JAMPJournal of Applied Mathematics and Physics2327-4352Scientific Research Publishing10.4236/jamp.2016.410194JAMP-71549ArticlesPhysics&Mathematics Solving of Klein-Gordon by Two Methods of Numerical Analysis JosephBonazebi Yindoula1AlphonseMassamba2GabrielBissanga1Laboratory of Numerical Analysis, Kibernetics and Applications, University Marien NGOUABI, Brazzaville, CongoDivision of Physics, Brazzaville Institute of Technology, Brazzaville, Congo13102016041019161929August 15, 2016Accepted: October 23, October 27, 2016© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

In this paper, the Decomposion Laplace-Adomian method and He-Laplace method are used to construct the solution of Klein-Gordon equation.

Laplace-Adomian Method He-Laplace Method Klein-Gordon Equation
1. Introduction

In field theory, the description of the free partide for the wave function in quantum physics obeys to Klein-Gordon equation  . In addition, it also appears in nonlinear optics and plasma physics.

In sum, the Klein-Gordon equation rises in physics in linear and non linear forms. In this paper we examine the Klein-Gordon equation, using the Laplace-Adomian de- composition method and He-Laplace method to get the exact solution. The Klein- Gordon equation is described as:

where are constants (spin zero) charged field, is a source term and

is a nonlinear function of.

2. Describing of Both Method2.1. The Laplace Transform [<xref ref-type="bibr" rid="scirp.71549-ref2">2</xref>]

Let’s note the laplace transform by

From (1), we have:

2.2. Laplace-Adomian Decomposition Method (LADM) [<xref ref-type="bibr" rid="scirp.71549-ref3">3</xref>] - [<xref ref-type="bibr" rid="scirp.71549-ref6">6</xref>]

Suppose that we need to solve the following equation:

subject to initial conditions:

E is a Banach space, where is a linear or a nonlinear operator, and u is the unknown function.

Let’s suppose that operator F can be decomposed under the following form:

where is linear, N nonlinear. Let’s suppose that L is inversible to the sense of Adomian with as inverse.

From above, by applying the Laplace transform to both sides of Equation (4), we have:

From the Equation (7), it follows:

and this equation gives

So, from the above Equation (9), we can write:

We have now:

We research solution of (4) in the following series expansion form

and we consider

where are the Adomian polynomials of and it can be calculated by formula given below.

Using Equation (12) and Equation (13) in Equation (11) we have:

From (15), we have the following Adomian algorithm:

and we obtain the Adomian algorithm:

Remark

In order overcome the short coming, we assume that can be divided into the sum of two parts namely and. Therefore,we get:

Instead of the iteration procedure Equation (17) we suggest the following modifi- cation

The solution through the modified Laplace decomposition method highly depends upon the choice of and.

2.3. He-Laplace Method [<xref ref-type="bibr" rid="scirp.71549-ref7">7</xref>]

We consider a general nonlinear non homogeneous partial differential equation with initial conditions of the form

N represents the general nonlinear differential operateur and is the source term.

Taking the Laplace transform on both sides of (20), we obtain:

Û

Applying the initial conditions given in (22), we have:

Operating the inverse Laplace transform on both sides of (23), we have

Now, we apply the homotopy perturbation method

and the non linear term can be decomposed as

for some He’s polynomials that are given by

Sustituding Equation (25) and Equation (26) in Equation (24), we get

Comparing the coefficients of like powers of p, we have the following approxima- tions:

3. Illustrative Examples

To demonstrate the applicability of the above-presented method, we have applied it to two linear and two non linear partial differential equations. These examples have been chosen because they have been widely discussed in literature.

3.1. Example 1

Consider the following linear Klein-Gordon equation

3.1.1. Application of the LADM

Applying the Laplace transform on both side of Equation (30) with the initial con- ditions, we have:

The inverse Laplace transform give us:

Û

We suppose that solution of (30) has the following form:

From (34) and (33). we have:

This result garantee that the following Adomian algorithm is:

Consequently,we obtain:

So that the solution of (30) is given by

which is the exact solution of problem.

3.1.2. Application of the He-Laplace Method

Applying the Laplace transform on both side of Equation (30) with the initial con- ditions, we obtain:

By applying inverse Laplace transform, we have:

Now applying the homotopy perturbation method, we have:

Comparing the coefficient of like powers of p, we have

which gives us

So that, the solution is given by:

3.2. Exemple 2

Consider the following nonlinear Klein-Gordon equation

where.

3.2.1. Laplace-Adomian Method

Using the Laplace transform, we have

Û

by applying inverse Laplace transformation to Equation (48), we hace

Supposing that the solution of (46) has the following form:

and

Taking (50) and (51) in to (49), we obtain:

According to the standard Adomian algorithm (52), we need to chose

. Here, we choose by convenience So, we

have the following Adomian algorithm

then garantee that:

So the exact solution of (46) is

3.2.2. He-Laplace Method

Using the Laplace transform, we have:

Now, we apply the inverse Laplace transformation to Equation (46), we have:

Applying the homotopy perturbation method, we have:

where are He’s polynomials. The first few components of He’s polynomials are given by

Comparing the coefficients of the like powers of p, we have:

So that, the exact solution is given by:

4. Applications4.1. Problem 1

Consider the following linear Klein-Gordon equation

Application of the LADM

Using the Laplace transform, we have

Û

By appling the inverse Laplace transform, we have:

Û

Û

From above equation, we have the following modified Adomian allgorithm:

Equation (69) give us:

Thus

and the exact solution of Equation (64) is

4.2. Problem 2

Consider the following nonlinear Klein-Gordon equation

Application of the LADM

Using the Laplace transform from (73), we have:

Now, we apply the inverse Laplace transform, we have:

Thus

Denoting that the solution of (73) has the following form:

Taking (77) and (78) into (76), we have:

and we obtain the following Adomian algorithm:

Calculation

Thus

So that, the solution is given by:

which is the exact solution of the problem.

4.3. Problem 3

Consider the following nonlinear Klein-Gordon equation

Application of the LADM

Using the Laplace transform, we have:

The inverse Laplace transformation is applied to Equation (85) we get

As before, we defines the solution by the series

and can be defined by an infinite series

The nonlinear term is decomposed in term of Adomian polynomials

Substituting (87), (88) and (89) into both sides of Equation (86) we obtain

The recursive relation is defined by

(91) give us

Thus

and the exact solution of Equation (84) is

5. Conclusion

Through these examplles, we showed again the usefulness of Laplace-Adomian Decomposition method and the He-Laplace method, in the search of an approximate solution of Klein-Gordon equation holds for the accepted forms of strong interaction of antiparticles in modern physics.

Cite this paper

Yindoula, J.B., Massamba, A. and Bissanga, G. (2016) Solving of Klein-Gordon by Two Methods of Numerical Analysis. Journal of Applied Ma- thematics and Physics, 4, 1916-1929. http://dx.doi.org/10.4236/jamp.2016.410194

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