Exact Travelling Wave Solutions of the Nonlinear Klein-Gordon Equation Using the Modified Tanh-Function Expansion Method and a Direct Approach

الملخص

In this paper, we will use the tanh-function method with the aid of Maple software to find new and exact travelling wave solutions of the nonlinear Klein-Gordon equation. We will also rely on a direct algebraic method based on the Liénard equation to discover other new and exact solutions. The results include soliton solutions, periodic solutions, and rational function solutions. The new results obtained in this paper will be compared with the well-known results. Additionally, some 2D and 3D graphs of the obtained exact traveling wave solutions will be presented. Finally, the tanh-function expansion method used in this paper is straightforward and concise, and it can be applied to other nonlinear partial differential equations in mathematical physics.