Cauchy problem for matrix factorizations of the Helmholtz equation in the space R^m
DOI:
https://doi.org/10.5269/bspm.62831Resumen
In this paper, we consider the problem of recovering solutions for matrix factorizations of the Helmholtz equation in a three-dimensional bounded domain from their values on a part of the boundary of this domain, i.e., the Cauchy problem. An approximate solution to this problem is constructed based on the Carleman matrix method.
Referencias
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25. J. Bulnes, An unusual quantum entanglement consistent with Schrodinger's equation. Global and Stochastic Analysis, Global and Stochastic Analysis, 9(2), 1-9, (2022).
26. J. Bulnes, Solving the heat equation by solving an integro-differential equation, Global and Stochastic Analysis, 9(2), 1-9, (2022).
27. J. Hadamard, The Cauchy problem for linear partial differential equations of hyperbolic type, Nauka, Moscow, (1978).
28. K. Berdawood, A. Nachaoui, R. Saeed, M. Nachaoui, F. Aboud, An efficient D-N alternating algorithm for solving an inverse problem for Helmholtz equation, Discrete & Continuous Dynamical Systems-S, 14, 1-22, (2021). https://doi.org/10.3934/dcdss.2021013
29. L.A. Aizenberg, Carleman's formulas in complex analysis, Nauka, Novosibirsk, (1990).
30. M.M. Dzharbashyan, Integral Transformations and Representations of Functions in Complex Domain, Nauka, Moscow, 1926.
31. M.M. Lavrent'ev, On the Cauchy problem for second-order linear elliptic equations, Reports of the USSR Academy of Sciences, 112(2), 195-197, (1957).
32. M.M. Lavrent'ev, On some ill-posed problems of mathematical physics, Nauka, Novosibirsk, (1962).
33. N.H. Giang, T.-T. Nguyen, C.C. Tay, L.A. Phuong, T.-T. Dang, Towards predictive Vietnamese human resource migration by machine learning: a case study in northeast Asian countries, Axioms, 11 (4): 1-14, (2022). https://doi.org/10.3390/axioms11040151
34. N.N. Tarkhanov, On the Carleman matrix for elliptic systems, Reports of the USSR Academy of Sciences, 284(2), 294-297, (1985).
35. N.N. Tarkhanov, The Cauchy problem for solutions of elliptic equations, V. 7, Akad. Verl., Berlin, (1995).
36. P. Agarwal, A. C¸ etinkaya, Sh. Jain, I.O. Kiymaz, S-Generalized Mittag-Leffler function and its certain properties, Mathematical Sciences and Applications E-Notes, 7(2), 139-148, (2019). https://doi.org/10.36753/mathenot.578638
37. P.K. Kythe, Fundamental solutions for differential operators and applications, Birkhauser, Boston, (1996). https://doi.org/10.1007/978-1-4612-4106-5
38. Sh. Yarmukhamedov, On the Cauchy problem for the Laplace equation, Reports of the USSR Academy of Sciences, 235(2), 281-283, (1977).
39. Sh. Yarmukhamedov, On the extension of the solution of the Helmholtz equation, Reports of the Russian Academy of Sciences, 357(3), 320-323, (1997).
40. Sh. Yarmukhamedov, The Carleman function and the Cauchy problem for the Laplace equation, Siberian Mathematical Journal, 45(3), 702-719, (2004). https://doi.org/10.1023/B:SIMJ.0000028622.69605.c0
41. Sh. Yarmukhamedov, Representation of Harmonic Functions as Potentials and the Cauchy Problem, Math. Notes, 83(5), 763-778 (2008). https://doi.org/10.1134/S0001434608050131
42. T. Carleman, Les fonctions quasi analytiques, Gautier-Villars et Cie., Paris, (1926).
43. V.K. Ivanov, About incorrectly posed tasks, Math. Collect., 61, 211-223, (1963).
44. V.R. Ibrahimov, G.Yu. Mehdiyeva, M.N. Imanova, On the computation of double integrals by using some connection between the wave equation and the system of ODE, The Second Edition of the International Conference on Innovative Applied Energy (IAPE'20), 1-8, (2020).
45. V.R. Ibrahimov, G.Yu. Mehdiyeva, X.G. Yue, M.K.A. Kaabar, S. Noeiaghdam, D.A. Juraev, Novel symmetric numerical methods for solving symmetric mathematical problems, International Journal of Circuits, Systems and Signal Processing, 15, 1545-1557, (2021). https://doi.org/10.46300/9106.2021.15.167
46. Yu. Fayziev, Q. Buvaev, D. Juraev, N. Nuralieva, Sh. Sadullaeva, The inverse problem for determining the source function in the equation with the Riemann-Liouville fractional derivative, Global and Stochastic Analysis, 9(2), 1-10, 2022).
2. A. Shokri, M.M. Khalsaraei, S. Noeiaghdam, D.A. Juraev, A new divided difference interpolation method for twovariable functions, Global and Stochastic Analysis, 9(2), 1-8, (2022).
3. A.T. Ramazanova, On determining initial conditions of equations flexural-torsional vibrations of a bar, European Journal of Pure and Applied Mathematics, 12(1), 25-38, (2019). https://doi.org/10.29020/nybg.ejpam.v12i1.3350
4. A.T. Ramazanova, Necessary conditions for the existence of a saddle point in one optimal control problem for systems of hyperbolic equations, European Journal of Pure and Applied Mathematics, 14(4), 1402-1414, (2021). https://doi.org/10.29020/nybg.ejpam.v14i4.4135
5. B.C. Corcino, R.B. Corcino RB, B.A.A. Damgo, J.A.A. CaËœnete, Integral representation and explicit formula at rational arguments for Apostol - Tangent polynomials, Symmetry, 14(1), 1-10, (2022). https://doi.org/10.3390/sym14010053
6. D.A. Juraev, Regularization of the Cauchy problem for systems of elliptic type equations of first order, Uzbek Mathematical Journal, 2, 61-71, (2016).
7. D.A. Juraev, The Cauchy problem for matrix factorizations of the Helmholtz equation in an unbounded domain, Siberian Electronic Mathematical Reports, 14, 752-764, (2017).
8. D.A. Juraev, Cauchy problem for matrix factorizations of the Helmholtz equation, Ukrainian Mathematical Journal, 69(10), 1364-1371, (2017). https://doi.org/10.1007/s11253-018-1456-5
9. D.A. Juraev, On the Cauchy problem for matrix factorizations of the Helmholtz equation in a bounded domain, Siberian Electronic Mathematical Reports, 15, 11-20, (2018). https://doi.org/10.33048/semi.2018.15.151
10. D.A. Zhuraev, Cauchy problem for matrix factorizations of the Helmholtz equation, Ukrainian Mathematical Journal, 69(10), 1583-1592, (2018). https://doi.org/10.1007/s11253-018-1456-5
11. D.A. Juraev, The Cauchy problem for matrix factorizations of the Helmholtz equation in R 3, Journal of Universal Mathematics, 1(3), 312-319, (2018).
12. D.A. Juraev, On the Cauchy problem for matrix factorizations of the Helmholtz equation in an unbounded domain in R2, Siberian Electronic Mathematical Reports, 15, 1865-1877, (2018). https://doi.org/10.33048/semi.2018.15.151
13. D.A. Juraev, On a regularized solution of the Cauchy problem for matrix factorizations of the Helmholtz equation, Advanced Mathematical Models & Applications, 4(1), 86-96, (2019).
14. D.A. Juraev, On the Cauchy problem for matrix factorizations of the Helmholtz equation, Journal of Universal Mathematics, 2(2), 113-126, (2019). https://doi.org/10.33773/jum.543320
15. D.A. Juraev, The solution of the ill-posed Cauchy problem for matrix factorizations of the Helmholtz equation, Advanced Mathematical Models & Applications, 5(2), 205-221, (2020). https://doi.org/10.3390/axioms10020082
16. D.A. Juraev, S. Noeiaghdam, Regularization of the ill-posed Cauchy problem for matrix factorizations of the Helmholtz equation on the plane, Axioms, 10(2), 1-14, (2021). https://doi.org/10.3390/axioms10020082
17. D.A. Juraev, Solution of the ill-posed Cauchy problem for matrix factorizations of the Helmholtz equation on the plane, Global and Stochastic Analysis, 8(3), 1-17, (2021).
18. D.A. Juraev, S. Noeiaghdam, Modern problems of mathematical physics and their applications, Axioms, 11(2), 1-6, (2022). https://doi.org/10.3390/axioms11020045
19. D.A. Juraev, Y.S. Gasimov, On the regularization Cauchy problem for matrix factorizations of the Helmholtz equation in a multidimensional bounded domain, Azerbaijan Journal of Mathematics, 12(1), 142-161, (2022).
20. D.A. Juraev, S. Noeiaghdam, Modern problems of mathematical physics and their applications, Axioms, MDPI, Basel, Switzerland, (2022). https://doi.org/10.3390/axioms11020045
21. D.A. Juraev, On the solution of the Cauchy problem for matrix factorizations of the Helmholtz equation in a multidimensional spatial domain, Global and Stochastic Analysis, 9(2), 1-17, (2022).
22. E.V. Arbuzov, A.L. Bukhgeim, The Carleman formula for the Helmholtz equation, Siberian Mathematical Journal, 47(3), 518-526, (2006). https://doi.org/10.1007/s11202-006-0055-0
23. I.E. Niyozov, Regularization of a nonstandard Cauchy problem for a dynamic Lame system, Izvestiya Vysshikh Uchebnykh Zavedenii, 4, 54-63, (2020) https://doi.org/10.26907/0021-3446-2020-4-54-63
24. I.E. Niyozov, The Cauchy problem of couple-stress elasticity in R3, Global and Stochastic Analysis, 9(2), 1-16, (2022).
25. J. Bulnes, An unusual quantum entanglement consistent with Schrodinger's equation. Global and Stochastic Analysis, Global and Stochastic Analysis, 9(2), 1-9, (2022).
26. J. Bulnes, Solving the heat equation by solving an integro-differential equation, Global and Stochastic Analysis, 9(2), 1-9, (2022).
27. J. Hadamard, The Cauchy problem for linear partial differential equations of hyperbolic type, Nauka, Moscow, (1978).
28. K. Berdawood, A. Nachaoui, R. Saeed, M. Nachaoui, F. Aboud, An efficient D-N alternating algorithm for solving an inverse problem for Helmholtz equation, Discrete & Continuous Dynamical Systems-S, 14, 1-22, (2021). https://doi.org/10.3934/dcdss.2021013
29. L.A. Aizenberg, Carleman's formulas in complex analysis, Nauka, Novosibirsk, (1990).
30. M.M. Dzharbashyan, Integral Transformations and Representations of Functions in Complex Domain, Nauka, Moscow, 1926.
31. M.M. Lavrent'ev, On the Cauchy problem for second-order linear elliptic equations, Reports of the USSR Academy of Sciences, 112(2), 195-197, (1957).
32. M.M. Lavrent'ev, On some ill-posed problems of mathematical physics, Nauka, Novosibirsk, (1962).
33. N.H. Giang, T.-T. Nguyen, C.C. Tay, L.A. Phuong, T.-T. Dang, Towards predictive Vietnamese human resource migration by machine learning: a case study in northeast Asian countries, Axioms, 11 (4): 1-14, (2022). https://doi.org/10.3390/axioms11040151
34. N.N. Tarkhanov, On the Carleman matrix for elliptic systems, Reports of the USSR Academy of Sciences, 284(2), 294-297, (1985).
35. N.N. Tarkhanov, The Cauchy problem for solutions of elliptic equations, V. 7, Akad. Verl., Berlin, (1995).
36. P. Agarwal, A. C¸ etinkaya, Sh. Jain, I.O. Kiymaz, S-Generalized Mittag-Leffler function and its certain properties, Mathematical Sciences and Applications E-Notes, 7(2), 139-148, (2019). https://doi.org/10.36753/mathenot.578638
37. P.K. Kythe, Fundamental solutions for differential operators and applications, Birkhauser, Boston, (1996). https://doi.org/10.1007/978-1-4612-4106-5
38. Sh. Yarmukhamedov, On the Cauchy problem for the Laplace equation, Reports of the USSR Academy of Sciences, 235(2), 281-283, (1977).
39. Sh. Yarmukhamedov, On the extension of the solution of the Helmholtz equation, Reports of the Russian Academy of Sciences, 357(3), 320-323, (1997).
40. Sh. Yarmukhamedov, The Carleman function and the Cauchy problem for the Laplace equation, Siberian Mathematical Journal, 45(3), 702-719, (2004). https://doi.org/10.1023/B:SIMJ.0000028622.69605.c0
41. Sh. Yarmukhamedov, Representation of Harmonic Functions as Potentials and the Cauchy Problem, Math. Notes, 83(5), 763-778 (2008). https://doi.org/10.1134/S0001434608050131
42. T. Carleman, Les fonctions quasi analytiques, Gautier-Villars et Cie., Paris, (1926).
43. V.K. Ivanov, About incorrectly posed tasks, Math. Collect., 61, 211-223, (1963).
44. V.R. Ibrahimov, G.Yu. Mehdiyeva, M.N. Imanova, On the computation of double integrals by using some connection between the wave equation and the system of ODE, The Second Edition of the International Conference on Innovative Applied Energy (IAPE'20), 1-8, (2020).
45. V.R. Ibrahimov, G.Yu. Mehdiyeva, X.G. Yue, M.K.A. Kaabar, S. Noeiaghdam, D.A. Juraev, Novel symmetric numerical methods for solving symmetric mathematical problems, International Journal of Circuits, Systems and Signal Processing, 15, 1545-1557, (2021). https://doi.org/10.46300/9106.2021.15.167
46. Yu. Fayziev, Q. Buvaev, D. Juraev, N. Nuralieva, Sh. Sadullaeva, The inverse problem for determining the source function in the equation with the Riemann-Liouville fractional derivative, Global and Stochastic Analysis, 9(2), 1-10, 2022).
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2022-12-27
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