On an Integral Equation with the Riemann Function Kernel
This paper is concerned with a study of a special integral equation. This integral equation arises in many applied problems, including transmutation theory, inverse scattering problems, the solution of singular Sturm–Liouville and Shrödinger equations, and the representation of solutions of singular...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-04-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/11/4/166 |
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| author | Sergei Sitnik Abdul Ahad Arian |
| author_facet | Sergei Sitnik Abdul Ahad Arian |
| author_sort | Sergei Sitnik |
| collection | DOAJ |
| description | This paper is concerned with a study of a special integral equation. This integral equation arises in many applied problems, including transmutation theory, inverse scattering problems, the solution of singular Sturm–Liouville and Shrödinger equations, and the representation of solutions of singular Sturm–Liouville and Shrödinger equations. A special integral equation is derived and formulated using the Riemann function of a singular hyperbolic equation. In the paper, the existence of a unique solution to this equation is proven by the method of successive approximations. The results can be applied, for example, to representations of solutions to Sturm–Liouville equations with singular potentials, such as Bargmann and Miura potentials, and similiar. The treatment of problems with such potentials are very important in mathematical physics, and inverse, scattering and related problems. The estimates received do not contain any undefined constants, and for transmutation kernels all estimates are explicitly written. |
| first_indexed | 2024-03-09T11:09:53Z |
| format | Article |
| id | doaj.art-386899038c2c4753a3cd4eca0ca65e92 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-09T11:09:53Z |
| publishDate | 2022-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-386899038c2c4753a3cd4eca0ca65e922023-12-01T00:48:28ZengMDPI AGAxioms2075-16802022-04-0111416610.3390/axioms11040166On an Integral Equation with the Riemann Function KernelSergei Sitnik0Abdul Ahad Arian1Applied Mathematics and Computer Modelling, Institute of Engineering and Digital Technologies, Belgorod State National Research University, Pobedy Street, 85, 308015 Belgorod, RussiaApplied Mathematics and Computer Modelling, Institute of Engineering and Digital Technologies, Belgorod State National Research University, Pobedy Street, 85, 308015 Belgorod, RussiaThis paper is concerned with a study of a special integral equation. This integral equation arises in many applied problems, including transmutation theory, inverse scattering problems, the solution of singular Sturm–Liouville and Shrödinger equations, and the representation of solutions of singular Sturm–Liouville and Shrödinger equations. A special integral equation is derived and formulated using the Riemann function of a singular hyperbolic equation. In the paper, the existence of a unique solution to this equation is proven by the method of successive approximations. The results can be applied, for example, to representations of solutions to Sturm–Liouville equations with singular potentials, such as Bargmann and Miura potentials, and similiar. The treatment of problems with such potentials are very important in mathematical physics, and inverse, scattering and related problems. The estimates received do not contain any undefined constants, and for transmutation kernels all estimates are explicitly written.https://www.mdpi.com/2075-1680/11/4/166transmutationsSturm–Liouville operatorsingular potentialBargmann potentialsuccessive approximations |
| spellingShingle | Sergei Sitnik Abdul Ahad Arian On an Integral Equation with the Riemann Function Kernel Axioms transmutations Sturm–Liouville operator singular potential Bargmann potential successive approximations |
| title | On an Integral Equation with the Riemann Function Kernel |
| title_full | On an Integral Equation with the Riemann Function Kernel |
| title_fullStr | On an Integral Equation with the Riemann Function Kernel |
| title_full_unstemmed | On an Integral Equation with the Riemann Function Kernel |
| title_short | On an Integral Equation with the Riemann Function Kernel |
| title_sort | on an integral equation with the riemann function kernel |
| topic | transmutations Sturm–Liouville operator singular potential Bargmann potential successive approximations |
| url | https://www.mdpi.com/2075-1680/11/4/166 |
| work_keys_str_mv | AT sergeisitnik onanintegralequationwiththeriemannfunctionkernel AT abdulahadarian onanintegralequationwiththeriemannfunctionkernel |