Numerical Solution of Two Dimensional Fredholm Integral Equations of the Second Kind by the Barycentric Lagrange Function

This paper solves the two dimensional linear Fredholm integral equations of the second kind by combining the meshless barycentric Lagrange interpolation functions and the Gauss-Legendre quadrature formula. Inspired by this thought, we convert the equations into the associated algebraic equations. The results of the numerical examples are given to illustrate that the approximated method is feasible and efficient.


Introduction
Many of the mathematical physics problems and the engineering problems can be transformed into solving Fredholm integral equations [1] [2] [3]. In this paper, we pay attention to the two dimensional linear Fredholm integral equations  mensional FIEs, such as the radial basis functions method [4], Haar wavelets [5] and integral mean value method [6] [7]. Berrut specializes the barycentric Lagrangian interpolation formula [8], and some other authors present the corresponding numerical stability analyses in the literatures [9] [10] [11]. The authors take advantage of the equally spaced barycentric Lagrange polynomial to solve one dimensional linear Volterra-Fredholm integro-differential equations in [12].
There exists an intrinsic problem that polynomial interpolation is ill-posed at the equispaced nodes. This paper presents a modified Lagrange interpolation method with Chebyshev nodes to solve two dimensional linear Fredholm integral equations of the second kind.
This paper is constructed as follows: Section 2, we display the barycentric interpolation function. Section 3, we transform the two dimensional FIEs into the algebraic equations by utilizing the barycentric function and the composite Gauss-

Barycentric Interpolation Function
Barycentric Lagrange interpolation function (BLIF) is a variant of Lagrange interpolation function (LIF).

One Dimensional Barycentric Interpolation Function
The one dimensional barycentric Lagrange formula about continuous function which satisfy the property tions. And j w are the barycentric interpolation weight functions and j w only depend on the distribution of nodes.
The barycentric Lagrange interpolation formula is stable forward when we choose the Chebyshev points as interpolating points [9]. At the same time, the weight functions are simplified as ( )

Two Dimensional Barycentric Interpolation Function
Basing on the one dimensional barycentric Lagrange function, we define the two In the practical calculation, we take the second Chebyshev nodes ( )

The Barycentric Method of Two Dimensional Fredholm Integral Equation
We enter (5) into (1) Now, we deal with the integral part and let  to (1). The whole process is called discrete collocation method whose nature is the Nyström iterative approach. We can analyze the existence, uniqueness and convergence of the approximate solution under the theoretical framework of the Nyström from [13]. Once we obtain ij u , we can get the values of the discrete collocation solution ( ) , u x y at any interior points.
The specific Algorithm is as follows Step 1: Construct two dimensional Chebyshev nodes ( ) Step 2: Approximate integral operator K in (10) by using composite Gauss-Legendre quadrature formula, Step 3: Solve the algebraic Equations (11) using the above steps and Gaussian elimination method.

Numerical Example
In    Table 3, we find that the errors first reduce and remain stable finally at the Chebyshev nodes. In reverse, the errors of the BLIF method increase at the equal nodes with the increasing of m, especially the method is invalid when m = 32. In Table 4, we list the accurate solution is ( , ) x y u x y e + = .  method is more efficient than the LIF method which is agreement with the theoretical analysis.

Conclusion
In this paper, we solve two dimensional linear Fredholm integral equations of the second kind by means of the barycentric Lagrange interpolation method.
The modified Lagrange method with Chebyshev nodes transforms the equations into linear algebraic equations, and the corresponding numerical solutions are stable forward. The numerical results also demonstrate that the barycentric method is a simple and powerful technique. Furthermore, the barycentric method can extend to solve high dimensional FIEs.