Open Access
Issue
Wuhan Univ. J. Nat. Sci.
Volume 27, Number 4, August 2022
Page(s) 313 - 320
DOI https://doi.org/10.1051/wujns/2022274313
Published online 26 September 2022

© Wuhan University 2022

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

The mathematical formulation and subsequent assistance in the resolution of physical and other problems involving functions of multiple variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, and electrodynamics, are made possible by the use of partial differential equations[1-3]. After the first report by Cajori Florian in the year 1928 on partial differential equations (PDEs), partial differentiation and integration, there have been different kinds of PDEs. A large number of mathematical models in mathematical physics can be described by the one-dimensional Klein-Gordon equations. It has attracted much attention in studying classical and quantum mechanics, solitons and condensed matter physics[4,5].

For such problems, it is almost impossible to get the analytical solutions[6-9]. Thus one should consider numerical approximations to the Klein-Gordon equations. A variety of numerical techniques have been developed and compared for solving the Klein-Gordon equations[10-14]. These numerical techniques are based on the finite difference schemes[15] or spectral and pseudo-spectral methods[16,17].

To avoid the mesh generation, the traditional radial-basis-function-based meshless methods have attracted the attention of researchers [18-20]. Based on the Thin Plate Splines radial basis functions and the cubic B-spline scaling functions, Dehghan and his coworkers[21,22] proposed numerical schemes to solve the one-dimensional nonlinear Klein-Gordon equation with quadratic and cubic nonlinearity. Dehghan and Mohammadi[23] proposed two numerical meshless techniques based on radial basis functions and the method of generalized moving least squares for simulation of coupled Klein-Gordon-Schrodinger equations. The spectral meshless radial point interpolation technique is applied to obtain the solution of two- and three-dimensional coupled Klein-Gordon-Schrodinger equations by Shivanian and Jafarabadi [24]. Very recently, Ahmad et al [25] proposed a local meshless differential quadrature collocation method based on radial basis functions for the numerical simulation of one-dimensional Klein-Gordon equations. These traditional numerical techniques are based on two-level finite difference approximations. A direct meshless method, which belongs to the one-level type, is promising in dealing with such one-dimensional Klein-Gordon equations.

In this paper, we propose a direct meshless method with one-level approximation, based on the radial basis functions, for the one-dimensional Klein-Gordon equations. Since the Klein-Gordon equation is time-dependent, we present two schemes for the basis functions from radial and non-radial aspects. The radial aspect is similar to the traditional radial basis functions and the non-radial aspect is a newly-proposed scheme. The first scheme is fulfilled by considering time variable as normal space variable to construct an "isotropic" space-time radial basis function. The other scheme considered a realistic relationship between space variable and time variable which is not radial. The time-dependent variable is treated regularly during the whole solution process and the Klein-Gordon equations can be solved in a direct way.

The structure of this paper is organized as follows. Followed by Section 1, we describe the one-dimensional Klein-Gordon equations with initial and boundary conditions. Section 2 introduces the space-time radial and non-radial basis functions. Followed by Section 3, we present the methodology of the direct meshless method (DMM) for the one-dimensional Klein-Gordon equations under initial condition and boundary conditions. Several numerical examples are presented to validate the accuracy and stability of the proposed algorithms in Section 4. Some conclusions are given in Section 5 with some additional remarks.

1 One-Dimensional Klein-Gordon Equation

In this paper, we consider the general mathematical formulation of one-dimensional Klein-Gordon equation

(1)

in terms with the initial conditions

(2)

and boundary conditions

(3)

2 Formulation of the Space-Time Radial and Non-Radial Basis Functions

By using traditional numerical techniques, Eqs. (1)-(3) can be solved by using the two-level finite difference approximations or integral transform methods. In order to overcome the two-level strategy, direct meshless methods by using space-time radial and non-radial basis functions were worthy to be proposed.

As is known to all, the radial basis functions (RBFs) are "isotropic" for Euclidean spaces. For steady-state problems, the approximate solution can be written as a linear combination of RBFs with 2D or higher dimensions. Take the famous Multiquadric (MQ) RBF as an example

(4)

where is the Euclidean distance between two points and , is the RBF shape parameter.

However, there is only one space variable for the one-dimensional Klein-Gordon equation, the traditional RBFs are inapplicable in the direct sense. For this reason, we propose a simple meshless method by combining the space variable and time variable from the perspective of radial and non-radial. More specifically, the interval is evenly divided into segments firstly with corresponding finess . The time variable is evenly chosen from the initial time to a final time as with time-step . The corresponding configuration of the space-time coordinate is shown in Fig. 1. Then the space-time radial basis function can be constructed as

(5)

thumbnail Fig.1 Configuration of the space-time coordinates

"○" stands for the value of space variable [x], "•" stands for the value

of time variable [t] and "×" stands for the point [(x,t)]

is the Euclidean distance between two points two points and . Besides, it is possible to construct the space-time non-radial basis function which has the following expression

(6)

where is a parameter which reflects a realistic relationship between space variable and time variable .

We note that the space-time non-radial basis function which is product of two positive definite functions on space dimension and time dimension is investigated in Refs.[26-27]. For the MQ case, one has

(7)

However, the corresponding numerical results are not well in dealing with the problems in this research.

For two-dimensional cases, the space-time radial and non-radial basis functions can be easily obtained

(8)

(9)

Here is the Euclidean distance between two points two points and .

3 Implementation of the Direct Meshless Method (DMM)

Here, we consider the initial boundary value problem Eqs. (1)-(3) to illustrate the direct meshless method (DMM). Based on the definition of space-time radial and non-radial basis functions, Eqs. (1)-(3) can be solved directly in a one level approximation. The approximate solution of the function has the form

(10)

with the unknown coefficients.

To illustrate the direct meshless method, we choose collocation points on the whole physical domain which include internal points , initial boundary points and boundary points . According to the traditional collocation approach, substituting Eq. (10) into Eqs. (1)-(3), we have the following equations

(11)

(12)

(13)

(14)

where . We note that the initial boundary points are used twice to cope with initial conditions. Thus, the number of total collocation points .

Hence we should seek for the solution of the following linear algebraic system

(15)

where

(16)

is known matrix. and are vectors.

Eq. (15) can be solved by the backslash computation in MATLAB codes. From the above procedures, we can find that the implementation of the proposed direct meshless method is very simple.

4 Numerical Simulations

To compare with the previous literatures, we consider using the maximum error(ML), absolute error and root mean square error (RMSE) [28,29] defined as below:

(17)

where is the analytical solution at test points and is the numerical solutions at the test points . is the number of test points on the physical domain. The optimal choice of RBF parameter is beyond the scope of our current research. For more details about this topic, readers can be referred to Refs.[30,31] and references therein. The shape parameter is chosen prior to numerical results.

For simplicity, we denote the space-time radial basis function Eq. (5) and space-time non-radial basis function Eq. (6) as DMM1 and DMM2, respectively.

4.1 Example 1

Here, we consider an example of the one-dimensional Klein-Gordon equation with parameters and , i.e., . The corresponding exact solution is given as

(18)

which has a series form

(19)

with the following initial conditions

(20)

and boundary conditions

(21)

The corresponding source function is .

Numerical results of the DMM are listed in Table 1 with the final time T=1. We note that our time step , which leads to less computations, is far larger than the one in Refs.[25, 32]. It is clear from Table 1 that the two schemes of the DMM give the same or better accuracy compared to the numerical procedures reported in Refs.[25, 32].

Table 2 shows the time convergence rate in terms of the and error norms for the different time step sizes for fixed point parameter . From Table 2, we can find that the two schemes of the DMM perform better than the forward Euler difference formula (FEDF ) in Ref.[25].

For fixed parameter , the numerical results obtained by the DMM for a long range of shape parameter value , are shown in Fig. 2 for Example 1. Less sensitivity to the selection of the shape parameter in the case of the DMM2, in comparison to the DMM1, can be observed from Fig. 2. But the quasi-optimal parameter for the DMM1 is near the number .

thumbnail Fig.2 Shape parameter versus the ML of the DMM1 (a) and DMM2 (b)

The numerical results obtained by the DMM for the point parameter value are shown in Fig. 3. It can be seen that solutions of the two DMM schemes consistently converge very quickly from Fig.3. For example, the convergence rate for the square plate is about 9 for both DMM1 and DMM2 before reaching the minimum relative error value.

thumbnail Fig.3 Point parameter versus the ML of the DMM1 (a) and DMM2 (b)

Table 1

Numerical comparison of the maximum errors (ML) for example 1

Table 2

Time convergence results of the maximum errors and errors for example 1

4.2 Example 2

In this example, we consider the one-dimensional Klein-Gordon equation with parameters , and , i.e., . The corresponding exact solution is given as

(22)

with the following initial conditions

(23)

and boundary conditions

(24)

and the source function .

Numerical results of the DMM are listed in Table 3 with the final time . We note that our time step , which leads to less computations, is larger than the one in Refs.[25,32,33]. It is clear from Table 3 that the two schemes of the DMM give the same or better accuracy compared to the numerical procedures reported in Refs.[25,32,33]. Meanwhile the DMM results are more stable than the other methods for different time

For fixed parameter , the numerical results obtained by the DMM for a long range of shape parameter value , are shown in Fig. 4. Less sensitivity to the selection of the shape parameter in the case of the DMM1, as well as DMM2, can be observed from Fig. 4. The quasi-optimal parameter for the DMM1 is near the number which is similar to the previous example.

thumbnail Fig.4 Shape parameter versus the ML of the DMM1 (a) and DMM2 (b)

The numerical results obtained by the DMM for the point parameter value are shown in Fig. 5. It is shown that the DMM solutions consistently converge very quickly. Before reaching the minimum relative error value, the convergence rate for the square plate is about 9 for both DMM1 and DMM2. Table 4 shows the time convergence rate in terms of the and error norms for the different time step sizes , . From Table 4, we can find that the two schemes of the DMM perform better than the FEDF in Ref.[25].

thumbnail Fig.5 Point parameter versus the ML of the DMM1 (a) and DMM2 (b)

Table 3

Numerical comparison of the maximum errors for example 2

Table 4

Time convergence results of the maximum errors and errors for example 2

5 Conclusion

In this paper, a new direct meshless method is proposed for the one-dimensional Klein-Gordon equations. Two schemes are proposed for the basis functions from radial and non-radial aspects. The first scheme is fulfilled by considering time variable as normal space variable to construct an "sotropic" space-time radial basis function. The other scheme considered a realistic relationship between space variable and time variable which is not radial. Both schemes for the proposed meshless method are simple, accurate, stable, easy-to-program and efficient for the Klein-Gordon equations. More importantly, the proposed method can be used to nonlinear problems accompanied with iteration methods. The theory of our DMM procedure can be directly applied to wave propagation, transient heat transfer and thermo-elastic problems with high dimensions. Also, it is promising in dealing with fractional equations [34,35].

Moreover, there is much theoretical investigation that needs to be done in this area of numerical analysis. This will be studied in the near future.

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All Tables

Table 1

Numerical comparison of the maximum errors (ML) for example 1

Table 2

Time convergence results of the maximum errors and errors for example 1

Table 3

Numerical comparison of the maximum errors for example 2

Table 4

Time convergence results of the maximum errors and errors for example 2

All Figures

thumbnail Fig.1 Configuration of the space-time coordinates

"○" stands for the value of space variable [x], "•" stands for the value

of time variable [t] and "×" stands for the point [(x,t)]

In the text
thumbnail Fig.2 Shape parameter versus the ML of the DMM1 (a) and DMM2 (b)
In the text
thumbnail Fig.3 Point parameter versus the ML of the DMM1 (a) and DMM2 (b)
In the text
thumbnail Fig.4 Shape parameter versus the ML of the DMM1 (a) and DMM2 (b)
In the text
thumbnail Fig.5 Point parameter versus the ML of the DMM1 (a) and DMM2 (b)
In the text

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