On the Cauchy problem for the Helmholtz equation
Keywords:
Integral formula, Matrix factorization, Helmholtz equation, Bounded domain, Cauchy problemAbstract
The work is devoted to the study of continuation and evaluation of the stability of the solution of the Cauchy problem for the Laplace equation in a domain from its known values on the smooth part of the boundary of the domain. The problem under consideration belongs to the problems of mathematical physics, in which there is no continuous dependence of solutions on the initial data. When solving applied problems, it is necessary to find not only an approximate solution, but also a derivative of the approximate solution.
References
Tikhonov, A. N., & Arsenin V. Y. (1974). Methods for solving ill-posed problems. Nauka, Moscow.
Carleman, T. (1926). Les fonctions quasi analytiques. Gautier-Villars et Cie., Paris.
Goluzin, G. M., & Krylov, V. M. (1933) The generalized Carleman formula and its application to the analytic continuation of functions. Sbornik: Mathematics, 40 (2), 144-149.
Yarmukhamedov, Sh. (1997). On the extension of the solution of the Helmholtz equation. Reports of the Russian Academy of Sciences, 357(3), 320-323.
Tarkhanov, N. N. (1995). The Cauchy problem for solutions of elliptic equations. V. 7, Akad. Verl., Berlin.
Juraev, D. A. (2014). The construction of the fundamental solution of the Helmholtz equation. Reports of the Academy of Sciences of the Republic of Uzbekistan, (4), 14-17.
Juraev, D. A. (2016). Regularization of the Cauchy problem for systems of elliptic type equations of first order. Uzbek Mathematical Journal, (2), 61-71.
Juraev, D. A., & Noeiaghdam, S. (2021). Regularization of the ill-posed Cauchy problem for matrix factorizations of the Helmholtz equation on the plane. Axioms, 10(2), 1-14.
Juraev D. A. (2021). 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.
Juraev, D. A., & Gasimov, Y. S. (2022). Regularization of the ill-posed Cauchy problem for matrix factorizations of the Helmholtz equation on the plane. Azerbaijan Journal of Mathematics, 12(1), 142-161.
Juraev, D. A., & Noeiaghdam, S. (2022). Modern problems of mathematical physics and their applications. Axioms, 11(2), 1-6.
Juraev, D.A., & Noeiaghdam, S. (2022). Modern Problems of Mathematical Physics and Their Applications. Axioms, MDPI. Switzerland. 1-352.
Juraev, D. A. (2022). 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.
Juraev, D. A. & Cavalcanti, M. M. (2023). Cauchy problem for matrix factorizations of the Helmholtz equation in the space Rm, Boletim da Sociedade Paranaense de Matematica, 41(3s.), 1–12.
Juraev, D. A. (2023) The Cauchy problem for matrix factorization of the Helmholtz equation in a multidimensional unbounded domain, Boletim da Sociedade Paranaense de Matematica, 41(3s, 1-18.
Juraev, D. A., Ibrahimov, V. & Agarwal, P. (2023) Regularization of the Cauchy problem for matrix factorizations of the Helmholtz equation on a two-dimensional bounded domain, Palestine Journal of Mathematics, 12(1), 381-403.
Juraev, D. A. (2023) Fundamental solution for the Helmholtz equation, Engineering Applications, 2(2), 164-175.
Juraev, D. A., Jalalov, M. J. & Ibrahimov, V. R. (2023) On the formulation of the Cauchy problem for matrix factorizations of the Helmholtz equation, Engineering Applications, 2(2), 176-189.
