terms ( [12,35] and see the references therein).…”

“…Advances in Mathematical Physics Moreover, RDTM reduces the size of the calculations, and it solves the equations straightforwardly and directly without using Adomian's polynomial, perturbation, discretization, linearization or any other transformation, and restrictive conditions [35][36][37]. As a result, the RDTM can overcome the aforementioned constraints and restrictions of perturbation approaches as well as high computing complexity, allowing us to accurately evaluate nonlinear equations.…”

“…Paper [40] presents the RDTM for finding the exact and approximate analytical solution for the conformable time-fractional Swift-Hohenberg (S-H) equations with and without dispersive terms. For extra information, see [12,35,[41][42][43] and the references therein.…”

Approximate Analytical Solution to Nonlinear Delay Differential Equations by Using Sumudu Iterative Method

“…Recently, many researchers established and investigated various analytical and numerical methods to obtain approximate/exact solutions for nonlinear DEs as well as DDEs [6][7][8]. The variational iteration method (VIM) was employed by the authors of the publication [9][10][11][12] to discover a rough solution to nonlinear DDEs.…”

Analytical Solutions to Two-Dimensional Nonlinear Telegraph Equations Using the Conformable Triple Laplace Transform Iterative Method

“…Fractional partial differential equations (FPDEs) have become increasingly important in recent years for modeling a wide range of applications in real-world sciences and engineering, including fluid dynamics, mathematical biology, electrical circuits, optics, and quantum mechanics [2]. As a result, many researchers have focused on solving FPDEs in recent decades [3,4]. Since many physical and mechanical systems contain internal damping, which makes it impossible to derive equations describing the physical behavior of a non-conservative system using the traditional energy-based approach, fractional deriva-tive formulations can be used to model them more accurately.…”