Numerical Solution for Third-Order Two-Point Boundary Value Problems with the Barycentric Rational Interpolation Collocation Method

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]. .is 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]. .e third-order differential equation has a wide range of applications and important theoretical values in many scientific fields, such as applied mathematics and physics. .erefore, 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 twopoint boundary value problem,


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][2][3][4]. Barycentric rational interpolation not only has high interpolation accuracy on special distributed nodes but also has high interpolation accuracy for equidistant nodes [5][6][7]. is 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][16][17]. e third-order differential equation has a wide range of applications and important theoretical values in many scientific fields, such as applied mathematics and physics. erefore, the third-order boundary value problem has been widely concerned by many scholars [18][19][20]. In this paper, we consider the numerical solution of the third-order twopoint 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. e 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. erefore, the barycentric rational interpolation collocation method is an effective method for solving boundary value problems of differential equations.

Formula of the Barycentric Interpolation Collocation Method
Discretize the interval [a, b] into n uniform parts with h � ((b − a)/n), and suppose u 1 , u 2 , . . . , u n is the function value of an unknown function u at discrete nodes where By changing the polynomial P i (x) into the Lagrange interpolation form as and combining (4) and (5) together, we get where en, we get where 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 u(x) at the nodes x 1 , x 2 , . . . , x n can be expressed as and then (12) can be written in the matrix form as where By using the barycentric interpolation function as equation (1) can be written in the numerical form as 2 Journal of Mathematics By using the notation of the differential matrix, (15) can also be denoted as or the simple matrix form Boundary conditions (2) can be divided into where

Convergence and Error Analysis
In this section, we will consider the error problem of equidistant interpolation nodes: Let u(x) be the solution of (1); for any 0 ≤ d ≤ n, suppose P(x i ), i � 0, 1, . . . , n − d, to be the barycentric interpolation function at the point x i , x i+1 , . . . , x i+d ; then, we have where en, the error function is defined as and where Taking the numerical form, and combining (24) and (1), we get where

Lemma 1. For e(x) defined in (23), we have
Let u(x) be the solution of (1) and u n (x) be the numerical solution; then, we have u ″ ′ n x k + pu n ″ x k + qu n ′ x k + ru n x k � f x k , k � 0, 1, 2, . . . , . . . n . . . , e results can be obtained in [1]. Based on Lemma 1, we can get the following theorem.
Add column 2, column 3,. . ., column n to column 1, and we have which means where M j (x) is the element of matrix L − 1 , we have e proof is completed.

Numerical Example
As an example, we consider the two-point boundary value problem: For this problem, we can find a function f(x) such that the analysis solution is where λ is a freely selected parameter. Substituting (38) into (36), we get (39) For different values of d 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.
In Table 1, the convergence rate of equidistant nodes with different d is O(h d− 2 ); in Table 2, the convergence rate of the Chebyshev point of the second kind with different d is For different values of λ and different number of nodes, we can calculate the corresponding relative error; some of the data are shown in Tables 3 and 4. From Tables 3 and 4, we can find that, for different values of λ, the convergence rate can reach O(h d+2 )(d ≥ 2) both for equidistant and nonequidistant nodes.

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 O(h d− 2 ) 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 O(h d− 2 ) for the equidistant nodes and Chebyshev point of the second kind with d ≥ 2, 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 e data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.