Abstract

The numerical solution for a kind of third-order boundary value problems is discussed. With the barycentric rational interpolation collocation method, the matrix form of the third-order two-point boundary value problem is obtained, and the convergence and error analysis are obtained. In addition, some numerical examples are reported to confirm the theoretical analysis.

1. Introduction

Differential equations can give full play to their mathematical advantages in various disciplines. Combining the theory of differential equations with practical problems can build models of practical problems. Many engineering and physical problems can be transformed into the initial boundary value problems of differential equations. In these problems, only a few simple cases can be solved analytically, and most engineering problems need to be solved by numerical methods. Compared with polynomial interpolation, rational function interpolation has higher interpolation accuracy and can effectively overcome the instability of interpolation [1–4]. Barycentric rational interpolation not only has high interpolation accuracy on special distributed nodes but also has high interpolation accuracy for equidistant nodes [5–7]. This method has been used to solve certain problems such as Volterra integral equations [2, 8, 9], delay Volterra integrodifferential equations [10, 11], plane elastic problems [12], nonlinear problems [13], heat conduction equation [14], and so on [15–17].

The third-order differential equation has a wide range of applications and important theoretical values in many scientific fields, such as applied mathematics and physics. Therefore, the third-order boundary value problem has been widely concerned by many scholars [18–20]. In this paper, we consider the numerical solution of the third-order two-point boundary value problem,by the barycentric rational interpolation collocation method.

Barycentric rational interpolation collocation method means using barycentric interpolation polynomials to find the differential matrix of a function at each discrete point; thus, the solution of the differential equation can be obtained by matrix operation. The barycentric rational interpolation has excellent numerical stability and high approximation accuracy, and the barycentric rational interpolation formula has a compact calculation formula of all order derivatives. Therefore, the barycentric rational interpolation collocation method is an effective method for solving boundary value problems of differential equations.

2. Formula of the Barycentric Interpolation Collocation Method

Discretize the interval into uniform parts with , and suppose is the function value of an unknown function at discrete nodes .

For any , , is the interpolation function at the point ; then, we have , andwhere

By changing the polynomial into the Lagrange interpolation form asand combining (4) and (5) together, we getwhere .

Then, we getwhere its basis function is

For the equidistant point, the weight function is

For the Chebyshev point of the second kind,the weight function is

By formula (8), the m-th order derivative of at the nodes can be expressed asand then (12) can be written in the matrix form aswhere and .

By using the barycentric interpolation function asequation (1) can be written in the numerical form as

By using the notation of the differential matrix, (15) can also be denoted asor the simple matrix form

Boundary conditions (2) can be divided intowhere ,

3. Convergence and Error Analysis

In this section, we will consider the error problem of equidistant interpolation nodes:

Let be the solution of (1); for any , suppose , to be the barycentric interpolation function at the point ; then, we have , andwhere

Then, the error function is defined asandwhere .

Taking the numerical form,and combining (24) and (1), we getwhere .

Lemma 1. For defined in (23), we have

Let be the solution of (1) and be the numerical solution; then, we haveand

The results can be obtained in [1].

Based on Lemma 1, we can get the following theorem.

Theorem 1. Let , , and ; then, we have

Proof. Let Add column 2, column 3,…, column n to column 1, and we haveThen, we have with , , where
Bywhich meanswhere is the element of matrix , we haveThe proof is completed.

4. Numerical Example

As an example, we consider the two-point boundary value problem:

For this problem, we can find a function such that the analysis solution iswhere is a freely selected parameter.

Substituting (38) into (36), we get

For different values of and different number of nodes, we can calculate the corresponding relative error and convergence rate; some of the data are shown in Tables 1 and 2.

Table 1 
Errors and convergence rate of the equidistant nodes with different .
Table 2 
Errors and convergence rate of the Chebyshev point with different .

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 .

For different values of and different number of nodes, we can calculate the corresponding relative error; some of the data are shown in Tables 3 and 4.

Table 3 
Errors and convergence rate of the equidistant nodes with different .
Table 4 
Errors and convergence rate of the Chebyshev point with different .

From Tables 3 and 4, we can find that, for different values of , the convergence rate can reach both for equidistant and nonequidistant nodes.

5. Conclusion

In this paper, the barycentric rational collocation method for solving third-order two-point boundary value equations is presented, and 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 Chebyshev point of the second kind with so the barycentric rational collocation method is an effective method. Compared with other methods, the advantage of this method is that the matrix equation can be easily obtained, the program is simple, and high computational accuracy can be obtained by using a few points.

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 there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The support from the Industry-University Cooperation Collaborative Education Project (201801123024) is gratefully acknowledged.