Application of Iterative Three-Dimensional Laplace Transform Method for 2-Dimensional Nonlinear Klein-Gordon Equation

Abstract

In the present study, the exact analytical solutions of the 2-dimensional nonlinear Klein-Gordon equation (NLKGE) is investigated using the 3-dimensional Laplace transform method in conjunction with the Daftardar-Gejji and Jafari Method (Iterative method). Through this method, the linear part of the problem is solved by using the 3-dimensional Laplace transform method, while the noise terms from the nonlinear part of the equation disappear through a successive iteration process of the Daftardar-Gejji and Jafari Method (DJM), where a single iteration gives the exact solution to the problem. Three test modeling problems from mathematical physics nonlinear Klein-Gordon equations are taken to confirm the performance and efficiency of the presented technique. For each illustrative example, the convergence of the novel iterative approach is shown. The findings also suggest that the proposed method could be applied to other types of nonlinear partial differential equation systems. HIGHLIGHTS Partial differential equations play a crucial role in the formulation of fundamental laws of nature and the mathematical analysis in different fields of studies The Klein-Gordon equation (KGE) is a nonlinear hyperbolic class of partial differential equations that occur in relativistic quantum mechanics and field theory, both of which are crucial to high-energy physicists The 3-dimensional Laplace transform method provides rapid convergence of the exact solution without any restrictive assumptions about the solution for linear differential equations The Laplace Transform method is often combined with other methods like the Daftardar-Gejji and Jafari (the new Iterative) method to solve complicated nonlinear differential equations The new iterative technique yields a series that can be summed up to get an analytical formula or used to build an appropriate approximation with a faster convergent series solution GRAPHICAL ABSTRACT