This article examines inverse retrospective problems, which have important practical applications in various fields where it is necessary to determine the initial state or previous state of dynamic objects based on the available information about the characteristics of the field at the current time. Particular attention is paid to inverse and ill-posed problems for parabolic equations, which are among the most complex and practically significant classes of inverse problems. It is noted that retrospective inverse problems for parabolic equations have various formulations based on the analysis of initial-boundary problems for the equation.

inverse retrospective problems, parabolic equations, ill-posed problems, hyperbolic equations, retrospective inverse problems, initial-boundary value problems, heat equation, temperature measurement, field, Fredholm integral equation of the first kind, complete orthogonal system, Weierstrass theorem.

Vatulyan A. O. Inverse problems in the mechanics of a deformable solid. – M.: Fizmatlit, 2007. – 224 p.

Denisov A. M.Introduction to the theory of inverse problems. – M.: MSU, 1994. – 208 p.

Romanov V. G.Inverse problems of mathematical physics. – M.: Nauka, 1984. – 264 p.

Kabanikhin S. I.Projection-difference methods for determining the coefficients of hyperbolic equations. – Novosibirsk: Nauka, 1988. – 168 p.

Vatulyan A. O. Dragilev V. M., Dragileva L. L.Restoration of dynamic stresses acting on a viscoelastic layer // Akust. magazine 2005. T. 51. No. 6. – P. 742–748.

Lavrentyev M.M. Ill-posed problems for differential equations. tutorial. - NSU, 1981. - 75 p.

Karimov Sh.T., Yulbarsov H. Goursat problem for one third-order pseudoparabolic equation with the Bessel operator. Materials of the IX international scientific conference “Modern problems of mathematics and physics” dedicated to the 70th anniversary of corresponding member. Academy of Sciences of the Republic of Belarus K.B. Sabitova, September 12 - 15, Sterlitamak, 2021

Karimov Sh.T., Mamadalieva Sh. Solution of the coefficient inverse problem for a hyperbolic equation by reducing it to the Gelfand-Levitan equation of the first kind. Fars Int J Edu Soc Sci Hum 10(12), 2022; Volume-10, Issue-12, pp. 142-151.

Karimov Sh.T. On some generalizations of the properties of the Erdelyi-Kober operator and their application. Vestnik KRAUNTS. Phys.-math. Sciences. 2017. No. 2(18). C. 20-40. DOI: 10.18454/2079-6641-2017-18-2-20-40.

Karimov Sh.T. New properties of the generalized Erdelyi–Kober operator and their applications. Reports of the Academy of Sciences of the Republic of Uzbekistan. – 2014. -No. 5 -S. 11-13

Lavrentyev M. M., Romanov V. G., Shishatsky S. P. Ill-posed problems of mathematical physics and analysis. – M: Nauka, 1980. – 286 s.

Tikhonov A. N., Arsenin V. Ya. Methods for solving ill-posed problems. – M.: Nauka, 1986. – 287 p.

Vatulyan A. O. Inverse problems in the mechanics of a deformable solid. – M.: Fizmatlit, 2007. – 224 p.

Denisov A. M. Introduction to the theory of inverse problems. – M.: MSU, 1994. – 208 p.

Lattes R., Lyons J.-L. Quasi-inversion method and its applications. – M.: Mir, 1970. – 336 p.