EXACT SOLUTION OF NONLINEAR KLEIN-GORDON EQUATIONS WITH QUADRATIC NONLINEARITY BY MODIFIED ADOMIAN DECOMPOSITION

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 threedimensional surface and contour plots.


Introduction
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 where α, β and γ are constant, x = x + y + z and is the laplacian of u.Equation ( 1) and the given initial values is a generalised three-dimensional case of NK-GE.The nonlinear term g(u) could be a polynomial function in u or any other function.In application, the nonlinear term is a nonlinear force.For g(u) = sin u equation (1) becomes sine-Gordon equation, which could also be sinh-Gordon equation, double sinh-Gordon equation depending on the form of g(u).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.These include solid state physics, nonlinear optics, quantum theory, radiation theory, thermal equilibrium, field theory, harmonic oscillation of clocks in supersonic flow, you name it.
Many powerful methods have been used to investigate this class of partial differential equation equation.Some of which are the symplectic and multi-symplectic finite differnce scheme, Spectral and pseudo-spectral methods etc.Recently, [12] modified the exponentiation and expansion method and applied it to obtain exact solution to coupled Klein-Gordon-Zakharov equation.The solution were expressed in terms of hyperbolic, trigonomeric, exponential and rational functions.Other methods that gives exact solutions to NK-GE are inverse scattering method, the tanh-function method, F-expansion method etc.
Numerical solutions to NK-GE has also been studied.[14] used ADM with different version of Adomian polynomial to obtain numerical solutions to this class of equation.[9] compared solutions by ADM and Talor Matrix method obtained from the Telegraph equations that describes wave propagation in electric signal in cable transmission and Klein-Gordon equation.
[8] applied the radial basis function based on collocation method.[11] used the laplace decomposition method to solve nonlinear partial differential equation.
In this work, we show, for the first time, that ADM [1] studied in [2], [3], [4], [5], [6] can also be use to obtain exact solution to NK-GE.In the next section we present the generalised modified ADM to equation ( 1), we illustrate with examples how to apply it and we conclude.

The Concept of Modified ADM On NK-GE
By ADM equation ( 1) can be given as where L is a second order differential operator (in this case L tt = ∂ 2 ∂t 2 ), R is the remaining linear operator (in this case R = α∇ 2 u + β u).N is a nonlinear differential operator (N = γg(u)) and f ( x,t) is the source term.See [2], [3], [4], [5], [6] and the references there in for more details.Suppose L −1 tt of L tt exist the solution of equation ( 2) is given as and where is the Adomian polynomial given as Applying equations ( 3), ( 4) and (5) on equation (2), equation (1) becomes where We modify f ( x,t) by applying Talor series expansion on it.This result to a form putting equation (8) in equation ( 7) and applying the traditional concept ADM [1], we quickly identify where ω( x,t) represent the term arising from the source term and/or given initial conditions which must satisfy the origal equation (1).And subsequent terms given as ) continuing in this order, we have the recurrence relation as Adomian polynomials of frequently occuring nonlinear terms are given explicitely in [4] and [5].

Main results
In this section, we apply the concept of modified ADM to NK-GE with quadratic nonlinearity.

Example 1
In this example, we consider the NK-GE (1) with quadratic nonlinearity in one-dimensional space.The constants are α = −1, β = 0, γ = 1 in the interval −2 x 2. 0 < t 6, g(u) = u 2 and f ( x,t) = −x cost + x 2 cos 2 t.The initial conditions are u( x, 0) = x, u t ( x, 0) = 0 and the exact solution, as given in [8] is Applying equations ( 3)- (13) in this example, we instantly identify that and so on.The A k of u 2 are clearly given in [4] and [6].Consequently, the following are obtain as and so on.And The exact solution, as stated in [14], is Also applying equations ( 3)-(13) in the example, we have and the A k are as given in [4] and [6].Consequently,

2
cost which is the exact solution of example 1 as shown in equation (14).The physical behavior in form of 3D and contour plots of exact solution and Modified ADM solution are shown in Figures 1 -4.And, the slight variation, if at all, in the plots is as a result of considering a series of order 20.Example Also in this example, we consider the NK-GE (1) with quadratic nonlinearity still in onedimensional space.The constants are α = −1, β = π 2 4 , γ = 1 in the interval −3 x 3. 0 < t 4, g(u) = u 2 and f ( x,t) = x 2 sin 2 πt 2 .The initial conditions are u( x, 0) = 0, u t ( x, 0) = πt 2 .

x sin πt 2 which
is the exact solution of example 2 as given in equation(15).Figures5 -8shows the contour plots and 3D physical behavior of the exact and modified ADM solutions.The not-too obvious variation in the plots between the two solutions is as a result of taking only a series of order 20 in the modified ADM solution