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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.

Many of the mathematical physics problems and the engineering problems can be transformed into solving Fredholm integral equations [

where

There are some researches for obtaining the numerical solutions of two dimensional FIEs, such as the radial basis functions method [

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.

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

The one dimensional barycentric Lagrange formula about continuous function

where

which satisfy the property

and

The barycentric Lagrange interpolation formula is stable forward when we choose the Chebyshev points as interpolating points [

weight functions are simplified as

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

where

where

In the practical calculation, we take the second Chebyshev nodes

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

then taking the collocation points

Now, we deal with the integral part and let

where

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

where

The specific Algorithm is as follows

Step 1: Construct two dimensional Chebyshev nodes

Step 2: Approximate integral operator

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

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

and

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

where

Node numbers | The BLIF method | The LIF method | ||
---|---|---|---|---|

AE | RE | AE | RE | |

m = n = 8 | 2.9984e−11 | 4.1823e−12 | 1.1735e−10 | 1.6369e−11 |

m = n = 16 | 3.9568e−08 | 2.9075e−09 | 2.4235e−03 | 1.7808e−04 |

Node numbers | The BLIF method | The LIF method | ||
---|---|---|---|---|

AE | RE | AE | RE | |

m = n = 8 | 4.6649e−13 | 6.6148e−14 | 7.0170e−11 | 9.9500e−12 |

m = n = 16 | 3.7665e−14 | 2.8191e−15 | 6.2815e−05 | 4.7015e−06 |

Nodes (x, y) | The Chebyshev nodes | The equidistant nodes | |||
---|---|---|---|---|---|

m = 8 | m = 16 | m = 32 | m = 8 | m = 16 | |

(0.1, 0.1) | 3.8528e−11 | 7.4940e−16 | 2.4980e−16 | 3.6538e−11 | 9.3405e−07 |

(0.3, 0.3) | 8.1330e−12 | 1.6653e−15 | 1.6653e−15 | 1.3789e−11 | 2.6547e−06 |

(0.5, 0.5) | 5.5622e−14 | 2.5535e−15 | 2.4425e−15 | 8.3724e−12 | 3.9562e−06 |

(0.7, 0.7) | 4.0140e−12 | 2.4425e−15 | 1.2212e−15 | 5.8561e−12 | 4.6331e−06 |

(0.9, 0.9) | 5.8634e−12 | 2.9976e−15 | 2.9976e−15 | 1.5618e−11 | 4.5786e−06 |

Nodes (x, y) | The BLIF method | The LIF method | ||
---|---|---|---|---|

m = 8 | m = 16 | m = 8 | m = 16 | |

(2^{−}^{1}, 2^{−}^{1}) | 1.1935e−13 | 1.8203e−10 | 8.3724e−12 | 3.9562e−06 |

(2^{−}^{2}, 2^{−}^{2}) | 4.6897e−11 | 2.0025e−11 | 5.1702e−11 | 2.2540e−06 |

(2^{−}^{3}, 2^{−}^{3}) | 2.2128e−11 | 9.7228e−14 | 1.9650e−11 | 1.1632e−06 |

(2^{−}^{4}, 2^{−}^{4}) | 2.7555e−11 | 7.9936e−15 | 2.6306e−11 | 5.8616e−07 |

(2^{−}^{5}, 2^{−}^{5}) | 7.5557e−12 | 1.1727e−15 | 8.1812e−12 | 2.9365e−07 |

(2^{−}^{6}, 2^{−}^{6}) | 1.6780e−11 | 1.0755e−16 | 1.7093e−11 | 1.4690e−07 |

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

where

the accurate solution is

Nodes (x, y) | The BLIF method | The LIF method | ||
---|---|---|---|---|

m = 8 | m = 16 | m = 8 | m = 16 | |

(0.1, 0.1) | 2.2598e−09 | 6.6613e−16 | 2.2597e−09 | 2.4946e−08 |

(0.3, 0.3) | 4.5070e−09 | 6.6613e−16 | 4.5078e−09 | 5.3816e−07 |

(0.5, 0.5) | 1.3426e−09 | 1.3323e−15 | 1.3395e−09 | 1.9373e−06 |

(0.7, 0.7) | 8.2825e−09 | 0.0000e−00 | 8.2724e−09 | 4.4205e−06 |

(0.9, 0.9) | 1.2460e−08 | 1.7764e−15 | 1.2495e−08 | 1.7531e−05 |

Nodes (x, y) | The BLIF method | The LIF method | ||
---|---|---|---|---|

m = 8 | m = 16 | m = 8 | m = 16 | |

(2^{−}^{1}, 2^{−}^{1}) | 1.3426e−09 | 1.3323e−15 | 1.3395e−09 | 1.9373e−06 |

(2^{−}^{2}, 2^{−}^{2}) | 1.4203e−09 | 1.1102e−15 | 1.4208e−09 | 3.3037e−07 |

(2^{−}^{3}, 2^{−}^{3}) | 3.1271e−09 | 1.5543e−15 | 3.1270e−09 | 4.6671e−08 |

(2^{−}^{4}, 2^{−}^{4}) | 1.7789e−10 | 1.5543e−15 | 1.7788e−10 | 6.3278e−09 |

(2^{−}^{5}, 2^{−}^{5}) | 1.1286e−09 | 4.4409e−15 | 1.1287e−09 | 8.1572e−10 |

(2^{−}^{6}, 2^{−}^{6}) | 1.0338e−09 | 1.1102e−15 | 1.0338e−09 | 1.0351e−10 |

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

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.

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).

Liu, H.Y., Huang, J. and Pan, Y.B. (2017) Numerical Solution of Two Dimensional Fredholm Integral Equations of the Second Kind by the Barycentric Lagrange Function. Journal of Applied Mathematics and Physics, 5, 259-266. https://doi.org/10.4236/jamp.2017.52023