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

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1. 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 (FIEs) of the second kind

(1)

where and are non-zero continuous functions on the bounded region, and the kernel function , is the undetermined function. Usually, it’s difficult to obtain the analytic solutions of integral equations, so the numerical solutions of the integral equations are necessarily needed.

There are some researches for obtaining the numerical solutions of two dimensional 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- Legendre quadrature formula. Section 4, the numerical examples illustrate the efficiency and applicability of our method via comparing with the Lagrange method.

2. Barycentric Interpolation Function

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

2.1. One Dimensional Barycentric Interpolation Function

The one dimensional barycentric Lagrange formula about continuous function at the nodes is introduced as

(2)

where are the function values of, and are the barycetric Lagrange interpolation basis functions

(3)

which satisfy the property, where is the Kronecker-delta functions. And are the barycentric interpolation weight functions

(4)

and 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, where for and for.

2.2. Two Dimensional Barycentric Interpolation Function

Basing on the one dimensional barycentric Lagrange function, we define the two dimensional barycentric Lagrange interpolation function about a continuous function as follows

(5)

where, and the tensor product points are obtained by subdividing and into the calculation nodes and, respectively. The two dimensional barycentric interpolation basis functions satisfy

(6)

where and are equal to (4). The above equation has the following property

(7)

In the practical calculation, we take the second Chebyshev nodes, , , , as the interpolation nodes.

3. The Barycentric Method of Two Dimensional Fredholm Integral Equation

We enter (5) into (1) to approximate two dimensional Fredholm integral equation

(8)

then taking the collocation points into (8), and exchanging the integral and summation sign

(9)

Now, we deal with the integral part and let and , then we can easily have

(10)

where,

. Then, we apply the composite

Gauss-Legendre quadrature formula to approximate integral (10). Then the Equation (9) becomes

(11)

where; are the quadrature coefficients and points of the composite Gauss-Legendre quadrature formula, respectively. Finally, we transform the Equation (1) into algebraic equations. The solution of the Equation (11) is close 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, we can get the values of the discrete collocation solution at any interior points.

The specific Algorithm is as follows

Step 1: Construct two dimensional Chebyshev nodes, , ,

Step 2: Approximate integral operator in (10) by using composite Gauss- Legendre quadrature formula,

Step 3: Solve the algebraic Equations (11) using the above steps and Gaussian elimination method.

4. Numerical Example

In this section, we present two numerical examples to verify the effectiveness and accuracy of the barycentric method. Defining the absolute error (AE) and the relative error (RE) as

(12)

(13)

and, represent the exact and approximate solution of Equa- tion (1), respectively.

Example 1. For the following two dimensional linear Fredholm integral equation of the second kind

(14)

where, the accurate solution is

.

Table 1 and Table 2 are the absolute and relative errors of the barycentric Lagrange interpolation function method and the Lagrangian interpolation function method at the equidistant nodes and Chebyshev nodes, respectively, where m, n represent the interpolation node numbers and M, N are the quadrature node numbers. Usually, the accuracy of the BLIF method is higher than the LIF method, and the numerical results at the Chebyshev nodes are superior to equidistant nodes. From Table 1, we can see that the errors increase with m varying from 8 to 16. In Table 2, the numerical results of the BLIF are stable with Chebyshev nodes and those of the LIF method are oscillatory. From 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

Table 1. The absolute and relative errors of equal nodes, M = 8, N = 7, example 1.

Table 2. The absolute and relative errors of Chebyshev nodes, M = 8, N = 7, example 1.

Table 3. The absolute errors of the barycentric Lagrange method, M = 8, N = 7, example 1.

Table 4. The numerical results with Chebyshev nodes, M = 8, N = 7, example 1.

the numerical results of the BLIF method and the LIF method. We can conclude that the barycentric Lagrange method is more efficient and stable than the Lagrange method.

Example 2. Consider two dimensional Fredholm integral equation of the following form

(15)

where

the accurate solution is.

Table 5 and Table 6 reveal that the absolute and relative errors of the barycentric and Lagrangian methods with at the interior points. Generally, the approximate solutions of barycentric method are more accurate than Lagrange method, especially for large node numbers. In the tables, the absolute errors are decreasing by barycentric method and increasing by the Lagrange method with the increasing of Chebyshev node numbers. In a word, the BLIF

Table 5. The absolute errors with Chebyshev nodes, M = N = 11, example 2.

Table 6. The absolute errors with Chebyshev nodes, M = N = 11, example 2.

method is more efficient than the LIF method which is agreement with the theoretical analysis.

5. 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.

Acknowledgements

The authors are very grateful to the reviewers. This work was partially supported by the financial support from National Natural Science Foundation of China (Grant no. 11371079).

NOTES

*This work was supported by the National Natural Science Foundation of China (11371079).

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