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Numerical solution of the two-dimensional first kind Fredholm integral equations using a regularized collocation method

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Abstract

In this study, we present an advanced numerical model for solving the 2D first kind Fredholm integral equations, which are well known to be ill-posed problems. This numerical approach is built on the quadrature formula with the Lavrentiev regularization method. Under some essential assumptions, a comprehensive theoretical explanation of the presented numerical approach is provided. Finally, various numerical examples support the theoretical findings and demonstrate the accuracy of our method.

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References

  • Altürk A (2016) The regularization-homotopy method for the two-dimensional fredholm integral equations of the first kind. Math Comput Appl 21(2):9

    MathSciNet  Google Scholar 

  • Chen Z, Micchelli CA, Xu Y (2015) Multiscale methods for Fredholm integral equations, vol 28. Cambridge University Press, United Kingdom

    Book  MATH  Google Scholar 

  • Davis PJ, Rabinowitz P (2007) Methods of numerical integration. Courier Corporation, Mineola, New York

    MATH  Google Scholar 

  • Engl HW, Hanke M, Neubauer A (1996) Regularization of inverse problems, vol 375. Springer, Boston, London

    Book  MATH  Google Scholar 

  • Fattahzadeh F (2019) Approximate solution of two-dimensional fredholm integral equation of the first kind using wavelet base method. Int J Appl Comput Math 5(5):1–12

    Article  MathSciNet  MATH  Google Scholar 

  • Gautschi W (2004) Orthogonal polynomials: computation and approximation. OUP Oxford, Oxford

    Book  MATH  Google Scholar 

  • Hofmann B (1986) Regularization for Applied Inverse and Ill-posed Problems, volume 85 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics]. BSB BG Teubner Verlagsgesellschaft, Leipzig

  • Kanwal RP (2013) Linear integral equations. Springer, New York

    Book  MATH  Google Scholar 

  • Kirsch A (2011) An introduction to the mathematical theory of inverse problems, vol 120. Springer, Switzerland

    MATH  Google Scholar 

  • Kress R, Maz’ya V, Kozlov V (1989) Linear integral equations, vol 82. Springer, New York

    Google Scholar 

  • Lavrentiev MM (2013) Some improperly posed problems of mathematical physics, vol 11. Springer, Berlin

    Google Scholar 

  • Lu S, Pereverzev SV (2013) Regularization theory for ill-posed problems. de Gruyter, Berlin, Boston

    Book  MATH  Google Scholar 

  • Lu Y, Shen L, Xu Y (2010) Integral equation models for image restoration: high accuracy methods and fast algorithms. Inverse Prob 26(4):045006

    Article  MathSciNet  MATH  Google Scholar 

  • Maleknejad K, Sohrabi S (2007) Numerical solution of fredholm integral equations of the first kind by using legendre wavelets. Appl Math Comput 186(1):836–843

    MathSciNet  MATH  Google Scholar 

  • Mirzaee F, Hoseini AA (2014) A computational method based on hybrid of block-pulse functions and taylor series for solving two-dimensional nonlinear integral equations. Alex Eng J 53(1):185–190

    Article  Google Scholar 

  • Mirzaee F, Samadyar N (2019) Numerical solution based on two-dimensional orthonormal bernstein polynomials for solving some classes of two-dimensional nonlinear integral equations of fractional order. Appl Math Comput 344:191–203

    MathSciNet  MATH  Google Scholar 

  • Mirzaee F, Samadyar N (2020) Numerical solution of two dimensional stochastic volterra-fredholm integral equations via operational matrix method based on hat functions. SeMA J 77(3):227–241

    Article  MathSciNet  MATH  Google Scholar 

  • Nair MT (2009) Linear operator equations: approximation and regularization. World Scientific, Singapore

    Book  MATH  Google Scholar 

  • Nair MT (2012) Quadrature based collocation methods for integral equations of the first kind. Adv Comput Math 36(2):315–329

    Article  MathSciNet  MATH  Google Scholar 

  • Nair MT, Pereverzev SV (2007) Regularized collocation method for fredholm integral equations of the first kind. J Complex 23(4–6):454–467

    Article  MathSciNet  MATH  Google Scholar 

  • Neggal B, Boussetila N, Rebbani F (2016) Projected tikhonov regularization method for fredholm integral equations of the first kind. J Inequ Appl 2016(1):1–21

    MathSciNet  MATH  Google Scholar 

  • Pachpatte BG (2011) Multidimensional integral equations and inequalities, vol 9. Springer, Paris

    Book  MATH  Google Scholar 

  • Pereverzev S, Schock E (2005) On the adaptive selection of the parameter in regularization of ill-posed problems. SIAM J Numer Anal 43(5):2060–2076

    Article  MathSciNet  MATH  Google Scholar 

  • Samadyar N, Mirzaee F (2019) Numerical solution of two-dimensional weakly singular stochastic integral equations on non-rectangular domains via radial basis functions. Eng Anal Boundary Elem 101:27–36

    Article  MathSciNet  MATH  Google Scholar 

  • Samadyar N, Mirzaee F (2020) Numerical solution of two-dimensional stochastic fredholm integral equations on hypercube domains via meshfree approach. J Comput Appl Math 377:112875

    Article  MathSciNet  MATH  Google Scholar 

  • Tahami M, Askari Hemmat A, Yousefi S (2016) Numerical solution of two-dimensional first kind fredholm integral equations by using linear legendre wavelet. Int J Wavelets Multiresolut Inf Process 14(01):1650004

    Article  MathSciNet  MATH  Google Scholar 

  • Tahar B, Nadjib B, Faouzia R (2021) A variant of projection-regularization method for ill-posed linear operator equations. Int J Comput Methods 18(04):2150008

    Article  MathSciNet  MATH  Google Scholar 

  • Tautenhahn U (2002) On the method of lavrentiev regularization for nonlinear ill-posed problems. Inverse Prob 18(1):191

    Article  MathSciNet  MATH  Google Scholar 

  • Tikhonov AN, Arsenin VY (1977) Solutions of ill-posed problems. New York 1(30):487

    Google Scholar 

  • Torabi SM, Tari Marzabad A (2016) Numerical solution of two-dimensional integral equations of the first kind by multi-step methods. Comput Methods Differ Equ 4(2):128–138

    MathSciNet  MATH  Google Scholar 

  • Wazwaz A-M (2011) Linear and nonlinear integral equations, vol 639. Springer, Berlin

    Book  MATH  Google Scholar 

  • Yang S, Luo X, Li F, Long G (2013) A fast multiscale galerkin method for the first kind ill-posed integral equations via iterated regularization. Appl Math Comput 219(21):10527–10537

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Faculty of Science and Technology, Mohammed Cherif Messaadia University, P. O. Box 1553, Souk Ahras, 41000, Algeria

    Tahar Bechouat

  2. Department of Mathematics, Faculty MISM, University 8 Mai 1945 Guelma, P. O. Box 401, Guelma, 24000, Algeria

    Nadjib Boussetila

  3. Laboratory of Applied Mathematics, University Badji Mokhtar Annaba, P. O. Box 12, Annaba, 23000, Algeria

    Nadjib Boussetila

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  1. Tahar Bechouat
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  2. Nadjib Boussetila
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Correspondence to Tahar Bechouat.

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Communicated by Hui Liang.

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Bechouat, T., Boussetila, N. Numerical solution of the two-dimensional first kind Fredholm integral equations using a regularized collocation method. Comp. Appl. Math. 42, 267 (2023). https://doi.org/10.1007/s40314-023-02403-w

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  • DOI: https://doi.org/10.1007/s40314-023-02403-w

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