Numerical approximation of nonlinear Klein-Gordon (KG) equation with quadratic and cubic nonlinearity is performed using the element-free improved moving least squares Ritz (IMLS-Ritz) method. A regular arrangement of nodes is employed in this study for the numerical integration to compute the system equation. A functional formulation for the KG equation is established and discretized by the Ritz minimization procedure. Newmark’s integration scheme combined with an iterative technique is applied to the resulting nonlinear system equations. The effectiveness and efficiency of the IMLS-Ritz method for the KG equation have been testified through convergence analyses and comparison study between the present results and the exact solutions.

1. Introduction

The Klein-Gordon (KG) equation is essentially a relativistic version of the Schrödinger equation. It has wide applications in many scientific fields, such as quantum mechanics, solid state physics, and nonlinear optics [1]. Similar to the Schrödinger equation, the KG equation is considered as one of the important equations in mathematical physics, as well as kinds of solitons studies, especially in the investigation of solitons interactions for a collisionless plasma and the recurrence of initial states [2, 3].

As a kind of essential nonlinear PDEs, the KG type equations have received considerable attention in deriving both analytical and numerical solutions by using different types of methods, such as the Adomian decomposition method [3, 4], the sine-cosine ansatz and the tanh methods [2, 5, 6], the auxiliary equation method, the Weierstrass elliptic function method, the elliptic equation rational expansion method, and the extended -function method [79]. In the process, various numerical schemes have also been developed based on different theories, such as the homotopy method [10], the cubic B-spline collocation method on a uniform mesh [11], and the approximation with thin plate splines (TPS) radial basis functions (RBF) based collocation approach [12].

To seek for an effective and efficient numerical technique, the meshless method has been successfully developed to solve partial differential equations that used to describe many physical and engineering problems. The advantages of these meshless methods are as follows: (i) solutions can be obtained with only a minimum of meshing or no meshing at all [1318]; (ii) a set of scattered nodes is used instead of meshing the entire domain of the problem. Several meshless methods have been proposed and can be chosen as an alternative to search for approximate solutions of the KG equations [19, 20]. Based on different approximation functions, various meshless methods were proposed, such as the element-free Galerkin (EFG) method [21], the moving least squares differential quadrature method [22], the radial point interpolation method [23], the smooth particle hydrodynamics methods [24], the radial basis function [25], the element-free kp-Ritz method [2630], the meshless local Petrov-Galerkin method [31], the reproducing kernel particle method [32], and the local Krigging method [33].

In this study, by combining the IMLS approximation and the Ritz procedure, the element-free IMLS-Ritz method for numerical solution of the nonlinear KG equation is presented. The cubic spline weight function and linear basis are employed in this study. A regular arrangement of nodes is employed for numerical integration to compute the system equation. A functional formulation for the KG equation is established and discretized by the Ritz procedure. The essential boundary conditions are imposed by the penalty method. Newmark’s integration scheme is employed to solve the nonlinear system equations. The applicability of the IMLS-Ritz method is examined on a few selected example problems. The accuracy of the presented method is also investigated by comparing the obtained numerical results with the existing analytical solutions.

2. Theoretical Formulation

2.1. Equivalent Functional of the One-Dimensional Nonlinear KG Equation

We consider the following KG equation including the nonlinear term assubject to the initial conditionand the boundary conditions where , denotes the wave displacement at position and time , , , and are known functions, and , , and are real numbers (). The function is to be determined when functions , , and are given; for the case of quadratic nonlinearity and for a cubic nonlinearity.

An equivalent functional is defined in the weighted integral form based on (1) with the initial condition in the following form:Using integration by parts and the divergence theorem, (4) yields the following expression:where the weight is set to be in this numerical study.

2.2. Improved Moving Least Squares Shape Functions

The IMLS approximation was proposed for construction of the shape functions in the element-free method. In one-dimensional IMLS approximation, for all , , we definewhere is an inner product and is the Hilbert space.

In , for the set of points and weight functions , if functions satisfy the conditionswe furnish the function set as a weighted orthogonal function set with a weight function about points . If are polynomials, the function set is called a weighted orthogonal polynomials set with a weight function about points .

Consider an equation system from MLS approximation:where is the moment matrix. Then, (8) can be expressed as

If the basis function set , , is a weighted orthogonal function set about points , that is, ifthen (8) becomes

Subsequently, coefficients can be determined accordingly:that is,where

From (8) and (12), the expression of approximation function iswhere is the shape function and

The abovementioned formulation details an IMLS approximation in which coefficients are obtained directly. It is, therefore, avoiding forming an ill-conditioned or singular equation system.

From (16), we havewhich represents the shape function of the IMLS approximation corresponding to node . From (17), the partial derivatives of lead to

The weighted orthogonal basis function set is formed by using the Schmidt method as

Moreover, using the Schmidt method, the weighted orthogonal basis function set can be formed from the monomial basis function. For example, for the monomial basis functionthe weighted orthogonal basis function set can be generated by

When the weighted orthogonal basis functions in (20) and (21) are used, there exist fewer coefficients in the trial function.

3. The Ritz Minimization Procedure and Discretion Implementation

In the present work, the penalty method is used to modify the constructed functional in implementing the specified Dirichlet boundary conditions for a domain bounded by . We use a penalty parameter to penalize the difference between the displacement of the IMLS approximation and the prescribed displacement on the essential boundary. The penalty function can be expressed aswhere is the penalty parameter and is the specified function on the Dirichlet boundary . Normally, is chosen as which is case dependent.

The resulting functional enforcing the Dirichlet boundary conditions for the KG equation isSubstituting (5) and (22) into the functional of (23), we have the modified functional

The approximation of the field function can be obtained from (15) as follows:where

Substituting (25) into (24) and applying the Ritz minimization procedure to the maximum energy function that yields the following matrix form: where

To solve the above nonlinear system, time discretization of (28) is forming with Newmark’s integration scheme. According to the fundamental assumptions of Newmark’s integrationwe have where and are redefined as parameters here to influence the accuracy and stability of the integration.

The dynamic form of (28) at can be written asSubstituting (31) into (32), we have the full discretized equation

By solving the above iteration equations, we can obtain numerical solutions to the one-dimensional nonlinear Klein-Gordon equation.

4. Numerical Results and Discussion

Three selected examples are included with their numerical solutions obtained by the presented method for the nonlinear KG equation. The problems are solved using regular node arrangements. The convergence study is carried out for the results of the KG equation. The accuracy and efficiency of the IMLS-Ritz method are compared with available analytical solutions by evaluating the -norm and errors defined aswhere and present the exact solution and numerical approximation, respectively.

4.1. Example

Consider the KG equation (1) with quadratic nonlinearity (), by taking the parameters , , , and .

The exact solution of the equation is given as [1]

The corresponding initial conditions and Dirichlet boundary function can be extracted from the analytical solution directly as

In the present example, the numerical solutions are obtained as the penalty factor and . We examine the convergence of the element-free IMLS-Ritz method by varying the number of nodes () from 11 to 201. The -norm and errors of with CPU times are computed at with and tabulated in Table 1. We found that both -norm and errors arise as increases. This may be due to that once convergent result has been obtained, in this case on , the additional arranged nodes will cause errors being accumulated. Based on this observation, the following analysis will be performed using for accuracy consideration. We also investigated the influence of on the accuracy of the IMLS-Ritz method. As illustrated in Table 2, by varying from 2 to 3, accurate results can be furnished when . Furthermore, the predicted results are compared with the available exact solutions at and illustrated in Figure 1. It is apparent that a close agreement is obtained from the illustrated results. The computed results of for a time history are also predicted between  s and  s () (see Figure 2(a)). The corresponding absolute error contour is plotted in Figure 2(b). From the presented results, we can conclude that the approximate solutions generated by the IMLS-Ritz method agree well with the analytical results.

4.2. Example

In the present numerical example, we consider KG in (1) with a quadratic nonlinearity (), by taking the parameters , , , and . The initial conditions are described by

The exact solution of the equation is given as [1]

The corresponding Dirichlet boundary function can be extracted from the analytical solution directly as

In this analysis, numerical solutions are predicted and compared with the analytical solutions at , , , and the penalty factor . Table 3 presents the convergence patterns of the IMLS-Ritz results by varying from 6 to 101. A similar convergence trend is observed in Example ; that is, convergent results can be obtained from to 21; then, the errors are accumulated as increases. Table 4 illustrates the values of -norm and errors as varying from 2 to 3.5. A growing trend of -norm and errors is observed from Table 4, and the CPU time rises oscillatory as increases. As presented in Figure 3, the comparison study shows that the IMLS-Ritz method provides very similar solutions to the exact results. In Figure 4, the absolute errors of at a selected time point () and the absolute error contour on a time period () are exhibited at . Figure 5 is plotted at for comparison with Figure 4. Although the increase in number of nodes has been identified to be unaided in enhancing the accuracy of the approximation, it influences the smoothness of the solutions indeed.

4.3. Example

Consider the nonlinear Klein-Gordon equation (1) with a cubic nonlinearity (), by taking parameters as , , , and . The initial conditions are given bywhere and and .

The exact solution of the equation is given as [4]

The IMLS-Ritz computation is carried out by setting , the penalty factor , and . The -norm and errors of are computed with the number of nodes varied from 13 to 201. The results are tabulated in Table 5. It is apparent that both -norm and errors decrease as increases, indicating that convergent results are obtained by the IMLS-Ritz method. From Table 6, the results of numerical analysis suggested that satisfied accuracy can be achieved when . In Figure 6, the numerical and analytical solutions are plotted on a time point () and a time period (). From the comparison results, we can conclude that the IMLS-Ritz method provides very similar solutions to the exact results. In Figures 7 and 8, the absolute errors of at a selected time point () and the absolute error contour on a time period () are depicted at and , respectively. As expected, more accurate results can be obtained as increases in this example. From the results presented in both tables and figures, it is evident that the IMLS-Ritz values almost coincide with the exact solutions.

5. Conclusion

In this paper, an element-free IMLS-Ritz method and its numerical implementation on three examples of nonlinear KG equation have been presented. The effectiveness and efficiency of the IMLS-Ritz method for KG equation have been testified through convergence and comparison studies. From the numerical results, it is concluded that the agreement of the IMLS-Ritz solutions with the exact results is excellent. Due to difficulties of constructing analytical solutions for many nonlinear PDEs, the element-free IMLS-Ritz method will have great advantages for solving them through simple implementation with high accuracy.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


The work described in this paper was partly supported by the National Natural Science Foundation of China (Grant no. 61272098 and Grant no. 11402142), Open Research Fund Program for State Key Laboratory of Estuarine and Coastal Research (SKLEC201207), and Open Research Fund Program for Shandong Province Key Laboratory of Marine Ecology Environment and Disaster Prevention (2012011).