Carleman's Formula of a Solution of the Poisson Equation
Keywords:
Poisson equations, ill-posed problem, regular solution, Carleman -Yarmuhamedov function, Green's formula, Carleman formula, Mittag-Le¬er entire function
Abstract
We suggest an explicit continuation formula for are a solution to the Cauchy problem for the Poisson equation in a domain from its values and the values of its normal derivative on part of the boundary. We construct an continuation formula of this problem based on the Carleman-Yarmuhamedov function method.
References
1. Alessandrini G., Rondi L., Rosset E., and Vessella S., The stability for the Cauchy problem for elliptic equations, Inverse problems, 25 (2009), p. 123004.
2. Belgacem F. B., Why is the Cauchy problem severely ill-posed?, Inverse Problems, 23 (2007), p. 823.
3. Blum J., Numerical simulation and optimal control in plasma physics, New York, NY; John Wiley and Sons Inc., 1989.
4. David Gilarg Neil S.Trudinger Elliptik Partial Differential Equations of Second Order Springer – Verlag BerlinHeidelberg New York Tokyo 1983
5. Goluzin G. M. and Krylov V.I., "A generalized Carleman fornula and its application to analytic continuation of functions,"Mat.Sb., 40, No.2, (1933), p.144-149 .
6. Dzhrbashyan M.M., Integral Transformations and Representations of Functions in a Complex Domain [in Russian], Nauka, Moskow (1966).
7. Inglese G., An inverse problem in corrosion detection, Inverse problems, 13 (1997), p. 977.
8. Latt es R. and Lions J., The method of quasi-reversibility: applications to partial differential equations, no. 18 in Translated from the French edition and edited by Richard Bellman. Modern Analytic and Computational Methods in Science and Mathematics, American Elsevier Publishing Co., Inc., New York½ 1969.
9. Lavrent'ev M.M., On Some Ill-Posed Problems of Mathematical Physics [in Russian], Sibirsk. Otdel. Akad. Nauk SSSR. Novosibirsk (1962).
10. Mergelyan S.N. "Harmonic approximation and approximate solution of the Cauchy problem for the Laplace equation."Uspekhi Mat. Nauk, 11.No.5. (1956), p.3-26 .
11. Hadamard J. The Cauchy Problem for Linear Partial Dierential Equations of Huperbolic Type [Russian translation], Mir, Moscow (1978), 352 p.
12. Carleman T., Les founctions quasi analitiques, Gauthier-Villars,Paris, 1926.
13. Colli-Franzone P., Guerri L., Tentoni L., Viganotti L., Baru S., Spaggiari S., and Taccardi B., A mathematical procedure for solving the inverse potential problem of electrocardiography. Analysis of the time-space accuracy from in vitro experimental data, Mathematical Biosciences, 77 (1985), pp. 353396.
14. Tikhonov A. N. and Arsenin V. Y., Solutions of ill-posed problems, Winston, 1977.
15. Fasino D. and Inglese G., An inverse Robin problem for Laplace's equation: theoretical results and numerical methods, Inverse problems, 15 (1999), p. 41.
16. Fok V.A. and Kuni F.M., "On the introduction of a 'suppressing' function in dispersion relations,"Dokl. Akad. Nauk SSSR, 127, No. 6, (1959), p. 1195-1198.
17. Xiaoze Hu, Lin Mu,and Xin Ye A Simple nite element method of the Cauchy problem for Poisson equation , International journal of Numerical analysis and modeling, 14:4-5 (2017), p.591603
18. Yarmukhamedov Sh., A Carleman function and the Cauchy problem for the Laplace equation. Sibir.Math.Zhur. 45, No.3,(2004), p.702-719
2. Belgacem F. B., Why is the Cauchy problem severely ill-posed?, Inverse Problems, 23 (2007), p. 823.
3. Blum J., Numerical simulation and optimal control in plasma physics, New York, NY; John Wiley and Sons Inc., 1989.
4. David Gilarg Neil S.Trudinger Elliptik Partial Differential Equations of Second Order Springer – Verlag BerlinHeidelberg New York Tokyo 1983
5. Goluzin G. M. and Krylov V.I., "A generalized Carleman fornula and its application to analytic continuation of functions,"Mat.Sb., 40, No.2, (1933), p.144-149 .
6. Dzhrbashyan M.M., Integral Transformations and Representations of Functions in a Complex Domain [in Russian], Nauka, Moskow (1966).
7. Inglese G., An inverse problem in corrosion detection, Inverse problems, 13 (1997), p. 977.
8. Latt es R. and Lions J., The method of quasi-reversibility: applications to partial differential equations, no. 18 in Translated from the French edition and edited by Richard Bellman. Modern Analytic and Computational Methods in Science and Mathematics, American Elsevier Publishing Co., Inc., New York½ 1969.
9. Lavrent'ev M.M., On Some Ill-Posed Problems of Mathematical Physics [in Russian], Sibirsk. Otdel. Akad. Nauk SSSR. Novosibirsk (1962).
10. Mergelyan S.N. "Harmonic approximation and approximate solution of the Cauchy problem for the Laplace equation."Uspekhi Mat. Nauk, 11.No.5. (1956), p.3-26 .
11. Hadamard J. The Cauchy Problem for Linear Partial Dierential Equations of Huperbolic Type [Russian translation], Mir, Moscow (1978), 352 p.
12. Carleman T., Les founctions quasi analitiques, Gauthier-Villars,Paris, 1926.
13. Colli-Franzone P., Guerri L., Tentoni L., Viganotti L., Baru S., Spaggiari S., and Taccardi B., A mathematical procedure for solving the inverse potential problem of electrocardiography. Analysis of the time-space accuracy from in vitro experimental data, Mathematical Biosciences, 77 (1985), pp. 353396.
14. Tikhonov A. N. and Arsenin V. Y., Solutions of ill-posed problems, Winston, 1977.
15. Fasino D. and Inglese G., An inverse Robin problem for Laplace's equation: theoretical results and numerical methods, Inverse problems, 15 (1999), p. 41.
16. Fok V.A. and Kuni F.M., "On the introduction of a 'suppressing' function in dispersion relations,"Dokl. Akad. Nauk SSSR, 127, No. 6, (1959), p. 1195-1198.
17. Xiaoze Hu, Lin Mu,and Xin Ye A Simple nite element method of the Cauchy problem for Poisson equation , International journal of Numerical analysis and modeling, 14:4-5 (2017), p.591603
18. Yarmukhamedov Sh., A Carleman function and the Cauchy problem for the Laplace equation. Sibir.Math.Zhur. 45, No.3,(2004), p.702-719
Published
2021-07-31
How to Cite
[1]
Ermamatova, Z.E. 2021. Carleman’s Formula of a Solution of the Poisson Equation. International Journal on Integrated Education. 4, 7 (Jul. 2021), 112-117. DOI:https://doi.org/10.17605/ijie.v4i7.2071.
Section
Articles
Copyright (c) 2021 Zuxro E. Ermamatova
This work is licensed under a Creative Commons Attribution 4.0 International License.
In submitting the manuscript to the International Journal on Integrated Education (IJIE), the authors certify that:
- They are authorized by their co-authors to enter into these arrangements.
- The work described has not been formally published before, except in the form of an abstract or as part of a published lecture, review, thesis, or overlay journal.
- That it is not under consideration for publication elsewhere,
- The publication has been approved by the author(s) and by responsible authorities – tacitly or explicitly – of the institutes where the work has been carried out.
- They secure the right to reproduce any material that has already been published or copyrighted elsewhere.
- They agree to the following license and copyright agreement.
License and Copyright Agreement
Authors who publish with International Journal on Integrated Education (IJIE) agree to the following terms:
- Authors retain copyright and grant the International Journal on Integrated Education (IJIE) right of first publication with the work simultaneously licensed under Creative Commons Attribution License (CC BY 4.0) that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
- Authors can enter into separate, additional contractual arrangements for the non-exclusive distribution of the International Journal on Integrated Education (IJIE) published version of the work (e.g., post it to an institutional repository or edit it in a book), with an acknowledgment of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) before and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
