# Fundamental solution for the Helmholtz equation

## Main Article Content

## Abstract

This paper deals with the construction of a family of fundamental solutions of the Helmholtz equation, parameterized by an entire function with certain properties. Functions that possess these properties are called the Carleman function. On the basis of the proved lemmas for the Helmholtz equation in two-dimensional and three-dimensional bounded domains, in what follows we will find a regularized solution of the Cauchy problem already for a multidimensional domain.

## Article Details

*Engineering Applications*,

*2*(2), 164–175. Retrieved from https://publish.mersin.edu.tr/index.php/enap/article/view/965

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

## References

Carleman T. (1926). Les fonctions quasi analytiques. Gautier-Villars et Cie., Paris.

Yarmukhamedov, Sh. (1977). On the Cauchy problem for the Laplace equation”, Dokl. AN SSSR, 235:2, 281-283.

Lavrent’ev, M. M. (1957). On the Cauchy problem for second-order linear elliptic equations. Reports of the USSR Academy of Sciences. 112(2), 195-197.

Lavrent’ev, M. M. (1962). On some ill-posed problems of mathematical physics. Nauka, Novosibirsk.

Tikhonov A. N. & Arsenin V. Ya. (1974). Methods for solving ill-posed problems. Nauka, Moscow.

Tikhonov, A. N. & Arsenin, V. Y. (1977). Solutions of ill-posed problems. New York: Winston.

Lavrent’ev M. M., Romanov V. G., & Shishatsky S. P. (1980). Ill-posed problems of mathematical physics and analysis. Nauka, Moscow.

Hadamard, J. (1978). The Cauchy problem for linear partial differential equations of hyperbolic type. Nauka, Moscow.

Yarmukhamedov, Sh. (1985). Green's formula in an infinite region and its application. Dokl. AN SSSR, 305-308.

Aizenberg, L. A. (1990). Carleman’s formulas in complex analysis. Nauka, Novosibirsk.

Shlapunov, A. A. E. (1992). The Cauchy problem for Laplace's equation. Siberian Mathematical Journal, 33(3), 534-542.

Tarkhanov, N. N. (1985). Stability of the solutions of elliptic systems. Functional Analysis and Its Applications, 19(3), 245-247.

Tarkhanov, N. N. (1995). The Cauchy problem for solutions of elliptic equations. V. 7, Akad. Verl., Berlin.

Arbuzov, E. V., & Bukhgeim, A. L. V. (2006). The Carleman formula for the Helmholtz equation on the plane. Siberian Mathematical Journal, 47(3), 425-432.

Ikehata, M. (2001). Inverse conductivity problem in the infinite slab. Inverse Problems, 17, 437-454.

Ikehata, M. (2007). Probe method and a Carleman function. Inverse Problems, 23, 1871–1894.

Niyozov, I.E. (2015). On the continuation of the solution of systems of equations of the theory of elasticity. Uzb. Math J. (3), 95-105.

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). 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.

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., & 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. (2023). Fundamental solution for the Helmholtz equation in the plane. Advanced Engineering Days (AED), 6, 179-182.