A Barycentric Interpolation Collocation Method for Linear Nonlocal Boundary Value Problems

This paper is devoted to the numerical treatment of a class of higher-order multi-point boundary value problems(BVPs). The method is proposed based on the Lagrange interpolation collocation method, but it avoids the numerical instability of Lagrange interpolation. Numerical results obtained by present method compare with other methods show that the present method is simple and accurate for higher-order multi-point BVPs, and it is effective for solving six order or higher order multi-point BVPs.


Introduction
Collocation method as a numerical calculation method for solving differential equations, it has many merits, such as calculation formula is simple, program implementation is convenient.Use Lagrange interpolation collocation method to solve differential equations, when select too many nodes, Lagrange collocation formula will be not numerical instability, the famous Runge phenomenon illustrates the problem [1].But barycentric interpolation collocation method has excellent numerical instability [2].In this paper, we select suitable number of the second kind Chebyshev point as interpolation nodes, use barycentric interpolation collocation method establish the dierential matrix of function to solve multi-point boundary value problems(BVPs).As we know, boundary value problems(BVPs) arise in many fields(see [11][12][13][14][15][16]). In [3], Henderson and Kunkel proved the uniqueness of solutions for the following linear differential equations with nonlocal boundary conditions: where a i (x) ∈ C [a, b], m j are positive integers such that Lin and Wu use the reproducing kernel to solve the following boundary value problems(BVPs).In [6], Li use another reproducing kernel method to solve this problem.
In this work, we use barycentric interpolation to solve (1) , several numerical examples are given to demonstrate the efficiency of the present method.The present method compared with the others methods, reveals that the present method is more effective and convenient.

The Differential Matrix Of Barycentric Interpolation
According to the ideas of the collocation method, the all-order derivatives of function at nodes can be approximate as linear weighted sum of the function value at node.Consider the function u(x) which defined in interval [a, b], the values of function u(x) at nodes, a = x 1 < x 2 < . . .< x n = b, u i = u(x i ), i = 1, 2, . . ., n, and the all-order derivatives of u(x) at the nodes can be expressed as the linear weighted sum of the function value, Written in matrix form is, There, T is the column vector of the m order derivatives m of unknown function at nodes, matrix D (m) is m order differential matrix of unknown function, the element of D (m) , D (m)  i j is the weighted coefficient, u (m) = [u (m)  1 , u (m) 2 , . . ., u (m) n ] T is the value of unknown function at nodes.Barycentric interpolation primary function is denoted by L j (x), the barycentric interpolation of u(x) can be expressed as, So, the one order and two order derivatives of u(x) can be expressed as, Barycentric interpolation primary function is, There, w j = 1 ∏ j k x j −x k , j = 1, 2, . . ., n is barycentric interpolation weight, it based on the distribution of interpolation nodes.Multiply x − x j , (i j) at both sides of (5) at the same time, after deformation is, To facilitate, let, Calculate derivative about x at both sides of (6) at the same time, we get, Calculate derivative about x at both sides of (7) , Take formula (10)∼( 12) into the formula ( 8) and ( 9), we know L j (x i ) = 0(i j), so we can get, Now, we can get one and two order differential matrix, Using mathematical induction, we can get the recursion formula of m order differential matrix,

The Barycentric Interpolation Collocation Formula of Multi-Point BVPs
Think about fuction (??) Where . ., u n as the value of function u(x) at disperse nodes x 1 , x 2 . . ., x n , using the barycentric interpolation collocation can get approximate function u(x), Take formula (20) into the differential equation, we can get, Let ( 21) accurate established at disperse nodes, we can get n equations, i.e.
Writ ( 23) in matrix form is

Applying Method of Initial Boundary Conditions
Use collocation method to solve the differential equation problem, the key is how to use the initial conditions.
There are three method.The first method is displacement method.That is, we use the functions of ( 24) displace from the first to m functions of (23).The second method is supplemental method.That is, the functions of (24) add after functions of (23).The third method is elimination method.That is, we get u 1 , u n from (24), i.e. use u 2 , u 3 . . ., u n−1 express u 1 and u n , and then take them into from the second to n − 1 functions of (23).About multi-point BVPs,we always use displacement method and supplemental method.

Numerical Experiment
In this section, six numerical examples are studied to demonstrate the accuracy of the present method.
The exact solution is u T (x) = 1 + sinhx.The numerical results are presented in Table 1. Figure 1.
The exact solution is u T (x) = sinx.The numerical results are presented in Table 2. Figure 2. Example 3 [7]  Considering following fourth-order boundary value problems.The exact solution is u T (x) = x(x − 1 4 )(x − 1 2 )e −4xln4 .The numerical results are presented in Table 3. Figure 3. Example 4 [7] Considering following fourth-order boundary value problems.
The exact solution is u T (x) = (x 2 − 1)sinx.The numerical results are presented in Table 5. Figure 5.The exact solution is u T (x) = x(1 − x)e x .The numerical results are presented in Table 6. Figure 6.

Discussion
In this paper, we devote to the numerical treatment of a class of higher-order multi-point solving higher-order multi-point BVPs.The numerical results demonstrate that the method is quite accurate and efficient for linear higher-order multi-point BVPs.This makes it easy to solve the higher-order multi-point BVPs.It is worthy to note that this method can be generalized to more higher order BVPs.All computations are performed by the MatlabR2013a software package.

Figure 5 .Figure 6 .
Figure 5. Present numerical method for Example 5, the first picture shows the numerical error when we select 16 nodes, the second picture is 21 nodes

Table 1 .
Comparison of absolute errors for Example 1

Table 4 .
Comparison of errors for Example 4

Table 5 .
The numerical results of Example 5Example 6 Considering following eighth-order boundary value problems.