Linear barycentric rational collocation method for solving biharmonic equation

: Two - dimensional biharmonic boundary - value problems are considered by the linear barycentric rational collocation method, and the unknown function is approximated by the barycentric rational poly -nomial. With the help of matrix form, the linear equations of the discrete biharmonic equation are changed into a matrix equation. From the convergence rate of barycentric rational polynomial, we present the convergence rate of linear barycentric rational collocation method for biharmonic equation. Finally, several numerical examples are provided to validate the theoretical analysis.


Introduction
In this article, we pay our attention to the numerical solution of biharmonic equation: or u x y g x y u x y x u x y y h x y x y , , , ∈ ∂. Biharmonic equation [1] is widely used in electrostatics, mechanical engineering, theoretical physics, and so on. There are a lot of numerical methods to solve biharmonic equation such as finite difference method, finite element method, boundary element method, spectral method, and so on.
There are some advantages of the collocation method [2] to solve the partial differential equation, such as meshless, no integrals, and easy to program. Barycentric formula can be obtained by the Lagrange interpolation formulae [3][4][5], which have been used to solve certain problems such as delay Volterra integro-differential equations [6,7], Volterra integral equations [8], boundary value problems [9], convection-diffusion equations [10,11], and so on. Generally, the interpolation nodes of barycentric Lagrange interpolation such as second kind of Chebyshev point is not the equidistant node. In order to obtain the equidistant node of the barycentric formulae, Floater and Hormann [12], Floater et al. [13], Klein and Berrut [14,15] have proposed a rational interpolation scheme which has high numerical stability and interpolation accuracy on both equidistant nodes and non-equidistant nodes. In recent articles, Wang et al. [16][17][18][19] successfully applied the barycentric interpolation collocation method to solve initial value problems, plane elasticity problems [20], incompressible plane problems, telegraph equation [21], beam force vibration equation [22], and non-linear problems, which have expanded the application fields of the collocation method. For the two-dimensional biharmonic boundary problems, a new spectral collocation method [23] and depression of order [24] are reported to numerically solve it.
Based on the one-dimensional linear barycentric rational interpolation, two-dimensional barycentric rational interpolation polynomial is constructed, then barycentric rational interpolation collocation method is obtained to solve biharmonic equation. With the help of vector form, the discrete linear equation of twodimensional biharmonic equation is changed into matrix equation which can be coded easily. Moreover, the error estimate of linear barycentric rational interpolation for biharmonic equation is obtained and the convergence rate is also presented.
This article is organized as follows: In Section 2, the differentiation matrix and barycentric rational interpolation collocation scheme for biharmonic equation are presented, then the matrix form of collocation scheme is obtained. In Section 3, the convergence rate is proved. Finally, some numerical examples are listed to illustrate our theorem.

Differentiation matrix of biharmonic equation
is the basis function, and J i I k d i k ; , and u y L y u , Taking (8) into equation (1), we obtain L x L y u L x L y u L x L y u f x y , .
By taking x x j = in (9), then we change equation ( and equation (10) where f x y f f , , According to mathematical induction, we obtain the recurrence formula of k-order (k 2 ≥ ) differential matrix as First, the Kronecker product of matrix A a ij m n ( ) where u u y f f y j m k n , , 1, 2, , ; 1, 2, , j j k j j k ( ) ( ) = = = … = … and I I , m n and identity matrix of m n , , then we have and where and U u u , , , , , , , , , , , , , , , , , , , , and ⊗ is the Kronecker product of matrix.
For the boundary condition of (2), u x y g x y h x y , , , , , we obtain the discrete x a b a n j j n 1 1 , 1, , , and its weight function is And the second kind of Chebyshev point is and its weight function is

Convergence and error analysis
For one-dimensional function u x ( ) is approximated by rational function r x ( ). Its error functional is defined as where and The following theorem has been proved by Jean-Paul Berrut, see [13].
and J k I j d k j : By the error term of barycentric rational interpolation for two-dimensional function, we have The following theorem can be proved similarly as Li and Cheng [11].
By the error formulae of barycentric rational interpolation This corollary can be obtained similarly as Theorem 3.2, here we omit it.
Combining (8) and (1) In the following theorem, the main result is presented. and where u x y , to test our algorithm. Figure 1 shows the error estimate of equidistant nodes with barycentric Lagrange interpolation collocation method, and Figure 2 shows the error estimate of Chebyshev nodes with barycentric Lagrange interpolation collocation method. Figure 3 shows the error estimate of barycentric rational Lagrange interpolation collocation method with equidistant nodes, and Figure 4 shows the barycentric rational interpolation collocation method of the error estimate of Chebyshev nodes. From Figures 3 and 4, we know that the barycentric rational interpolation collocation method has higher accuracy under the condition of Chebyshev nodes. Table 3 shows condition number of equidistant nodes with different d 1 and d 2 . In Table 4, condition number of Chebyshev nodes with different d 1 and d 2 is shown.

Its analytical solution is
In this example, we test the linear barycentric rational collocation method with the equidistant nodes,  Tables 5 and 6, which is out of our goal of this article.
We choose m n d d 10; 10; 8; to test our algorithm. Figure 5 shows the error estimate of equidistant nodes with barycentric Lagrange interpolation collocation method, and Figure 6 shows the error estimate of Chebyshev nodes with barycentric Lagrange interpolation collocation method. Figure 7 shows the error estimate of equidistant nodes with rational barycentric rational interpolation collocation method, and Figure 8 shows the error estimate of Chebyshev nodes. From Figures 7 and 8, we know that the barycentric rational interpolation collocation method has higher accuracy under the condition of Chebyshev nodes.       Linear barycentric rational collocation method for solving biharmonic equation  597   10; 8 1 2 . and u y u y u x u x 0, 1, , 0 , 1 0.
Its analytical solution is u x y πx πy , sin 2 sin 2 .
In this example, we test the linear barycentric rational with the equidistant nodes, and Table 7 Figure 9 shows the error estimate of equidistant nodes with barycentric Lagrange interpolation collocation method, and Figure 10 shows the error estimate of Chebyshev nodes with barycentric Lagrange interpolation collocation method. Figure 11 shows the error estimate of equidistant nodes with rational barycentric rational interpolation collocation method, and Figure 12 shows the error estimate of Chebyshev nodes. From Figures 11 and 12, we know that the barycentric rational interpolation collocation method has higher accuracy under the condition of Chebyshev nodes. .
Linear barycentric rational collocation method for solving biharmonic equation  599

Conclusion
In this article, we have presented the linear barycentric collocation methods to solve the two-dimensional elliptic boundary value problems. ( ), this is an interesting phenomenon which will be investigated in future works.
Acknowledgments: The author gratefully acknowledges the helpful comments and suggestions of the reviewers, which have improved the presentation.