Abstract
In this paper, we used the modified Adomian decomposition method (ADM) to obtain exact solution to Nonlinear Klein-Gordon equation (NK-GE) with quadratic nonlinearity. The paper contains an introduction and the concept of modified ADM for a generalized three-dimensional NK-GE. And, we applied this concept to obtain exact solution to two one-dimensional NK-GE with quadratic nonlinearity. The modified method is based on Taylors series expansion of the source term and implementation on any computer algebra software (Maple, Mathematica, etc) is simple. We discovered that the results of the examples considered are the same as the series solution of those obtained by using any known analytical method. Furthemore, we depicted our findings in three-dimensional surface and contour plots.
Highlights
IntroductionNonlinear Klein-Gordon equation (NK-GE) are partial differential equations used to model various space-time phenomena in physics and engineering which can be presented in general form as (1)
Nonlinear Klein-Gordon equation (NK-GE) are partial differential equations used to model various space-time phenomena in physics and engineering which can be presented in general form as (1)utt + α∇2u + β u + γg(u) = f (x,t) xεΩ = [a, b] ⊂ R, 0 < t T with u(x, 0) = h1(x), ut(x, 0) = h2(x) where α, β and γ are constant, x = x + y + z and ∇2u = ∂ 2u ∂ x2 + ∂ 2u ∂ y2∂ 2u ∂ z2 is the laplacian of u
We discovered that the results of the examples considered are the same as the series solution of those obtained by using any known analytical method
Summary
NK-GE are partial differential equations used to model various space-time phenomena in physics and engineering which can be presented in general form as (1). For a wide class of g(u), NK-GE has several Hamiltonian quantities [11] As it is mention in [12], [15], [8], [7], [13] and [16], NK-GE has several types of nonlinear terms, which in general, plays a significant role in very many scientific applications. [12] modified the exponentiation and expansion method and applied it to obtain exact solution to coupled Klein-Gordon-Zakharov equation. [14] used ADM with different version of Adomian polynomial to obtain numerical solutions to this class of equation. We present the generalised modified ADM to equation (1), we illustrate with examples how to apply it and we conclude
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