Linear Barycentric Rational Method for Two-Point Boundary Value Equations

Linear barycentric rational method for solving two-point boundary value equations is presented. -e 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 collocationmethod for solving two-point boundary value problems is proved. Several numerical examples are provided to validate the theoretical analysis.


Introduction
e 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. e 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. e linear barycentric rational method (LBRM) [1][2][3] has been used to solve certain problems such as delay Volterra integro-differential equations [4], Volterra integral equations [5][6][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: (Tu)(x) ≔ u ″ (x) + qu(x) � f(x), x ∈ (a, b), (1) u(a) � u ℓ , u(b) � u r . (2) Let the interval [a, b] be partitioned into n uniform part with h � (b − a)/n and x 0 , x 1 , . . . , x n with its related func- where Change the polynomial P i (x) into the Lagrange interpolation form as Combining (7) and (5) together, we get where w k � i∈J k where its basis function is For the equidistant point, its weight function is e Chebyshev point of the second kind is and its weight function is Consider the barycentric interpolation function as and the numerical scheme is given as By using the notation of the differential matrix, equation (13) is denoted as matrices in the form of n j�0 D (2) ij u j + q n j�0 δ ij u j � f x i , (14) where i � 1, 2, . . . , n. Equation (13) is written as matrices in the form of where and . . , f(x n )] T . Using interpolation formulas, boundary conditions can be discretized into n j�0 D (1) 1j u j � a,

Convergence and Error Analysis
With the error function of difference formula and where Combining (20) and (1), we have where e following Lemma has been proved by Jean-Paul Berrut in [13].
Lemma 1 (see [13]). For e(x) defined in (18), we have Let u(x) be the solution of (1) and u n (x) is the numerical solution, then we have and e results can be obtained in the reference of [14]. Based on the above lemma, we derive the following theorem.
2 Journal of Mathematics and Putting column 2, column 3, column n added to column 1, we have which means the matrix D (2) is the singular matrix. Similarly we have and then we assume which means where M j (x) is the element of matrix L − 1 . en we have e 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.

(35)
In this example, we consider the two-point boundary value equations with the boundary condition y(0) � y(1) � 0. In Table 1 Tables 1 and 2 for d � 1, respectively, and we will give exact analysis in other paper.

Concluding Remarks
In this paper, the numerical approximation of linear barycentric rational collocation method for solving two-point boundary value equations is presented. e matrix form of the algorithm is given for the simple calculation; with the help of Newton formula, the error function of the Data Availability e data that support the findings of this study are available from the corresponding author upon reasonable request.