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A new approach to the numerical solution of Fredholm integral equations using least squares-support vector regression

Author

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  • Parand, K.
  • Aghaei, A.A.
  • Jani, M.
  • Ghodsi, A.

Abstract

In this paper, we develop a machine learning method with the Least Squares Support Vector Regression (LS-SVR) for the numerical solution of Fredholm integral equations. Two different approaches are proposed for training the network by using the shifted Legendre kernel, the collocation and Galerkin LS-SVR approaches. As with the standard LS-SVR for known dataset regression, the formulation of the method gives rise to an optimization problem. An equivalent system of algebraic equations is then derived and in linear cases discussed in terms of the sparsity of the matrices and computational efficiency. Finally, the method is carried out on some numerical examples, including nonlinear and multidimensional cases to show the accuracy and efficiency of the method.

Suggested Citation

  • Parand, K. & Aghaei, A.A. & Jani, M. & Ghodsi, A., 2021. "A new approach to the numerical solution of Fredholm integral equations using least squares-support vector regression," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 180(C), pages 114-128.
    • Handle: RePEc:eee:matcom:v:180:y:2021:i:c:p:114-128
    DOI: 10.1016/j.matcom.2020.08.010
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    References listed on IDEAS

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    1. Keller, Alexander & Dahm, Ken, 2019. "Integral equations and machine learning," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 161(C), pages 2-12.
    2. Assari, Pouria & Dehghan, Mehdi, 2019. "A meshless local discrete Galerkin (MLDG) scheme for numerically solving two-dimensional nonlinear Volterra integral equations," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 249-265.
    3. Bellour, A. & Sbibih, D. & Zidna, A., 2016. "Two cubic spline methods for solving Fredholm integral equations," Applied Mathematics and Computation, Elsevier, vol. 276(C), pages 1-11.
    4. Justin Sirignano & Konstantinos Spiliopoulos, 2017. "DGM: A deep learning algorithm for solving partial differential equations," Papers 1708.07469, arXiv.org, revised Sep 2018.
    5. Yousefi, S. & Razzaghi, M., 2005. "Legendre wavelets method for the nonlinear Volterra–Fredholm integral equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 70(1), pages 1-8.
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    Cited by:

    1. Pakniyat, A. & Parand, K. & Jani, M., 2021. "Least squares support vector regression for differential equations on unbounded domains," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    2. Sun, Hongli & Lu, Yanfei, 2024. "A novel approach for solving linear Fredholm integro-differential equations via LS-SVM algorithm," Applied Mathematics and Computation, Elsevier, vol. 470(C).
    3. Hajimohammadi, Zeinab & Parand, Kourosh, 2021. "Numerical learning approximation of time-fractional sub diffusion model on a semi-infinite domain," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    4. Ahadian, P. & Parand, K., 2022. "Support vector regression for the temperature-stimulated drug release," Chaos, Solitons & Fractals, Elsevier, vol. 165(P2).
    5. Rahimkhani, P. & Ordokhani, Y., 2022. "Chelyshkov least squares support vector regression for nonlinear stochastic differential equations by variable fractional Brownian motion," Chaos, Solitons & Fractals, Elsevier, vol. 163(C).
    6. Bhaumik, Bivas & De, Soumen & Changdar, Satyasaran, 2024. "Deep learning based solution of nonlinear partial differential equations arising in the process of arterial blood flow," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 217(C), pages 21-36.

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