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.
Highlights
Nonlinear phenomena, which appear in many areas of scientific fields such as solid-state physics, plasma physics, fluid dynamics, mathematical biology, and chemical kinetics, can be modeled by partial differential equations
Duan et al [35] proposed a numerical model based on the lattice Boltzmann method to obtain the numerical solutions of the two-dimensional generalized sine-Gordon equation, and the method was Advances in Mathematical Physics extended to solve the nonlinear hyperbolic telegraph equation as indicated in [36]
The main aim of this study is to obtain the approximate analytical solutions for the two-dimensional nonlinear sineGordon equation (TDNLSGE), since most of the research focused on the numerical solutions for this problem
Summary
Nonlinear phenomena, which appear in many areas of scientific fields such as solid-state physics, plasma physics, fluid dynamics, mathematical biology, and chemical kinetics, can be modeled by partial differential equations. The nonlinear sine-Gordon equation (SGE), a type of hyperbolic partial differential equation, is often used to describe and simulate the physical phenomena in a variety of fields of engineering and science, such as nonlinear waves, propagation of fluxions, and dislocation of metals, for details see [10] and the references therein. As one of the crucial equations in nonlinear science, the sine-Gordon equation has been constantly investigated and solved numerically and analytically in recent years [10, 14,15,16,17,18]. Duan et al [35] proposed a numerical model based on the lattice Boltzmann method to obtain the numerical solutions of the two-dimensional generalized sine-Gordon equation, and the method was Advances in Mathematical Physics extended to solve the nonlinear hyperbolic telegraph equation as indicated in [36]
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