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Publications (10 of 13) Show all publications
Ashyraliyev, M. & Ashyralyyeva, M. A. (2024). A stable difference scheme for the solution of a source identification problem for telegraph-parabolic equations. BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS SERIES, 115(3), 46-54
Open this publication in new window or tab >>A stable difference scheme for the solution of a source identification problem for telegraph-parabolic equations
2024 (English)In: BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS SERIES, ISSN 2518-7929, Vol. 115, no 3, p. 46-54Article in journal (Refereed) Published
Abstract [en]

In the present paper, we construct a first order of accuracy difference scheme for the approximate solution of the inverse problem for telegraph-parabolic equations with an unknown spacewise dependent source term. The unique solvability of constructed difference scheme and the stability estimates for its solution were obtained. The proofs are based on the spectral representation of the self-adjoint positive definite operator in a Hilbert space.

Place, publisher, year, edition, pages
KARAGANDA STATE UNIV, 2024
Keywords
Difference scheme, source identification problem, telegraph-parabolic equation, stability estimates
National Category
Mathematics
Identifiers
urn:nbn:se:mdh:diva-69426 (URN)10.31489/2024M3/46-54 (DOI)001330178000006 ()2-s2.0-85206336885 (Scopus ID)
Available from: 2024-12-11 Created: 2024-12-11 Last updated: 2024-12-20Bibliographically approved
Arjmand, D. & Ashyraliyev, M. (2024). Efficient low rank approximations for parabolic control problems with unknown heat source. Journal of Computational and Applied Mathematics, 450, Article ID 115959.
Open this publication in new window or tab >>Efficient low rank approximations for parabolic control problems with unknown heat source
2024 (English)In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 450, article id 115959Article in journal (Refereed) Published
Abstract [en]

An inverse problem of finding an unknown heat source for a class of linear parabolic equations is considered. Such problems can typically be converted to a direct problem with non-local conditions in time instead of an initial value problem. Standard ways of solving these non-local problems include direct temporal and spatial discretization as well as the shooting method, which may be computationally expensive in higher dimensions. In the present article, we present approaches based on low-rank approximation via Arnoldi algorithm to bypass the computational limitations of the mentioned classical methods. Regardless of the dimension of the problem, we prove that the Arnoldi approach can be effectively used to turn the inverse problem into a simple initial value problem at the cost of only computing one-dimensional matrix functions while still retaining the same accuracy as the classical approaches. Numerical results in dimensions d=1,2,3 are provided to validate the theoretical findings and to demonstrate the efficiency of the method for growing dimensions.

Place, publisher, year, edition, pages
Elsevier B.V., 2024
Keywords
Arnoldi algorithm, Control problems, Heat equation, Inverse problems, Low rank approximations, Parabolic PDEs, Approximation algorithms, Approximation theory, Initial value problems, Numerical methods, Partial differential equations, Direct problems, Heat sources, Initial-value problem, Linear parabolic equation, Parabolics
National Category
Mathematics
Identifiers
urn:nbn:se:mdh:diva-66731 (URN)10.1016/j.cam.2024.115959 (DOI)001266419100001 ()2-s2.0-85193820326 (Scopus ID)
Available from: 2024-05-29 Created: 2024-05-29 Last updated: 2024-12-20Bibliographically approved
Ashyraliyev, M. & Ashyralyyeva, M. (2024). Stable difference schemes for hyperbolic–parabolic equations with unknown parameter. Boletín de la Sociedad Matematica Mexicana, 30(1), Article ID 14.
Open this publication in new window or tab >>Stable difference schemes for hyperbolic–parabolic equations with unknown parameter
2024 (English)In: Boletín de la Sociedad Matematica Mexicana, ISSN 1405-213X, Vol. 30, no 1, article id 14Article in journal (Refereed) Published
Abstract [en]

In the present paper, we study the first and second order of accuracy difference schemes for the approximate solution of the inverse problem for hyperbolic–parabolic equations with unknown time-independent source term. The unique solvability of constructed difference schemes and the stability estimates for their solutions are obtained. The proofs are based on the spectral representation of the self-adjoint positive definite operator in a Hilbert space.

Place, publisher, year, edition, pages
Birkhauser, 2024
Keywords
Difference schemes, Hyperbolic–parabolic equation, Source identification problem, Stability estimates
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:mdh:diva-65362 (URN)10.1007/s40590-023-00585-1 (DOI)001137943400001 ()2-s2.0-85181692212 (Scopus ID)
Available from: 2024-01-17 Created: 2024-01-17 Last updated: 2024-12-20Bibliographically approved
Ashyraliyev, M. & Ashyralyyeva, M. (2021). A Note on a Hyperbolic-Parabolic Problem with Involution. In: Springer Proceedings in Mathematics and Statistics, Volume 351: . Paper presented at 4th International Conference on Analysis and Applied Mathematics ICAAM 2018, Mersin, Turkey, September 6-9, 2018 (pp. 213-221). , 351, Article ID 262329.
Open this publication in new window or tab >>A Note on a Hyperbolic-Parabolic Problem with Involution
2021 (English)In: Springer Proceedings in Mathematics and Statistics, Volume 351, 2021, Vol. 351, p. 213-221, article id 262329Conference paper, Published paper (Refereed)
Abstract [en]

In the present paper, a boundary value problem for a one-dimensional hyperbolic-parabolic equation with involution and the Dirichlet condition is studied. The stability estimates for the solution of the hyperbolic-parabolic problem are established. The first order of accuracy stable difference scheme for the approximate solution of the problem under consideration is constructed. Numerical algorithm for implementation of this scheme is presented. Numerical results are provided for a simple test problem.

National Category
Computational Mathematics Mathematical Analysis
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-56223 (URN)10.1007/978-3-030-69292-6_16 (DOI)2-s2.0-85112221635 (Scopus ID)
Conference
4th International Conference on Analysis and Applied Mathematics ICAAM 2018, Mersin, Turkey, September 6-9, 2018
Available from: 2021-10-15 Created: 2021-10-15 Last updated: 2021-10-28Bibliographically approved
Ashyraliyev, M. & Ashyralyyeva, M. (2021). A note on the hyperbolic-parabolic identification problem with nonlocal conditions. Paper presented at 4th International Conference of Mathematical Sciences ICMS 2020, Istanbul, Turkey, June 17-21, 2020. AIP Conference Proceedings, 2334, Article ID 060001.
Open this publication in new window or tab >>A note on the hyperbolic-parabolic identification problem with nonlocal conditions
2021 (English)In: AIP Conference Proceedings, ISSN 0094-243X, E-ISSN 1551-7616, Vol. 2334, article id 060001Article in journal (Refereed) Published
Abstract [en]

In the present paper, we study a source identification problem for hyperbolic-parabolic equation with nonlocal conditions. The stability estimates for the solution of this source identification problem are established. Furthermore, we construct the second order of accuracy difference scheme for the approximate solution of the problem under consideration. The stability estimates for the solution of this difference scheme are presented.

National Category
Mathematical Analysis Computational Mathematics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-56226 (URN)10.1063/5.0042271 (DOI)000664201400042 ()2-s2.0-85102297253 (Scopus ID)
Conference
4th International Conference of Mathematical Sciences ICMS 2020, Istanbul, Turkey, June 17-21, 2020
Available from: 2021-10-15 Created: 2021-10-15 Last updated: 2021-10-28Bibliographically approved
Ashyraliyev, M., Ashyralyev, A. & Zvyagin, V. (2021). A Space-Dependent Source Identification Problem for Hyperbolic-Parabolic Equations. In: Springer Proceedings in Mathematics and Statistics, Volume 351: . Paper presented at 4th International Conference on Analysis and Applied Mathematics ICAAM 2018, Mersin, Turkey, September 6-9, 2018 (pp. 183-198). , 351
Open this publication in new window or tab >>A Space-Dependent Source Identification Problem for Hyperbolic-Parabolic Equations
2021 (English)In: Springer Proceedings in Mathematics and Statistics, Volume 351, 2021, Vol. 351, p. 183-198Conference paper, Published paper (Refereed)
Abstract [en]

In the present paper, a space-dependent source identification problem for the hyperbolic-parabolic equation with unknown parameter p $$ \left\{ \begin{array}{l} \displaystyle u''(t) + Au(t) = p + f(t), ~ 0<t<1, \\ \displaystyle u'(t) + Au(t) = p + g(t), ~ -1<t<0, \\ \displaystyle u(0^{+})=u(0^{-}), ~ u'(0^{+})=u'(0^{-}), \\ \displaystyle u(-1)=\varphi, ~ \int \limits _{0}^{1} u(z)dz=\psi \end{array} \right. $${u′′(t)+Au(t)=p+f(t),0<t<1,u′(t)+Au(t)=p+g(t),-1<t<0,u(0+)=u(0-),u′(0+)=u′(0-),u(-1)=φ,∫01u(z)dz=ψ in a Hilbert space H with self-adjoint positive definite operator A is investigated. The stability estimates for the solution of this identification problem are established. In applications, the stability estimates for the solutions of four space-dependent source identification hyperbolic-parabolic problems are obtained.

National Category
Mathematical Analysis Computational Mathematics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-56225 (URN)10.1007/978-3-030-69292-6_14 (DOI)2-s2.0-85112190160 (Scopus ID)
Conference
4th International Conference on Analysis and Applied Mathematics ICAAM 2018, Mersin, Turkey, September 6-9, 2018
Available from: 2021-10-15 Created: 2021-10-15 Last updated: 2021-10-28Bibliographically approved
Ashyraliyev, M. (2021). On hyperbolic-parabolic problems with involution and neumann boundary condition. International Journal of Applied Mathematics, 34(2), 363-376
Open this publication in new window or tab >>On hyperbolic-parabolic problems with involution and neumann boundary condition
2021 (English)In: International Journal of Applied Mathematics, ISSN 1311-1728, E-ISSN 1314-8060, Vol. 34, no 2, p. 363-376Article in journal (Refereed) Published
Abstract [en]

We study a nonlocal boundary value problem and a space-wise dependent source identification problem for one-dimensional hyperbolic-parabolic equation with involution and Neumann boundary condition. The stability estimates for the solutions of these two problems are established. The first order of accuracy stable difference schemes are constructed for the approximate solutions of the problems under consideration. Numerical results for two test problems are provided.

Keywords
Computational Theory and Mathematics, General Mathematics
National Category
Mathematical Analysis Computational Mathematics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-56216 (URN)10.12732/ijam.v34i2.12 (DOI)2-s2.0-85106586263 (Scopus ID)
Available from: 2021-10-15 Created: 2021-10-15 Last updated: 2023-05-17Bibliographically approved
Ashyraliyev, M., Ashyralyev, A. & Zvyagin, V. (2021). On the source identification problem for hyperbolic-parabolic equation with nonlocal conditions. Paper presented at 5th International Conference on Analysis and Applied Mathematics ICAAM 2020, Mersin, Turkey, September 23-30, 2020. AIP Conference Proceedings, 2325, Article ID 020016.
Open this publication in new window or tab >>On the source identification problem for hyperbolic-parabolic equation with nonlocal conditions
2021 (English)In: AIP Conference Proceedings, ISSN 0094-243X, E-ISSN 1551-7616, Vol. 2325, article id 020016Article in journal (Refereed) Published
Abstract [en]

In the present paper, we establish the well-posedness of an identification problem for determining the unknown space-dependent source term in the hyperbolic-parabolic equation with nonlocal conditions. The difference scheme is constructed for the approximate solution of this source identification problem. The stability estimates for the solution of the difference scheme are presented.

National Category
Mathematical Analysis Computational Mathematics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-56227 (URN)10.1063/5.0040269 (DOI)000653734600005 ()2-s2.0-85101657949 (Scopus ID)
Conference
5th International Conference on Analysis and Applied Mathematics ICAAM 2020, Mersin, Turkey, September 23-30, 2020
Available from: 2021-10-15 Created: 2021-10-15 Last updated: 2021-10-28Bibliographically approved
Ashyraliyev, M., Ashyralyyeva, M. & Ashyralyev, A. (2020). A note on the hyperbolic-parabolic identification problem with involution and Dirichlet boundary condition. Paper presented at 5th International Conference on Analysis and Applied Mathematics (ICAAM), Mersin, Turkey, September 23-30, 2020. Bulletin of the Karaganda University - Mathematics, 99(3), 120-129
Open this publication in new window or tab >>A note on the hyperbolic-parabolic identification problem with involution and Dirichlet boundary condition
2020 (English)In: Bulletin of the Karaganda University - Mathematics, ISSN 2518-7929, Vol. 99, no 3, p. 120-129Article in journal (Refereed) Published
Abstract [en]

In the present paper, a source identification problem for hyperbolic-parabolic equation with involution and Dirichlet condition is studied. The stability estimates for the solution of the source identification hyperbolic-parabolic problem are established. The first order of accuracy stable difference scheme is constructed for the approximate solution of the problem under consideration. Numerical results are given for a simple test problem.

National Category
Mathematical Analysis Computational Mathematics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-56221 (URN)10.31489/2020m3/120-129 (DOI)000580591000012 ()2-s2.0-85106600167 (Scopus ID)
Conference
5th International Conference on Analysis and Applied Mathematics (ICAAM), Mersin, Turkey, September 23-30, 2020
Available from: 2021-10-15 Created: 2021-10-15 Last updated: 2023-05-10Bibliographically approved
Ashyralyev, A., Ashyraliyev, M. & Ashyralyyeva, M. A. (2020). Identification Problem for Telegraph-Parabolic Equations. Computational Mathematics and Mathematical Physics, 60(8), 1294-1305
Open this publication in new window or tab >>Identification Problem for Telegraph-Parabolic Equations
2020 (English)In: Computational Mathematics and Mathematical Physics, ISSN 0965-5425, E-ISSN 1555-6662, Vol. 60, no 8, p. 1294-1305Article in journal (Refereed) Published
Abstract [en]

An identification problem for an equation of mixed telegraph-parabolic type with an unknown parameter depending on spatial variables is considered. The unique solvability of this problem is proved, and stability inequalities for its solution are established. As applications, stability estimates are obtained for the solutions of four identification problems for telegraph-parabolic equations with an unknown source depending on spatial variables.

Keywords
Computational Mathematics
National Category
Mathematical Analysis Computational Mathematics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-56214 (URN)10.1134/s0965542520080035 (DOI)000575902400005 ()2-s2.0-85092332413 (Scopus ID)
Available from: 2021-10-15 Created: 2021-10-15 Last updated: 2021-10-28Bibliographically approved
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Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0001-6708-3160

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