Application of Wavelet Transform to Urysohn-Type Equations
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...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/18/3999 |
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| author | V. Lukianenko M. Kozlova V. Belozub |
| author_facet | V. Lukianenko M. Kozlova V. Belozub |
| author_sort | V. Lukianenko |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-10T22:29:29Z |
| format | Article |
| id | doaj.art-e33b4789ad324155aa8ad2d860fc41fc |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T22:29:29Z |
| publishDate | 2023-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-e33b4789ad324155aa8ad2d860fc41fc2023-11-19T11:50:31ZengMDPI AGMathematics2227-73902023-09-011118399910.3390/math11183999Application of Wavelet Transform to Urysohn-Type EquationsV. Lukianenko0M. Kozlova1V. Belozub2Institute of Physics and Technology, V.I. Vernadsky Crimean Federal University, Simferopol 295007, RussiaInstitute of Physics and Technology, V.I. Vernadsky Crimean Federal University, Simferopol 295007, RussiaInstitute of Physics and Technology, V.I. Vernadsky Crimean Federal University, Simferopol 295007, RussiaThis 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.https://www.mdpi.com/2227-7390/11/18/3999convolution-type Urysohn equations of the first kindill-posed problemregularization algorithmsthe continuous wavelet transform |
| spellingShingle | V. Lukianenko M. Kozlova V. Belozub Application of Wavelet Transform to Urysohn-Type Equations Mathematics convolution-type Urysohn equations of the first kind ill-posed problem regularization algorithms the continuous wavelet transform |
| title | Application of Wavelet Transform to Urysohn-Type Equations |
| title_full | Application of Wavelet Transform to Urysohn-Type Equations |
| title_fullStr | Application of Wavelet Transform to Urysohn-Type Equations |
| title_full_unstemmed | Application of Wavelet Transform to Urysohn-Type Equations |
| title_short | Application of Wavelet Transform to Urysohn-Type Equations |
| title_sort | application of wavelet transform to urysohn type equations |
| topic | convolution-type Urysohn equations of the first kind ill-posed problem regularization algorithms the continuous wavelet transform |
| url | https://www.mdpi.com/2227-7390/11/18/3999 |
| work_keys_str_mv | AT vlukianenko applicationofwavelettransformtourysohntypeequations AT mkozlova applicationofwavelettransformtourysohntypeequations AT vbelozub applicationofwavelettransformtourysohntypeequations |