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Abstract
The generalized nonlinear Klien-Gordon equation plays an important role in quantum mechanics. In this paper, a new three-time level implicit approach based on cubic trigonometric B-spline is presented for the approximate solution of this equation with Dirichlet boundary conditions. The usual finite difference approach is used to discretize the time derivative while cubic trigonometric B-spline is applied as an interpolating function in the space dimension. Several examples are discussed to exhibit the feasibility and capability of the approach. The absolute errors and error norms are also computed at different times to assess the performance of the proposed approach and the results were found to be in good agreement with known solutions and with existing schemes in literature.
Citation: Mat Zin S, Abbas M, Abd Majid A, Md Ismail AI (2014) A New Trigonometric Spline Approach to Numerical Solution of Generalized Nonlinear Klien-Gordon Equation. PLoS ONE 9(5): e95774. https://doi.org/10.1371/journal.pone.0095774
Editor: Vladimir E. Bondarenko, Georgia State University, United States of America
Received: February 22, 2014; Accepted: March 28, 2014; Published: May 5, 2014
Copyright: © 2014 Mat Zin et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This study was fully supported by FRGS Grant of No. 203/PMATHS/6711324 from the School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Introduction
The generalized nonlinear Klien-Gordon (KG) equation arises in various problems in science and engineering. This paper focuses on the analysis and numerical solution of the generalized nonlinear KG equation, which is given in the following form [14]:(1)subject to initial conditions
(2)and with Dirichlet boundary conditions
(3)where
denotes the wave displacement at position and time
,
and
are real constants,
is a nonlinear function of
and
and
are known functions.
In particular, the KG equation is important in mathematical physics especially in quantum mechanics and it is well known as a soliton equation. A study on the interaction of soliton in collisionless plasma, the recurrence of initial states and examination of the nonlinear wave equations was in [1].
Several methods, in addition to several finite difference schemes, have been developed to solve the nonlinear KG equation. Jiminez and Vazquez [2] introduced four numerical schemes for solving the nonlinear KG equation. Ming and Guo utilized a Fourier collocation method for solving the nonlinear KG equation [3]. The KG equation was approximated using decomposition scheme by Deeba and Khuri [4] and using the Legendre spectral method by Guo et al [5]. Wong et. al. solved an initial value problem involving the nonlinear KG equation using fully implicit and discrete energy conserving finite difference scheme [6]. Wazwaz introduced the tanh and sine-cosine method to obtain compact and noncompact solutions for the nonlinear KG equation [7]. Sirendaoreji solved the nonlinear KG equation using the auxiliary equation method to construct new exact traveling wave solutions with quadratic and cubic nonlinerity [8]. Yucel solved the nonlinear KG equation using homotopy analysis method [9] and Chowdhury and Hashim solved the equation using homotopy-pertubation method [10].
B-spline functions can be used to solve numerically linear and non-linear differential equations. Caglar et. al. [11] has introduced a cubic B-spline interpolation method to solve the two-point boundary value problem. Hamid et al. [12] has introduced an alternative cubic trigonometric B-spline interpolation method to solve the same problem. Dehghan and Shokri [13] have obtained a numerical solution of the nonlinear KG equation using Thin Plate Splines radial basis functions. Khuri and Sayfy [14] have solved the generalized nonlinear KG equation using a finite element collocation approach based on third degree B-spline polynomials.
In this work, a new three-time level implicit approach which combines a finite difference approach and cubic trigonometric B-spline collocation method (CTBCM) is proposed to solve generalized nonlinear KG equation. The finite difference approach is proposed to discretize time derivative and cubic trigonometric collocation method is applied to interpolate the solutions at time. Two numerical experiments are carried out to calculate the numerical solutions, absolute errors,
error norms and order of convergence for each problem in order to show the accuracy of method.
Temporal Discretization
Consider a uniform mesh with grid points
to discretize the grid region
with
and
and
denote mesh space size and time step size respectively. The time derivative is approximated using the central finite difference formula
(4)
Using the approximation of equation (4), equation (1) becomes(5)
Using the weighted technique, the space derivatives of equation (5) becomes
(6)where
,
and the subscripts
and
are successive time levels.
After simplification, equation (6) leads to(7)
Trigonometric B-Spline Collocation method
In this section, CTBCM is used to solve Klien-Gordon equation. Cubic trigonometric B-spline are used to approximate the space derivatives. To construct the numerical solution, nodal points defined in the region
where
The approximate solution to the exact solution
is defined as [18]:
(8)where
are time dependent unknowns to be determined and
are cubic trigonometric B-spline basis function given as:
(9)where
Due to local support properties of B-spline basis function, there are only three non-zero basis functions
and
are included over subinterval
Thus, the approximation
and its derivatives with respect to x can be simplified as:
(10)where
The solution to problem (1) is obtained by substituting equation (10) into equation (7). Initially, time dependent unknowns are calculated and shown in section 3.1. Next, the following initial condition is substituted into the last term of equation (7) for computing
(10a)
Subsequently, the time dependent unknowns, for
should be generated. After simplification, equation (7) leads to
(11)where,
with
The system obtained on simplifying (11) consists of unknowns
in
linear equations at the time level
In order to obtain a unique solution, the equation (8) is applied to the boundary conditions given in equation (3) for two additional linear equations.
(12)
(13)
From equations (11), (12) and (13), the system can be written as(14)where,
Half explicit and half implicit scheme is produced by choosing to be 0.5. This scheme is known as Crank-Nicolson scheme. System (14) becomes a tri-diagonal matrix system of dimension
that can be solved using the Thomas Algorithm [17].
Initial vector 
The initial vectors can be obtained from the initial condition as well as boundary values of the derivatives of the initial condition [11], [15]:
(15)
This yields a matrix system:
(16)
The solution can be obtained by using the Thomas Algorithm [17].
Stability analysis using von Neumann method
The von Neumann analysis of stability considers the growth of error in a single Fourier mode [19]–[20](17)where
and
is the mode number. It is known that this method can be used to analyze the stability of linear scheme. All the nonlinear term in (11) are assumed to be zero [19]. Thus, equation (11) becomes
(18)where
and
Substituting the approximate solution (8) in equation (18) leads to
(19)where
In order to obtain the amplification factor the trial solution (17) is substituted in (19) and after some simplification, we obtain,
(20)where
and
Since
and
, the amplification factor of this scheme is
(21)where
and
Hence, this scheme is unconditionally stable.
Numerical Results and Discussions
In this section, the CTBCM is applied on two numerical problems. In order to measure the accuracy of the method, absolute errors and error norms are calculated using the following formulas [16]
(22)
(23)where
and
are analytical solution and approximate solution of proposed problem (1), respectively. The numerical order of convergence, p is obtained by using following formula [14]
(24)where
and
are the
at number of partition n and 2n respectively.
Problem.1
Consider the following nonlinear Klien-Gordon equation [13], [14](25)subject to the following boundary and initial conditions
The analytical solution of problem (25) is known to be and graphically shown in Fig. 1 (a). This problem is tested using different values of
and
to show the capability of the present method. The final time is taken to be
Fig. 1 (b) and Fig. 2 (a) show the approximate solution and Fig. 2 (b) shows the error of this problem with
and
Two cases of this problem are discussed where Case 1 and Case 2 consider
and
respectively. Numerical solutions, absolute errors and order of convergence of each case are tabulated in Tables 1–4 and Tables 5–8. The
error norms are compared to Dehghan and Shokri [13] and Khuri and Sayfy [14] in Table 3 and Table 7. The comparison indicates that the present method is more accurate. The order of convergence of the present problem is calculated by the use of the formula given in (24) and is tabulated in Table 4 and Table 8. An examination of these tables indicates the method has a nearly second order of convergence.
Problem.2
The following nonlinear Klien-Gordon equation which is also known as Sine-Gordon equation is considered [14](26)
It is subjected to initial conditions and boundary conditions as
The analytical solution of this problem is Fig. 3 (a) depicts a graph of this analytical solution. The final time is taken as
. The approximate solutions are calculated at time step size
with different mesh space size, h. Numerical solutions are recorded in Table 9 and graphical solutions are plotted in Fig. 3 (b) and Fig. 4 (a). Absolute errors are calculated and shown in Table 10 while the 3D error plot is depicted in Fig. 4 (b). Table 11 shows the comparison of
error norms between the present method with Khuri and Sayfy [14] method. This comparison shows that the present method gives better results.
Conclusions
In this work, Klien-Gordon equation has been successfully solved using CTBCM incorporating a finite difference scheme. Specifically, the central difference approach is used to discretize the time derivatives and cubic trigonometric B-spline is used to interpolate the solutions at displacement Well-known two test problem were solved using the proposed method and the solution obtained were in good agreement with the known solution. Accurate solutions at intermediate points can be easily obtained.
Acknowledgments
The authors are indebted to the anonymous reviewers for their helpful, valuable comments and suggestions in the improvement of this manuscript.
Author Contributions
Conceived and designed the experiments: SMZ MA AAM AIMI. Performed the experiments: SMZ MA. Analyzed the data: SMZ MA AAM AIMI. Contributed reagents/materials/analysis tools: SMZ. Wrote the paper: SMZ MA AAM AIMI. Calculations and employing B-spline method: SMZ MA. Solved the problem: SMZ MA AAM AIMI. Obtained the results: SMZ.
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