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Application of Wavelet Transform to Urysohn-Type Equations

Author

Listed:
  • V. Lukianenko

    (Institute of Physics and Technology, V.I. Vernadsky Crimean Federal University, Simferopol 295007, Russia
    These authors contributed equally to this work.)

  • M. Kozlova

    (Institute of Physics and Technology, V.I. Vernadsky Crimean Federal University, Simferopol 295007, Russia
    These authors contributed equally to this work.)

  • V. Belozub

    (Institute of Physics and Technology, V.I. Vernadsky Crimean Federal University, Simferopol 295007, Russia)

Abstract

This paper deals with convolution-type Urysohn equations of the first kind. Finding a solution for such equations is an ill-posed problem. For it to be solved, regularization algorithms and the continuous wavelet transform are used. Similar to the Fourier transform, the continuous wavelet transform is applied to convolution-type equations (based on the Fourier and wavelet transforms) and to Urysohn equations with unknown shift. The wavelet transform is preferable for the cases with approximated right-hand sides and for type 1 equations. We demonstrated that the application of the wavelet transform to Urysohn-type equations with unknown shift translates into a solution of a nonlinear equation with an oscillating kernel. Depending on the availability of a priori information, a combination of regularization and iterative algorithms with the use of close equations are effective for solving convolution-type equations based on the continuous wavelet transform and Urysohn equation.

Suggested Citation

  • V. Lukianenko & M. Kozlova & V. Belozub, 2023. "Application of Wavelet Transform to Urysohn-Type Equations," Mathematics, MDPI, vol. 11(18), pages 1-16, September.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:18:p:3999-:d:1244091
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    References listed on IDEAS

    as
    1. Rajagopalan Ramaswamy & Gunaseelan Mani & Arul Joseph Gnanaprakasam & Ola A. Ashour Abdelnaby & Stojan Radenović, 2022. "An Application of Urysohn Integral Equation via Complex Partial Metric Space," Mathematics, MDPI, vol. 10(12), pages 1-13, June.
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