Some solutions of the linear and nonlinear Klein–Gordon equations using the optimal homotopy asymptotic method
Introduction
Perturbation techniques have come to be considered as classical in solving nonlinear problems, specifically those that contain small parameters and therefore valid only for weakly nonlinear problems. These techniques are very effective in computing solutions, but the small parameter assumption, greatly restricts their applications.
In the last decade, a new breed of perturbation methods has emerged which is loosely based on Poincare’s homotopy applied in topology. The most fundamental work was done by Liao in 1992 with the introduction of the Homotopy Analysis Method (HAM) [8]. This was followed by the work of He in the form of the Homotopy Perturbation Method (HPM) [5] in 1998. Both methods were successful in solving nonlinear problems in science and engineering [9], [6], [13], [4].
Recently, Marinca et al. [7] introduced a new method known as the Optimal Homotopy Asymptotic Method (OHAM). The advantage of OHAM is in the built in convergence criteria similar to HAM but more flexible. In series of papers Marinca et al. [11], [12], [10] have applied this method successfully to obtain the solution of currently important problems in science, and have also shown its effectiveness, generalisation and reliability.
In many scientific and engineering applications, one of the corner stones of modeling are Partial Differential Equations (PDEs). Employed to model innumerable nonlinear phenomenon for instance in solid state physics, nonlinear optics, plasma physics, fluid dynamics, mathematical biology and chemical kinetics. The complexity of the equations though requires the use of numerical methods in most cases. The aim of our work is to investigate the effectiveness of the OHAM to solve PDEs. To this affect we consider the Klein–Gordon, time dependent equations, which are known to represent problems in classical and quantum mechanics, solitons and condensed matter physics [1], [15], [3].
Therefore, as mentioned earlier the objective of this paper is to show that the OHAM offers a reasonable, reliable solution to PDEs. To demonstrate this we intend to solve several examples in the succeeding sections considering the homogenous, non-homogenous, linear and nonlinear partial differential equations. We observe that the OHAM is not only useful for nonlinear differential equations but also useful for linear and nonlinear partial differential equations. Therefore, OHAM shows its validity and potential for the solution of nonlinear problems in science and engineering applications.
Section snippets
Basic formulation of OHAM
In this section we start by describing the basic idea of the OHAM [7], consider the boundary value problem of the formwhere A is a differential operator and B is boundary operator, u(r, t) is an unknown function, r and t denote spatial and temporal independent variables, respectively, Γ is the boundary of the domain Ω and f(r, t) is a known analytic function. According to the basic idea of OHAM, A can be divided intowhere L is a linear operator and N is
Application of OHAM to Klein–Gordon equations
Let us consider the following Klein–Gordon equation:with the initial and boundary conditions:where α, β and γ are known constants; when k = 2 we have quadratic nonlinearity and when k = 3 we have cubic nonlinearity; f(x, t), g1, g2, g3 and g4 are known functions, and the function u(x, t) is unknown.
To demonstrate the effectiveness of OHAM for PDEs and for time dependent problems, four special cases
Results and discussions
The formulation presented in Section 2 and illustration of the formulation in the examples given in Section 3 provides highly accurate solutions without spatial discretization for the problems. It is evident that there is no need of computing further higher order terms of u(x, t), when OHAM is used. Table 1, Table 3 show the absolute error between OHAM and the exact solution by ∣uEXACT − uOHAM∣. Table 1, Table 3 are constructed for spatial domain [0, 1] at time t = 1, t = 3, t = 6 and t = 9, t = 15, t = 20 for
Conclusions
In this paper, we applied a new powerful analytic technique, OHAM [7] for time dependent, linear and nonlinear partial differential equations. The main goal of this work was to demonstrate the effectiveness of OHAM. We have achieved this goal and verified by applying the formulation given in Section 2 to the Klein–Gordon equations. This approach is simple in applicability, as it does not require discretization like numerical methods. Furthermore, this method provides a convenient way to control
References (15)
- et al.
A decomposition method for solving the nonlinear Klein–Gordon equation
Journal of Computational Physics
(1996) - et al.
Solitary wave solution for a generalized Hirota–Satsuma coupled kdv equation by homotopy perturbation method
Physics Letters A
(2006) An approximation sol. technique depending upon an artificial parameter
Communications in Nonlinear Science and Numerical Simulation
(1998)Periodic solution and bifurcations of delay-differential equations
Physics Letters A
(2005)- et al.
Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer
International Communications in Heat and Mass Transfer
(2008) - et al.
An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate
Applied Mathematics Letters
(2009) The modified decomposition method for analytic treatment of differential equations
Applied Mathematics and Computation
(2006)