In this paper, the Decomposion Laplace-Adomian method and He-Laplace method are used to construct the solution of Klein-Gordon equation.
In field theory, the description of the free partide for the wave function in quantum physics obeys to Klein-Gordon equation [
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
Let’s note the laplace transform by
From (1), we have:
Suppose that we need to solve the following equation:
subject to initial conditions:
E is a Banach space, where
Let’s suppose that operator F can be decomposed under the following form:
where
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
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:
In order overcome the short coming, we assume that
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
We consider a general nonlinear non homogeneous partial differential equation with initial conditions of the form
N represents the general nonlinear differential operateur and
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
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:
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.
Consider the following linear Klein-Gordon equation
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.
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
Consider the following nonlinear Klein-Gordon equation
where
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
have the following Adomian algorithm
then garantee that:
So the exact solution of (46) is
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
Comparing the coefficients of the like powers of p, we have:
So that, the exact solution
Consider the following linear Klein-Gordon equation
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
Consider the following nonlinear Klein-Gordon equation
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
which is the exact solution of the problem.
Consider the following nonlinear Klein-Gordon equation
Using the Laplace transform, we have:
The inverse Laplace transformation is applied to Equation (85) we get
As before, we defines the solution
and
The nonlinear term
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
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.
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