Abstract
Nonlinear heat equation solved by the barycentric rational collocation method (BRCM) is presented. Direct linearization method and Newton linearization method are presented to transform the nonlinear heat conduction equation into linear equations. The matrix form of nonlinear heat conduction equation is also obtained. Several numerical examples are provided to valid our schemes.
1. Introduction
In this article, the nonlinear heat conduction equation is considered as
where is the nonlinear term of and , with initial condition and the boundary condition
Combining (1)–(3), we get nonlinear heat conduction equation.
The linear and nonlinear heat conduction equation is a second-order linear partial differential equation [1–6]. There are analytical and numerical methods to solve the linear/nonlinearheat conduction equation. For the numerical methods, there are the finite difference method, finite element method and boundary element method, the spectral method [7], etc.
The barycentric formula have been obtained by the Lagrange interpolation formula [8–10] and been used to solve Volterra equation and Volterra integro-differential equation [11–14]. Floater and Kai, Klein and Berrut [15–17] have proposed a rational interpolation scheme and get the equidistant node of the barycentric formula. In recent papers, references [18–25] have been extended the barycentric collocation methods to solve initial/boudary value problems and linear/nonlinear problems.
In this paper, the linear barycentric rational collocation method (LBRCM) for nonlinear heat conduction equation is presented. Direct linearization and Newton linearization are presented to transform the nonlinear heat conduction equation into linear equations. The matrix form of linearization scheme for nonlinear heat conduction equation is also obtained.
This paper is organized as follows: in Section 2, the linearization scheme for nonlinear heat conduction equation is given. Direct linearization and Newton linearization are constructed, and then the matrix form of LBRCM is also presented. At last, two numerical examples are listed to illustrate our numerical scheme.
2. Linearization
Direct linearization and Newton linearization are presented to change the nonlinear heat conduction equation into linear equation. By choosing the equidistant nodes and Chebyshev nodes as collocation point to solve the nonlinear heat conduction equation, the direct linearization scheme can be adopted as
Newton linearization scheme can be adopted as and then we get the Newton linearization as
We partition the interval into and into with and . Then, we have where and and .
Taking equation (7) into equation (1), we can get the LBRCM as
3. Numerical Examples
In the following, two examples are given to valid our algorithm.
Example 1. Consider with Its analysis solutions is Firstly, direct linearization of nonlinear heat conduction equation is given as with its direct linearization scheme is and its matrix form is where
In Figure 1, the analysis solution is presented. In the following Figures 2 and 3, errors of equidistant nodes and Chebyshev nodes for LBRCM with are given, respectively; from the figure, we know that the accuracy of Chebyshev nodes is higher than equidistant nodes.
In the following Figures 4 and 5, errors of equidistant nodes and Chebyshev nodes for BLCM with are given, respectively; from the figure, we know that the accuracy of Chebyshev nodes is also higher than equidistant nodes.
In Tables 1 and 2, the errors of direct linearization scheme by LBRCM with equidistant nodes for and are presented.
In Tables 3 and 4, the errors of the LBRCM with Chebyshev nodes for and are presented.
Secondly, Newton linearization of nonlinear heat conduction equation is given as with its Newton linearization scheme that is and its matrix form is where and are defined as (16).
In Tables 5 and 6, the errors of Newton linearization scheme by LBRCM with equidistant nodes for and are presented.
In Tables 7 and 8, the errors of Newton linearization by LBRCM with Chebyshev nodes for and are presented.
From the table above, we know that the accuracy of Newton linearization scheme is higher than direct linearization scheme.
Example 2. Consider with the condition Its analysis solutions is In this example, direct linearization scheme is presented with its direct linearization scheme is and its matrix form is where and are defined as (16).
In Figure 6, the analysis solution is presented. In the following Figures 7 and 8, errors of equidistant nodes and Chebyshev nodes for LBRCM with are given, respectively; from the figure, we know that the accuracy of Chebyshev nodes is higher than equidistant nodes.
In Tables 9 and 10, the errors of direct linearization by LBRCM with equidistant nodes for and are presented.
In Tables 11 and 12, the errors of direct linearization by LBRCM with Chebyshev nodes for and are presented.
4. Concluding Remarks
The nonlinear heat conduction equation is solved by LBRCM with direct linearization scheme and Newton linearization scheme, different from the finite difference methods with the time variable computed separately, while the LBRCM can get the space variable and time variable simultaneously. The matrix form of LBRCM with direct linearization scheme and Newton linearization scheme is given from the corresponding scheme, and numerical results are given to valid our scheme. In the table, the convergence of LBRCM for equidistant nodes and Chebyshev nodes has been given, while the theorem analysis of direct linearization scheme and Newton linearization scheme is out of goal in this paper which will be given in the near future.
Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Conflicts of Interest
The author declares that he/she has no conflicts of interest.
Acknowledgments
The work of Jin Li was supported by Natural Science Foundation of Hebei Province (Grant No. A2019209533).