Linear Barycentric Rational Method for Two-Point Boundary Value Equations

Abstract

Linear barycentric rational method for solving two-point boundary value equations is presented. The matrix form of the collocation method is also obtained. With the help of the convergence rate of the interpolation, the convergence rate of linear barycentric rational collocation method for solving two-point boundary value problems is proved. Several numerical examples are provided to validate the theoretical analysis.

1. Introduction

The analysis of many physical phenomena and engineering problems can be reduced to solving the boundary value problem of differential equation, most of which need to be solved by the numerical method. The barycentric interpolation method is a high precision calculation method, and a strong form of collocation that relies on differential equation, which has been studied extensively by many scholars. The linear barycentric rational method (LBRM) [1–3] has been used to solve certain problems such as delay Volterra integro-differential equations [4], Volterra integral equations [5–7], biharmonic equation [8], beam force vibration equation [9], boundary value problems [10], heat conduction problems [11], plane elastic problems [12], incompressible plane elastic problems [13], nonlinear problems [14], and so on [1, 15].

In this article, we pay our attention to the numerical solution of two-point boundary value problems:

Let the interval be partitioned into uniform part with and with its related function . For any , with , to be the interpolation function at the point , then we have , andwhere

Change the polynomial into the Lagrange interpolation form as

Combining (7) and (5) together, we getwhere and .

Then we getwhere its basis function is

For the equidistant point, its weight function is

The Chebyshev point of the second kind isand its weight function is

Consider the barycentric interpolation function asand the numerical scheme is given as

By using the notation of the differential matrix, equation (13) is denoted as matrices in the form ofwhere .

Equation (13) is written as matrices in the form ofwhere and . Using interpolation formulas, boundary conditions can be discretized into

2. Convergence and Error Analysis

With the error function of difference formulaandwhere . Taking the numerical scheme

Combining (20) and (1), we havewhere .

The following Lemma has been proved by Jean-Paul Berrut in [13].

Lemma 1 (see [13]). For defined in (18), we haveLet be the solution of (1) and is the numerical solution, then we haveand

The results can be obtained in the reference of [14].

Based on the above lemma, we derive the following theorem.

Theorem 1. Let , and , we have

Proof. As , whereandPutting column 2, column 3, column n added to column 1, we havewhich means the matrix is the singular matrix.
Similarly we haveand then we assume with , , where. .
Bywhich meanswhere is the element of matrix .
Then we haveThe proof is completed.
We know that the central difference method can achieve quadratic convergence and the convergence order is the same as that of d = 3. When d >3, the convergence of the barycentric rational method is better than that of the central difference method.

3. Numerical Example

Example 1. Consider the two-point boundary value:and its analysis solution isIn this example, we consider the two-point boundary value equations with the boundary condition . In Table 1, the convergence rate of equidistant nodes with different is ; in Table 2, the convergence rate of the Chebyshev point of the second kind with different is . From Theorem 1, the convergence rate is , and there are no convergence rates as . Here the convergence rate is and in Tables 1 and 2 for , respectively, and we will give exact analysis in other paper.

Example 2. Consider the two-point boundary value.with the boundary conditionand its analysis solution isIn this example, we consider the variable coefficient of two-point boundary value equations with the boundary condition . In Table 3, the convergence rate of equidistant nodes with different is ; in Table 4, the convergence rate of the Chebyshev point of second kind with different is .

4. Concluding Remarks

In this paper, the numerical approximation of linear barycentric rational collocation method for solving two-point boundary value equations is presented. The matrix form of the algorithm is given for the simple calculation; with the help of Newton formula, the error function of the convergence rate is also obtained. For the constant coefficient and variable coefficient of two-point boundary value equations, numerical results show that the convergence rate can reach for the equidistant nodes and for the Chebyshev point of the second kind with . For the special case of , there are still convergence rates with , and the analysis of this phenomenon will be presented in other papers.

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The support from the Nature Science Foundation of Shandong (No. ZR2019PA021) is gratefully acknowledged.

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Copyright

Copyright © 2021 Qian Ge and Xiaoping Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.