Development of a Finite-difference Regularized Solution of the One-Dimensional Inverse Problem of the Wave Process

Abdugany Dzhunusovich Satybaev; Yuliya Vladimirovna Anishchenko; Ainagul Zhylkychyevna Kokozova; Aliyma Torozhanovna Mamatkasymova; Guljamal Abdazovna Kaldybaeva

doi:doi:10.11648/j.ajam.20200802.13

Development of a Finite-difference Regularized Solution of the One-Dimensional Inverse Problem of the Wave Process

Abdugany Dzhunusovich Satybaev, Yuliya Vladimirovna Anishchenko, Ainagul Zhylkychyevna Kokozova, Aliyma Torozhanovna Mamatkasymova, Guljamal Abdazovna Kaldybaeva

Published in American Journal of Applied Mathematics (Volume 8, Issue 2)

Received: 3 March 2020 Accepted: 23 March 2020 Published: 13 April 2020

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Abstract

We consider a one-dimensional inverse problem for a partial differential equation of hyperbolic type with sources - the Dirac delta-function and the Heaviside theta-function. The generalized inverse problem is reduced to the inverse problem with data on the characteristics using the method of characteristics and the method of isolation of singularities. At the beginning, the inverse problem of the wave process with data on the characteristics with additional information for the inverse problem without small perturbations is solved by the finite-difference method. Then, for the inverse problem of the wave process with data on the characteristics with additional information with small perturbations, that is, with small changes is used by the finite-difference regularized method, which developed by one of the authors of this article. The convergence of the finite-difference regularized solution to the exact solution of the one-dimensional inverse problem of the wave process on the characteristics is shown, and the theorem on the convergence of the approximate solution to the exact solution is proved. An estimate is obtained for the convergence of the numerical regularized solution to the exact solution, which depends on the grid step, on the perturbations parameter, and on the norm of known functions. From the equivalence of the problems, the one-dimensional inverse problem of the wave process with sources - the Dirac delta-function and the Heaviside theta-function and the one-dimensional inverse problem of the wave process with data on the characteristics, it follows that the solution of the last problem will be the solution of the posed initial problem. An algorithm for solving a finite-difference regularized solution of a generalized one-dimensional inverse problem is constructed.

Published in	American Journal of Applied Mathematics (Volume 8, Issue 2)
DOI	10.11648/j.ajam.20200802.13
Page(s)	64-73
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

One-dimensional Inverse Problem, Wave Process, Dirac Delta-Function, Heaviside Theta-Function, Method of Characteristic, Method of Isolation of Singularities, Finite-Difference Regularized Solution

References

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Abdugany Dzhunusovich Satybaev, Yuliya Vladimirovna Anishchenko, Ainagul Zhylkychyevna Kokozova, Aliyma Torozhanovna Mamatkasymova, Guljamal Abdazovna Kaldybaeva. (2020). Development of a Finite-difference Regularized Solution of the One-Dimensional Inverse Problem of the Wave Process. American Journal of Applied Mathematics, 8(2), 64-73. https://doi.org/10.11648/j.ajam.20200802.13

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Abdugany Dzhunusovich Satybaev; Yuliya Vladimirovna Anishchenko; Ainagul Zhylkychyevna Kokozova; Aliyma Torozhanovna Mamatkasymova; Guljamal Abdazovna Kaldybaeva. Development of a Finite-difference Regularized Solution of the One-Dimensional Inverse Problem of the Wave Process. Am. J. Appl. Math. 2020, 8(2), 64-73. doi: 10.11648/j.ajam.20200802.13

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AMA Style

Abdugany Dzhunusovich Satybaev, Yuliya Vladimirovna Anishchenko, Ainagul Zhylkychyevna Kokozova, Aliyma Torozhanovna Mamatkasymova, Guljamal Abdazovna Kaldybaeva. Development of a Finite-difference Regularized Solution of the One-Dimensional Inverse Problem of the Wave Process. Am J Appl Math. 2020;8(2):64-73. doi: 10.11648/j.ajam.20200802.13

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@article{10.11648/j.ajam.20200802.13,
  author = {Abdugany Dzhunusovich Satybaev and Yuliya Vladimirovna Anishchenko and Ainagul Zhylkychyevna Kokozova and Aliyma Torozhanovna Mamatkasymova and Guljamal Abdazovna Kaldybaeva},
  title = {Development of a Finite-difference Regularized Solution of the One-Dimensional Inverse Problem of the Wave Process},
  journal = {American Journal of Applied Mathematics},
  volume = {8},
  number = {2},
  pages = {64-73},
  doi = {10.11648/j.ajam.20200802.13},
  url = {https://doi.org/10.11648/j.ajam.20200802.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200802.13},
  abstract = {We consider a one-dimensional inverse problem for a partial differential equation of hyperbolic type with sources - the Dirac delta-function and the Heaviside theta-function. The generalized inverse problem is reduced to the inverse problem with data on the characteristics using the method of characteristics and the method of isolation of singularities. At the beginning, the inverse problem of the wave process with data on the characteristics with additional information for the inverse problem without small perturbations is solved by the finite-difference method. Then, for the inverse problem of the wave process with data on the characteristics with additional information with small perturbations, that is, with small changes is used by the finite-difference regularized method, which developed by one of the authors of this article. The convergence of the finite-difference regularized solution to the exact solution of the one-dimensional inverse problem of the wave process on the characteristics is shown, and the theorem on the convergence of the approximate solution to the exact solution is proved. An estimate is obtained for the convergence of the numerical regularized solution to the exact solution, which depends on the grid step, on the perturbations parameter, and on the norm of known functions. From the equivalence of the problems, the one-dimensional inverse problem of the wave process with sources - the Dirac delta-function and the Heaviside theta-function and the one-dimensional inverse problem of the wave process with data on the characteristics, it follows that the solution of the last problem will be the solution of the posed initial problem. An algorithm for solving a finite-difference regularized solution of a generalized one-dimensional inverse problem is constructed.},
 year = {2020}
}

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TY  - JOUR
T1  - Development of a Finite-difference Regularized Solution of the One-Dimensional Inverse Problem of the Wave Process
AU  - Abdugany Dzhunusovich Satybaev
AU  - Yuliya Vladimirovna Anishchenko
AU  - Ainagul Zhylkychyevna Kokozova
AU  - Aliyma Torozhanovna Mamatkasymova
AU  - Guljamal Abdazovna Kaldybaeva
Y1  - 2020/04/13
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200802.13
DO  - 10.11648/j.ajam.20200802.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 64
EP  - 73
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200802.13
AB  - We consider a one-dimensional inverse problem for a partial differential equation of hyperbolic type with sources - the Dirac delta-function and the Heaviside theta-function. The generalized inverse problem is reduced to the inverse problem with data on the characteristics using the method of characteristics and the method of isolation of singularities. At the beginning, the inverse problem of the wave process with data on the characteristics with additional information for the inverse problem without small perturbations is solved by the finite-difference method. Then, for the inverse problem of the wave process with data on the characteristics with additional information with small perturbations, that is, with small changes is used by the finite-difference regularized method, which developed by one of the authors of this article. The convergence of the finite-difference regularized solution to the exact solution of the one-dimensional inverse problem of the wave process on the characteristics is shown, and the theorem on the convergence of the approximate solution to the exact solution is proved. An estimate is obtained for the convergence of the numerical regularized solution to the exact solution, which depends on the grid step, on the perturbations parameter, and on the norm of known functions. From the equivalence of the problems, the one-dimensional inverse problem of the wave process with sources - the Dirac delta-function and the Heaviside theta-function and the one-dimensional inverse problem of the wave process with data on the characteristics, it follows that the solution of the last problem will be the solution of the posed initial problem. An algorithm for solving a finite-difference regularized solution of a generalized one-dimensional inverse problem is constructed.
VL  - 8
IS  - 2
ER  -

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Author Information

Abdugany Dzhunusovich Satybaev

Department of Information Technology and Management, Faculty of Cybernetics and Information Technology, Osh Technological University, Osh, Kyrgyzstan
Yuliya Vladimirovna Anishchenko

Department of Information Technology and Management, Faculty of Cybernetics and Information Technology, Osh Technological University, Osh, Kyrgyzstan
Ainagul Zhylkychyevna Kokozova

Department of Information Technology and Management, Faculty of Cybernetics and Information Technology, Osh Technological University, Osh, Kyrgyzstan
Aliyma Torozhanovna Mamatkasymova

Department of Informatics, Naturally-technical Faculty, Osh Technological University, Osh, Kyrgyzstan
Guljamal Abdazovna Kaldybaeva

Department of Physics, Mathematics and Information Technology, Osh State University, Osh, Kyrgyzstan

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Abdugany Dzhunusovich Satybaev, Yuliya Vladimirovna Anishchenko, Ainagul Zhylkychyevna Kokozova, Aliyma Torozhanovna Mamatkasymova, Guljamal Abdazovna Kaldybaeva. (2020). Development of a Finite-difference Regularized Solution of the One-Dimensional Inverse Problem of the Wave Process. American Journal of Applied Mathematics, 8(2), 64-73. https://doi.org/10.11648/j.ajam.20200802.13

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ACS Style

Abdugany Dzhunusovich Satybaev; Yuliya Vladimirovna Anishchenko; Ainagul Zhylkychyevna Kokozova; Aliyma Torozhanovna Mamatkasymova; Guljamal Abdazovna Kaldybaeva. Development of a Finite-difference Regularized Solution of the One-Dimensional Inverse Problem of the Wave Process. Am. J. Appl. Math. 2020, 8(2), 64-73. doi: 10.11648/j.ajam.20200802.13

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AMA Style

Abdugany Dzhunusovich Satybaev, Yuliya Vladimirovna Anishchenko, Ainagul Zhylkychyevna Kokozova, Aliyma Torozhanovna Mamatkasymova, Guljamal Abdazovna Kaldybaeva. Development of a Finite-difference Regularized Solution of the One-Dimensional Inverse Problem of the Wave Process. Am J Appl Math. 2020;8(2):64-73. doi: 10.11648/j.ajam.20200802.13

Copy | Download

@article{10.11648/j.ajam.20200802.13,
  author = {Abdugany Dzhunusovich Satybaev and Yuliya Vladimirovna Anishchenko and Ainagul Zhylkychyevna Kokozova and Aliyma Torozhanovna Mamatkasymova and Guljamal Abdazovna Kaldybaeva},
  title = {Development of a Finite-difference Regularized Solution of the One-Dimensional Inverse Problem of the Wave Process},
  journal = {American Journal of Applied Mathematics},
  volume = {8},
  number = {2},
  pages = {64-73},
  doi = {10.11648/j.ajam.20200802.13},
  url = {https://doi.org/10.11648/j.ajam.20200802.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200802.13},
  abstract = {We consider a one-dimensional inverse problem for a partial differential equation of hyperbolic type with sources - the Dirac delta-function and the Heaviside theta-function. The generalized inverse problem is reduced to the inverse problem with data on the characteristics using the method of characteristics and the method of isolation of singularities. At the beginning, the inverse problem of the wave process with data on the characteristics with additional information for the inverse problem without small perturbations is solved by the finite-difference method. Then, for the inverse problem of the wave process with data on the characteristics with additional information with small perturbations, that is, with small changes is used by the finite-difference regularized method, which developed by one of the authors of this article. The convergence of the finite-difference regularized solution to the exact solution of the one-dimensional inverse problem of the wave process on the characteristics is shown, and the theorem on the convergence of the approximate solution to the exact solution is proved. An estimate is obtained for the convergence of the numerical regularized solution to the exact solution, which depends on the grid step, on the perturbations parameter, and on the norm of known functions. From the equivalence of the problems, the one-dimensional inverse problem of the wave process with sources - the Dirac delta-function and the Heaviside theta-function and the one-dimensional inverse problem of the wave process with data on the characteristics, it follows that the solution of the last problem will be the solution of the posed initial problem. An algorithm for solving a finite-difference regularized solution of a generalized one-dimensional inverse problem is constructed.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Development of a Finite-difference Regularized Solution of the One-Dimensional Inverse Problem of the Wave Process
AU  - Abdugany Dzhunusovich Satybaev
AU  - Yuliya Vladimirovna Anishchenko
AU  - Ainagul Zhylkychyevna Kokozova
AU  - Aliyma Torozhanovna Mamatkasymova
AU  - Guljamal Abdazovna Kaldybaeva
Y1  - 2020/04/13
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200802.13
DO  - 10.11648/j.ajam.20200802.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 64
EP  - 73
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200802.13
AB  - We consider a one-dimensional inverse problem for a partial differential equation of hyperbolic type with sources - the Dirac delta-function and the Heaviside theta-function. The generalized inverse problem is reduced to the inverse problem with data on the characteristics using the method of characteristics and the method of isolation of singularities. At the beginning, the inverse problem of the wave process with data on the characteristics with additional information for the inverse problem without small perturbations is solved by the finite-difference method. Then, for the inverse problem of the wave process with data on the characteristics with additional information with small perturbations, that is, with small changes is used by the finite-difference regularized method, which developed by one of the authors of this article. The convergence of the finite-difference regularized solution to the exact solution of the one-dimensional inverse problem of the wave process on the characteristics is shown, and the theorem on the convergence of the approximate solution to the exact solution is proved. An estimate is obtained for the convergence of the numerical regularized solution to the exact solution, which depends on the grid step, on the perturbations parameter, and on the norm of known functions. From the equivalence of the problems, the one-dimensional inverse problem of the wave process with sources - the Dirac delta-function and the Heaviside theta-function and the one-dimensional inverse problem of the wave process with data on the characteristics, it follows that the solution of the last problem will be the solution of the posed initial problem. An algorithm for solving a finite-difference regularized solution of a generalized one-dimensional inverse problem is constructed.
VL  - 8
IS  - 2
ER  -

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