Inverse Problem for the Integral Dynamic Models with Discontinuous Kernels
The objective of this paper was to present a new inverse problem statement and numerical method for the Volterra integral equations with piecewise continuous kernels. For such Volterra integral equations of the first kind, it is assumed that kernel discontinuity curves are the desired ones, but the...
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MDPI AG
2022-10-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/21/3945 |
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| author | Aleksandr N. Tynda Denis N. Sidorov |
| author_facet | Aleksandr N. Tynda Denis N. Sidorov |
| author_sort | Aleksandr N. Tynda |
| collection | DOAJ |
| description | The objective of this paper was to present a new inverse problem statement and numerical method for the Volterra integral equations with piecewise continuous kernels. For such Volterra integral equations of the first kind, it is assumed that kernel discontinuity curves are the desired ones, but the rest of the information is known. The resulting integral equation is nonlinear with respect to discontinuity curves which correspond to integration bounds. A direct method of discretization with a posteriori verification of calculations is proposed. The family of quadrature rules is employed for approximation purposes. It is shown that the arithmetic complexity of the proposed numerical method is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><msup><mi>N</mi><mn>3</mn></msup><mo>)</mo></mrow></semantics></math></inline-formula>. The method has first-order convergence. A generalization of the method is also proposed for the case of an arbitrary number of discontinuity curves. The illustrative examples are included to demonstrate the efficiency and accuracy of proposed solver. |
| first_indexed | 2024-03-09T18:52:31Z |
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| id | doaj.art-119e1b1b69fd4a44831379c861d8b368 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T18:52:31Z |
| publishDate | 2022-10-01 |
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| spelling | doaj.art-119e1b1b69fd4a44831379c861d8b3682023-11-24T05:42:27ZengMDPI AGMathematics2227-73902022-10-011021394510.3390/math10213945Inverse Problem for the Integral Dynamic Models with Discontinuous KernelsAleksandr N. Tynda0Denis N. Sidorov1Department of Mathematics, Penza State University, Krasnaya Str., 40, 440026 Penza, RussiaDepartment of Applied Mathematics, Energy Systems Institute of Siberian Branch of Russian Academy of Science, 664033 Irkutsk, RussiaThe objective of this paper was to present a new inverse problem statement and numerical method for the Volterra integral equations with piecewise continuous kernels. For such Volterra integral equations of the first kind, it is assumed that kernel discontinuity curves are the desired ones, but the rest of the information is known. The resulting integral equation is nonlinear with respect to discontinuity curves which correspond to integration bounds. A direct method of discretization with a posteriori verification of calculations is proposed. The family of quadrature rules is employed for approximation purposes. It is shown that the arithmetic complexity of the proposed numerical method is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><msup><mi>N</mi><mn>3</mn></msup><mo>)</mo></mrow></semantics></math></inline-formula>. The method has first-order convergence. A generalization of the method is also proposed for the case of an arbitrary number of discontinuity curves. The illustrative examples are included to demonstrate the efficiency and accuracy of proposed solver.https://www.mdpi.com/2227-7390/10/21/3945Volterra integral equation of the first kinddiscontinuous kernelsinverse problemunknown discontinuity curvesarithmetic complexity |
| spellingShingle | Aleksandr N. Tynda Denis N. Sidorov Inverse Problem for the Integral Dynamic Models with Discontinuous Kernels Mathematics Volterra integral equation of the first kind discontinuous kernels inverse problem unknown discontinuity curves arithmetic complexity |
| title | Inverse Problem for the Integral Dynamic Models with Discontinuous Kernels |
| title_full | Inverse Problem for the Integral Dynamic Models with Discontinuous Kernels |
| title_fullStr | Inverse Problem for the Integral Dynamic Models with Discontinuous Kernels |
| title_full_unstemmed | Inverse Problem for the Integral Dynamic Models with Discontinuous Kernels |
| title_short | Inverse Problem for the Integral Dynamic Models with Discontinuous Kernels |
| title_sort | inverse problem for the integral dynamic models with discontinuous kernels |
| topic | Volterra integral equation of the first kind discontinuous kernels inverse problem unknown discontinuity curves arithmetic complexity |
| url | https://www.mdpi.com/2227-7390/10/21/3945 |
| work_keys_str_mv | AT aleksandrntynda inverseproblemfortheintegraldynamicmodelswithdiscontinuouskernels AT denisnsidorov inverseproblemfortheintegraldynamicmodelswithdiscontinuouskernels |