Fredholm integral equations are an important class of integral equations widely used in science and engineering. This paper aims to numerically solve Fredholm integral equations of the second kind in two-dimensional complex domains using a combination of spectral methods and mapping techniques. First, we map the computational domain onto a rectangular domain through coordinate transformation. Then, within this rectangular domain, we employ the classical spectral method for numerical simulation. Our analysis focuses on discussing the existence, uniqueness, and convergence of numerical solutions. Numerical results demonstrate that the proposed method achieves high-order accuracy.

Conflicts of Interest

The authors declare no conflicts of interest.

Open Research

Data Availability Statement

The datasets generated during and/or analyzed in the current study are available from the corresponding author upon reasonable request.

References

1R. Farengo, Y. C. Lee, and P. N. Guzdar, “An Electromagnetic Integral Equation: Application to Microtearing Modes,” Physics of Fluids 26 (1983): 35153523.
Web of Science® Google Scholar
2M. Asadzadeh and K. Eriksson, “On Adaptive Finite Element Methods for Fredholm Integral Equations of the Second Kind,” SIAM Journal on Numerical Analysis 31 (1994): 831855.
10.1137/0731045
Google Scholar
3W. J. Xie and F. R. Lin, “A Fast Numerical Solution Method for Two Dimensional Fredholm Integral Equations of the Second Kind,” Applied Numerical Mathematics 59 (2009): 17091719.
10.1016/j.apnum.2009.01.009
Web of Science® Google Scholar
4B. L. Panigrahi, “Mixed Fourier Legendre Spectral Galerkin Methods for Two-Dimensional Fredholm Integral Equations of the Second Kind,” Applied Numerical Mathematics 168 (2021): 235250.
10.1016/j.apnum.2021.06.003
Web of Science® Google Scholar
5B. Alpert, G. Beylkin, R. Coifman, and V. Rokhlin, “Wavelets for the Fast Solution of Second-Kind Integral Equations,” SIAM Journal on Scientific Computing 14 (1993): 159184.
10.1137/0914010
Google Scholar
6K. E. Atkinson, A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind (SIAM, 1976).
Google Scholar
7C. T. Baker, The Numerical Treatment of Integral Equations (Clarendon Press, 1977).
Google Scholar
8L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations (Cambridge University Press, 1985).
10.1017/CBO9780511569609
Google Scholar
9S. Noeiaghdam and S. Micula, “A Novel Method for Solving Second Kind Volterra Integral Equations With Discontinuous Kernel,” Mathematics 9 (2021): 2172.
10.3390/math9172172
Web of Science® Google Scholar
10Y. Talaei, S. Micula, H. Hosseinzadeh, and S. Noeiaghdam, “A Novel Algorithm to Solve Nonlinear Fractional Quadratic Integral Equations,” AIMS Mathematics 7 (2022): 1323713257.
10.3934/math.2022730
Web of Science® Google Scholar
11J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed. (Springer-Verlag, 2001).
Google Scholar
12C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods. Fundamentals in Single Domains (Springer-Verlag, 2006).
10.1007/978-3-540-30726-6
Web of Science® Google Scholar
13B. Y. Guo, Spectral Methods and Their Applications (World Scientific, 1998).
10.1142/3662
Google Scholar
14J. Shen, T. Tang, and L. L. Wang, Spectral Methods: Algorithms, Analysis and Applications (Springer-Verlag, 2011).
10.1007/978-3-540-71041-7
Web of Science® Google Scholar
15B. Y. Guo and X. H. Yu, “Composite Spectral Method for Exterior Problems With Polygonal Obstacles,” Journal of Scientific Computing 59 (2014): 439472.
10.1007/s10915-013-9769-x
Web of Science® Google Scholar
16G. E. Karniadakis and S. J. Sherwin, Spectral/HP Element Methods for Computational Fluid Dynamics, 2nd ed. (Oxford University Press, 2005).
10.1093/acprof:oso/9780198528692.001.0001
Google Scholar
17J. Shen, L. L. Wang, and H. Y. Li, “A Triangular Spectral Element Method Using Fully Tensorial Rational Basis Functions,” SIAM Journal on Numerical Analysis 47 (2009): 1619–1650.
10.1137/070702023
Web of Science® Google Scholar
18B. L. Buzbee, F. W. Dorr, J. A. George, and G. H. Golub, “The Direct Solution of the Discrete Poisson Equation on Irregular Regions,” SIAM Journal on Numerical Analysis 8 (1971): 722–736.
10.1137/0708066
Web of Science® Google Scholar
19Y. Q. Gu and J. Shen, “Accurate and Efficient Spectral Methods for Elliptic PDEs in Complex Domains,” Journal of Scientific Computing 83 (2020): 42.
10.1007/s10915-020-01226-9
Web of Science® Google Scholar
20Y. Q. Gu and J. Shen, “An Efficient Spectral Method for Elliptic PDEs in Complex Domains With Circular Embedding,” SIAM Journal on Scientific Computing 43 (2021): A309A329.
10.1137/20M1345153
Web of Science® Google Scholar
21Y. Q. Gu and J. Shen, “A Fictitious Domain Spectral Method for Solving the Helmholtz Equation in Exterior Domains,” Journal of Scientific Computing 94 (2023): 46.
10.1007/s10915-023-02098-5
Web of Science® Google Scholar
22S. H. Lui, “Spectral Domain Embedding for Elliptic PDEs in Complex Domains,” Journal of Computational and Applied Mathematics 225 (2009): 541–557.
10.1016/j.cam.2008.08.034
Web of Science® Google Scholar
23W. J. Gordon and C. A. Hall, “Transfinite Element Methods: Blending-Function Interpolation Over Arbitrary Curved Element Domains,” Numerische Mathematik 21 (1973): 109129.
10.1007/BF01436298
Web of Science® Google Scholar
24S. A. Orszag, “Spectral Methods for Problems in Complex Geometries,” Journal of Computational Physics 37 (1980): 7092.
10.1016/0021-9991(80)90005-4
Web of Science® Google Scholar
25Z. Q. Wang, X. Wen, and G. Q. Yao, “An Efficient Fourier-Legendre Spectral-Galerkin Method for Elliptic Equations in 2D Complex Geometries,” Journal of Scientific Computing 95 (2023): 89.
10.1007/s10915-023-02207-4
Google Scholar
26G. Q. Yao, X. Wen, and Z. Q. Wang, “An Efficient Fourier-Laguerre Spectral-Galerkin Method for Exterior Problems of Two-Dimensional Complex Obstacles,” Applied Numerical Mathematics 193 (2023): 93108.
10.1016/j.apnum.2023.07.015
Web of Science® Google Scholar
27J. An, H. Y. Li, and Z. M. Zhang, “Spectral-Galerkin Approximation and Optimal Error Estimate for Biharmonic Eigenvalue Problems in Circular/Spherical/Elliptical Domains,” Numerical Algorithms 84 (2020): 427–455.
10.1007/s11075-019-00760-4
Web of Science® Google Scholar

Volume48, Issue12

August 2025

Pages 12282-12291

An Efficient Spectral Galerkin Method for Two-Dimensional Fredholm Integral Equations in Complex Domains

ABSTRACT

Conflicts of Interest

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References

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An Efficient Spectral Galerkin Method for Two-Dimensional Fredholm Integral Equations in Complex Domains

ABSTRACT

Conflicts of Interest

Open Research

Data Availability Statement

References

References

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