{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:59:11Z","timestamp":1760234351517,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,5,2]],"date-time":"2021-05-02T00:00:00Z","timestamp":1619913600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we present an explicit formula for the approximate solution of the Cauchy problem for the matrix factorizations of the Helmholtz equation in a bounded domain on the plane. Our formula for an approximate solution also includes the construction of a family of fundamental solutions for the Helmholtz operator on the plane. This family is parameterized by function K(w) which depends on the space dimension. In this paper, based on the results of previous works, the better results can be obtained by choosing the function K(w).<\/jats:p>","DOI":"10.3390\/axioms10020082","type":"journal-article","created":{"date-parts":[[2021,5,5]],"date-time":"2021-05-05T11:06:01Z","timestamp":1620212761000},"page":"82","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Regularization of the Ill-Posed Cauchy Problem for Matrix Factorizations of the Helmholtz Equation on the Plane"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1224-6764","authenticated-orcid":false,"given":"Davron Aslonqulovich","family":"Juraev","sequence":"first","affiliation":[{"name":"Higher Military Aviation School of the Republic of Uzbekistan, Karshi City 180100, Uzbekistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2307-0891","authenticated-orcid":false,"given":"Samad","family":"Noeiaghdam","sequence":"additional","affiliation":[{"name":"Industrial Mathematics Laboratory, Baikal School of BRICS, Irkutsk National Research Technical University, 664074 Irkutsk, Russia"},{"name":"Department of Applied Mathematics and Programming, South Ural State University, Lenin Prospect 76, 454080 Chelyabinsk, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,2]]},"reference":[{"key":"ref_1","first-page":"501","article-title":"On the solution of ill-posed problems and the method of regularization","volume":"151","author":"Tikhonov","year":"1963","journal-title":"Rep. USSR Acad. Sci."},{"key":"ref_2","first-page":"211","article-title":"About incorrectly posed tasks","volume":"61","author":"Ivanov","year":"1963","journal-title":"Math. Collect."},{"key":"ref_3","first-page":"195","article-title":"On the Cauchy problem for second-order linear elliptic equations","volume":"112","year":"1957","journal-title":"Rep. USSR Acad. Sci."},{"key":"ref_4","unstructured":"Lavrent\u2019ev, M.M. (1962). On Some Ill-Posed Problems of Mathematical Physics, Nauka."},{"key":"ref_5","unstructured":"Tarkhanov, N.N. (1995). The Cauchy Problem for Solutions of Elliptic Equations, Akad. Verl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1016\/j.asej.2015.08.018","article-title":"Fibonacci-regularization method for solving Cauchy integral equations of the first kind","volume":"8","author":"Araghi","year":"2017","journal-title":"Ain Shams Eng. J."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"105","DOI":"10.18576\/amis\/140114","article-title":"A novel approach to find optimal parameter in the homotopy-regularization method for solving integral equations","volume":"14","author":"Noeiaghdam","year":"2020","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_8","first-page":"1","article-title":"Homotopy regularization method to solve the singular Volterra integral equations of the first kind","volume":"11","author":"Noeiaghdam","year":"2018","journal-title":"Jordan J. Math. Stat."},{"key":"ref_9","unstructured":"Hadamard, J. (1978). The Cauchy Problem for Linear Partial Differential Equations of Hyperbolic Type, Nauka."},{"key":"ref_10","unstructured":"Bers, A., John, F., and Shekhter, M. (1966). Partial Differential Equations, Mir."},{"key":"ref_11","unstructured":"Aizenberg, L.A. (1990). Carleman\u2019s Formulas in Complex Analysis, Nauka."},{"key":"ref_12","unstructured":"Aleksidze, M.A. (1991). Fundamental Functions in Approximate Solutions of Boundary Value Problems, Nauka."},{"key":"ref_13","unstructured":"Carleman, T. (1926). Les Fonctions Quasi Analytiques, Gautier-Villars et Cie."},{"key":"ref_14","first-page":"531","article-title":"The solvability criterion for an ill-posed problem for elliptic systems","volume":"380","author":"Tarkhanov","year":"1989","journal-title":"Rep. USSR Acad. Sci."},{"key":"ref_15","first-page":"144","article-title":"The generalized Carleman formula and its application to the analytic continuation of functions","volume":"40","author":"Goluzin","year":"1993","journal-title":"Sb. Math."},{"key":"ref_16","first-page":"281","article-title":"On the Cauchy problem for the Laplace equation","volume":"235","author":"Yarmukhamedov","year":"1977","journal-title":"Rep. USSR Acad. Sci."},{"key":"ref_17","first-page":"320","article-title":"On the extension of the solution of the Helmholtz equation","volume":"357","author":"Yarmukhamedov","year":"1997","journal-title":"Rep. Russ. Acad. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"702","DOI":"10.1023\/B:SIMJ.0000028622.69605.c0","article-title":"The Carleman function and the Cauchy problem for the Laplace equation","volume":"45","author":"Yarmukhamedov","year":"2004","journal-title":"Sib. Math. J."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1134\/S0001434608050131","article-title":"Representation of Harmonic Functions as Potentials and the Cauchy Problem","volume":"83","author":"Yarmukhamedov","year":"2008","journal-title":"Math. Notes"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1088\/0266-5611\/17\/3\/305","article-title":"Inverse conductivity problem in the infinite slab","volume":"17","author":"Ikehata","year":"2001","journal-title":"Inverse Probl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1871","DOI":"10.1088\/0266-5611\/23\/5\/006","article-title":"Probe method and a Carleman function","volume":"23","author":"Ikehata","year":"2007","journal-title":"Inverse Probl."},{"key":"ref_22","first-page":"518","article-title":"The Carleman formula for the Helmholtz equation","volume":"47","author":"Arbuzov","year":"1991","journal-title":"Sib. Math. J."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1007\/BF01076639","article-title":"Stability of the solutions of elliptic systems","volume":"19","author":"Tarkhanov","year":"1985","journal-title":"Funct. Anal. Appl."},{"key":"ref_24","first-page":"294","article-title":"On the Carleman matrix for elliptic systems","volume":"284","author":"Tarkhanov","year":"1985","journal-title":"Rep. USSR Acad. Sci."},{"key":"ref_25","first-page":"953","article-title":"On the Cauchy problem for the Laplace equation","volume":"43","author":"Shlapunov","year":"1992","journal-title":"Sib. Math. J."},{"key":"ref_26","first-page":"217","article-title":"Boundary problems for Helmholtz equation and the Cauchy problem for Dirac operators","volume":"4","author":"Shlapunov","year":"2011","journal-title":"J. Sib. Fed. Univ. Math. Phys."},{"key":"ref_27","first-page":"95","article-title":"On the continuation of the solution of systems of equations of the theory of elasticity","volume":"3","author":"Niyozov","year":"2015","journal-title":"Uzb. Math. J."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"54","DOI":"10.26907\/0021-3446-2020-4-54-63","article-title":"Regularization of a nonstandard Cauchy problem for a dynamic Lame system","volume":"64","author":"Niyozov","year":"2020","journal-title":"Izv. Vyss. Uchebnykh Zaved."},{"key":"ref_29","first-page":"14","article-title":"The construction of the fundamental solution of the Helmholtz equation","volume":"2","author":"Juraev","year":"2012","journal-title":"Rep. Acad. Sci. Repub. Uzb."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Juraev, D.A. (2014). Regularization of the Cauchy Problem for Systems of Equations of Elliptic Type, LAP Lambert Academic Publishing.","DOI":"10.12737\/6323"},{"key":"ref_31","first-page":"61","article-title":"Regularization of the Cauchy problem for systems of elliptic type equations of first order","volume":"2","author":"Juraev","year":"2016","journal-title":"Uzb. Math. J."},{"key":"ref_32","first-page":"752","article-title":"The Cauchy problem for matrix factorizations of the Helmholtz equation in an unbounded domain","volume":"14","author":"Juraev","year":"2017","journal-title":"Sib. Electron. Math. Rep."},{"key":"ref_33","first-page":"1364","article-title":"Cauchy problem for matrix factorizations of the Helmholtz equation","volume":"69","author":"Juraev","year":"2017","journal-title":"Ukr. J."},{"key":"ref_34","first-page":"11","article-title":"On the Cauchy problem for matrix factorizations of the Helmholtz equation in a bounded domain","volume":"15","author":"Juraev","year":"2018","journal-title":"Sib. Electron. Math. Rep."},{"key":"ref_35","first-page":"1583","article-title":"Cauchy problem for matrix factorizations of the Helmholtz equation","volume":"69","author":"Zhuraev","year":"2018","journal-title":"Ukr. J."},{"key":"ref_36","first-page":"312","article-title":"The Cauchy problem for matrix factorizations of the Helmholtz equation in R3","volume":"1","author":"Juraev","year":"2018","journal-title":"J. Univers. Math."},{"key":"ref_37","first-page":"1865","article-title":"On the Cauchy problem for matrix factorizations of the Helmholtz equation in an unbounded domain in R2","volume":"15","author":"Juraev","year":"2018","journal-title":"Sib. Electron. Math. Rep."},{"key":"ref_38","first-page":"86","article-title":"On a regularized solution of the Cauchy problem for matrix factorizations of the Helmholtz equation","volume":"4","author":"Juraev","year":"2019","journal-title":"Adv. Math. Model. Appl."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"113","DOI":"10.33773\/jum.543320","article-title":"On the Cauchy problem for matrix factorizations of the Helmholtz equation","volume":"2","author":"Juraev","year":"2019","journal-title":"J. Univers. Math."},{"key":"ref_40","first-page":"205","article-title":"The solution of the ill-posed Cauchy problem for matrix factorizations of the Helmholtz equation","volume":"5","author":"Juraev","year":"2020","journal-title":"Adv. Math. Model. Appl."},{"key":"ref_41","unstructured":"Juraev, D.A. (2020). Ill-posed problems for first-order elliptic systems with constant coefficients. Abstracts of Communications of the Conference: Modern Stochastic Models and Problems of Actuarial Mathematics, Organizer\u2013Karshi State University."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"871","DOI":"10.1515\/jip-2013-0041","article-title":"Regularization of the continuation problem for elliptic equation","volume":"21","author":"Kabanikhin","year":"2013","journal-title":"J. Inverse Ill Posed Probl."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/2\/82\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:56:48Z","timestamp":1760162208000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/2\/82"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,2]]},"references-count":42,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,6]]}},"alternative-id":["axioms10020082"],"URL":"https:\/\/doi.org\/10.3390\/axioms10020082","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2021,5,2]]}}}