Abstract
Partial differential equations (PDEs) are widely used in mechanics, control processes, ecological and economic systems, chemical cycling systems, and epidemiology. Although there are some numerical methods for solving PDEs, simple and efficient methods have always been the direction that scholars strive to pursue. Based on this problem, we give the meshless barycentric interpolation collocation method (MBICM) for solving a class of PDEs. Four numerical experiments are carried out and compared with other methods; the accuracy of the numerical solution obtained by the present method is obviously improved.
1. Introduction
Consider the following partial differential equation:the initial conditions are expressed asand the boundary conditions are expressed asorwhere represents an unknown function at position and time . is a known constant. If is nonlinear, the equation is a nonlinear PDEs. If is linear, then the equation is linear PDEs.
PDEs are paid more attention, and there are many studies on the numerical solution of PDEs. For PDEs, a quintic polynomial spline technique is used for the numerical solutions of fourth-order PDEs with Caputo time fractional derivative on a finite domain [1]. The adaptive grid Haar wavelet collocation method is used for the numerical solution of parabolic PDEs [2]. The numerical stability of PDEs with piecewise constant arguments is considered [3] along with a quantum algorithm for preparing states that encode solutions of nonhomogeneous linear PDEs [4]. A time discretization scheme approximates the time fractional PDEs [5]. We also consider the application of a domain-type method of fundamental solutions together with a Picard iteration scheme for solving nonlinear elliptic PDEs [6] and a deep learning-based approach that can handle general high-dimensional parabolic PDEs [7]. Couple double Laplace transform with iterative method solves nonlinear Klein-Gordon equation subject to initial and boundary conditions [8]. The variational iteration method (VIM) and homotopy perturbation method (HPM) are used for various kinds of nonlinear problems ([9–12]). Numerical solutions to linear and nonlinear PDEs use the peridynamic differential operator [13]. Numerical collocation approach solves PDEs [14]. A modified method using wavelet intergrid operators known as new wavelet based full-approximation scheme similar to multigrid full-approximation scheme is developed for the numerical solution of nonlinear elliptic PDEs [15]. B-spline collocation method ([16–23]) had solved various kinds of nonlinear PDEs. Meshless Barycentric interpolation collocation method ([24–30]) is a high precision numerical method. This paper gives the meshless barycentric interpolation collocation method (MBICM) for solving a class of PDEs.
This paper is organized as follows. In Section 2, the meshless barycentric interpolation collocation method (MBICM) is introduced in detail. Several numerical examples are included to demonstrate the practicability of the method and some relevant figures, tables, and analyes are provided in Section 3. Some concluding remarks are given in Section 4.
2. Meshless Barycentric Interpolation Collocation Method
The MBICM uses the barycentric interpolation as the approximate function and uses the differential matrix of the barycentric interpolation to discrete the differential operator. It does not need to divide the calculation grid, and the calculation program is easy to write. It is an efficient and high precision meshless numerical method for calculating linear differential equations. For nonlinear PDEs, we construct a linearized iterative scheme to transform the equations into linear equations.
First, we introduce the MBICM for linear PDEs.
2.1. The Partial Differential Matrix of Barycentric Interpolation
For the nonlinear (1), we give an initial value and construct a linear iterative scheme
Next, we give meshless barycentric interpolation collocation method of (5).
In the interval takes different nodes, , respectively, in the interval takes different nodes, , respectively. These nodes can generate two dimensional nodes on the rectangular area , as follows: ; .
The barycenter interpolation form of function at the nodes ; can be expressed aswhere , . Moreover, is the barycentric interpolation basis function of nodes on interval , is the barycentric interpolation basis function of nodes on interval . is the function value of at the nodes ; .
According to formula (6), the partial derivative of order of function can be written asand the function values of the formula (7) at the nodes are
The nodes on the interval and constitute two groups of column vectors respectively. They are defined asthen the matrixes , above the region can be generated by the above column vectors , and the form of matrixes and is as follows:
The matrixes , are respectively written as column vectors , of dimension:
The relations between the components of the formula (8) and the components of formula (10) are as follows:
The function values of formula (6) and formula (7) at the node form column vectors , and they are as follows:
Therefore, formula (8) can be expressed in the following matrix form:in formula (12), is the Kronecker product of matrixes and , and it is also called order partial differential matrix at nodes ; . and are order differential matrixes formed by barycenter interpolation at node interval and interval , respectively.
Denote thatwhere and are order unit matrix and order unit matrix, respectively.
By using notation (6), (7), and (14), the calculation formula of BICM for model (1) can be written in following matrix form:
Here, is a symbol of diagonal matrix composed of vectors. If the linear iterative scheme (5) is convergent, then . In the calculation process, we set a calculation accuracy or , if , then the iteration stops, and at this point, approximate solution of nonlinear differential equations is obtained.
2.2. The Discrete Formula and Application Method of Initial Boundary Conditions
The discretization of initial boundary conditions requires the use of the barycenter interpolation formula (6). By acting on the initial boundary conditions, the discrete formulae of initial boundary conditions are given:
The boundary conditions are replaced by displacement method. The following is a description of the specific displacement process:
The first equation, the first equation, , and the first equation of the system of equations are replaced by equations of the initial condition (17); the first equation, the first equation, , and the first of the system of equations are replaced by equations of the initial condition (18); the first equation to the first equation of the system of equations are separated by equations of the boundary condition (19); the first equation to the first equation of the system of equations are replaced by equations of the boundary condition (20).
3. Numerical Experiment
Now, numerical examples of nonlinear Klein-Gordon equations, other types of equations, and systems of equations are solved by using MBICM. In the following numerical examples, we compare the errors of different methods to illustrate the accuracy of the barycentric interpolation collocation method. The errors are expressed by the error norm , , and , error norms and absolute error are expressed as follows:
Error Norm
Error Norm
Absolute Errorwhere and are exact and numerical solution, respectively.
Example 1 (see [31]). Consider the nonlinear Klein-Gordon equation (1) with and where , . Equation (24) satisfies the initial conditions:and with Dirichlet boundary conditionsThe exact solution for this problem is
In Table 1, we enumerate the error norms in different time and using different methods. By comparison, it is found that the present method is more accurate than other methods.
In Figure 1, the exact solution of obtained by the present method at with the number of nodes . Figure 3, the numerical and exact solutions of obtained by present method at different time and with the number of nodes . From this we can see that the numerical solution and the exact solution have high reunification, so the accuracy of the present method is high.
In Figure 2, the absolute error of obtained by present method at with the number of nodes . In Figure 4, the absolute errors of are obtained by the present method at different times , and with the number of nodes . We can get the absolute error of changing with at as the most stable.
Example 2 (see [31]). Consider the nonlinear Klein-Gordon equation (1) with and ,where , . Equation (28) satisfies the initial conditions:and with Dirichlet boundary conditionsThe exact solution for this problem is
In Table 2, we list the error norms in different times , and using different methods. By comparison, we found that the present method is much more accurate than other methods.
In Figure 5, the exact solution of is obtained by the present method at with the number of nodes . In Figure 7, numerical and exact solutions of are obtained by the present method at different times , and with the number of nodes . From this we can see that the numerical solution and the exact solution have high reunification, so the accuracy of the present method is high. Moreover, we find the monotonicity of function at different time.
In Figure 6, the absolute error of is obtained by the present method at with the number of nodes . In Figure 8, the absolute errors of are obtained by the present method at different times , and with the number of nodes . We can get the absolute errors of changing with at and are the most stable. But the absolute error of changing with fluctuates greatly at .
Example 3 (see [32]). We consider nonlinear Hunter-Saxton equation:where , . Equation (32) satisfies the initial conditions:and the boundary conditionsThe exact solution of this problem is as follows:
In Tables 3 and 4, the absolute errors of are obtained by the present method for different with the number of nodes at and , respectively. It can be found that the absolute error decreases with the decrease of time in the same number of nodes and .
In Figure 9, the numerical and exact solutions of are obtained by the present method at different times and with the number of nodes . From this we can see that the numerical solution and the exact solution have high reunification, so the accuracy of the present method is high. Moreover, the smaller the value of is, the faster the function changes with . In Figure 10, the absolute error of is obtained by the present method with the number of nodes .
In Figure 11, the logarithmic absolute errors of are obtained by the present method at different times and with the number of nodes . In Figure 12, the absolute errors of are obtained by the present method at different times and with the number of nodes . Both figures show that the errors fluctuate and then tend to be stable with the increase of at ; the errors fluctuate slightly and remain stable with the increase of at .
Example 4 (see [32]). For nonlinear Hunter-Saxton equation (32) with the following initial conditions The exact solution is as follows: where
In Table 5, there are the error norms of with different values nodes at and . We can see that the accuracy of is improved with the increase of the number of nodes; when the number of nodes increases to a certain number, the accuracy tends to be stable; it shows that the convergence of the present method is better. On the other hand, has the highest accuracy at .
In Figure 13, the numerical and exact solutions of are obtained by the present method at different times , and with the number of nodes . From this we can see that the numerical solution and the exact solution have high reunification. Moreover, the bigger the , the smaller the value. In Figure 14, the absolute error of is obtained by the present method with the number of nodes . The accuracy of the present method can be very high.
In Figure 15, the logarithmic absolute errors of are obtained by the present method at different times , and with the number of nodes . In Figure 16, the absolute errors of are obtained by the present method at different times , and with the number of nodes . We can get that the absolute error of function has obvious fluctuation at , and the absolute error of is always relatively stable at .
4. Conclusions and Remarks
In this paper, we first transform a nonlinear partial differential equation into a linear differential equation by using the direct linearization iterative method and then solve the linear partial differential equation by using the meshless barycentric interpolation collocation method. The error estimation and convergence analysis of the method are proved. Through four examples and their charts, it is found that this method has higher computation accuracy than the existing methods. Finally, this method is applied to nonlinear partial differential equations and the calculated accuracy is higher.
All computations are performed by the MatlabR2007b software packages.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This paper is supported by the Natural Science Foundation of Inner Mongolia , Inner Mongolia Maker Collaborative Innovation Center of Jining Normal University and the Numerical Analysis of Graduate Course Construction Project of Inner Mongolia University of Technology [grant number KC2014001], and the National Natural Science Foundation of China .