Uniqueness of solution of the inverse problem with overdetermination of the raised type for an abstract second-order differential equation

Almohamed M.

The Bulletin of Irkutsk State University. Series Mathematics, Tome 50 (2024), pp. 5-18

Résumé

We consider a linear inverse problem for a second-order abstract differential equation in a Banach space. The inhomogeneous term of the equation does not depend on time and is unknown. At the initial moment of time, the standard Cauchy conditions are given. An additional condition is specified at the final moment of time. This is a value of the second derivative of the main evolutionary function. For the studied problem, a uniqueness criterion of a solution is established. It is expressed in spectral terms. A simple sufficient condition for the solution uniqueness is noted. An example of the inverse problem for Poisson's equation in a cylindrical domain is considered.

Détail

Keywords: abstract differential equation of the second order, linear inverse problem, uniqueness criterion of solution, elementary solutions of inverse problem.

Bibliographie

[1] Almohamed M., “Uniqueness Criterion for Linear Inverse Problem with the Final Overdetermination of the Second Type”, Proc. Voronezh State University, Series: Physics. Mathematics, 2019, no. 3, 50–58 (in Russian)

[2] Almohamed M., “Reconstruction of the Inhomogeneous Term for Poisson's Equation with Special Boundary Conditions”, Sovremennyye Problemy Teorii Funktsiy i ikh Prilozheniya, Saratov, 2020, 30–32 (in Russian)

[3] Almohamed M., “On a Specific Inverse Problem for an Abstract Differential Equation of the Second Order”, Sovremennyye Problemy matematiki i Matematicheskogo Obrazovaniya, Saint Petersburg, 2024, 204–208 (in Russian)

[4] Amirov A. Kh., “Solvability of the Inverse Problems for Second-Order Equations”, Funct. Anal. Appl., 20:3 (1986), 236–237 | DOI | MR | Zbl

[5] Venttsel A. D., “On Boundary Conditions for Multidimensional Diffusion Processes”, Theory of Probability and its Applications, 4:2 (1959), 164–177 | DOI

[6] Denisov A. M., Elements of the Theory of Inverse Problems, Inverse and Ill-Posed Problems Series, 14, De Gruyter, 1999, iv+272 pp. | DOI | MR

[7] Iskenderov A. D., “Some Inverse Problems on Determining the Right-Hand Sides of Differential Equations”, Izvestia Academy of Sciences, Azerbaidzhan SSR, Series of Physical Technical and Mathematical Sciences, 1976, no. 2, 58–63 (in Russian) | Zbl

[8] Iskenderov A. D., Tagiev R. G., “An Inverse Problem of Determining of the Right-Hand Sides of Evolution Equations in a Banach Space”, Questions of Applied Mathematics and Cybernetics. Scientific works of the Azerbaijan University, 1979, no. 1, 51–56 (in Russian) | Zbl

[9] Ladyzhenskaya O. A., Ural'tseva N. N., Lineynyye i Kvazilineynyye Uravneniya Ellipticheskogo Tipa, 2-nd edition, Nauka Publ., M., 1973, 576 pp. (in Russian) | MR

[10] Nazarov A. I., Zadacha Venttselya i Yeye Obobshcheniya, Dr. sci. diss. abstr., Saint Petersburg, 2004 pp. (in Russian)

[11] Orlovskii D. G., “On a Problem of Determining the Parameter of an Evolution Equation”, Differ. Uravn., 26:9 (1990), 1614–1621 (in Russian) | MR

[12] Prilepko A. I., “Izbrannyye Voprosy v Obratnykh Zadachakh Matematicheskoy Fiziki”, Uslovno-Korrektnyye Zadachi Matematicheskoy Fiziki i Analiza, Novosibirsk, 1992, 151–162 (in Russian)

[13] Soloviev V. V., “Desidability of Inverse Problems for the Elliptic Equation in the Cylinder”, Bulletin of Moscow State Regional University, Series: Physics and Mathematics, 2012, no. 1, 27–38 (in Russian)

[14] Tikhonov A. N., Samarsky A. A., Uravneniya Matematicheskoy Fiziki, 7-th ed., MSU Publ., M.; Nauka Publ., 2004, 798 pp. (in Russian) | MR

[15] Tikhonov I. V., Almohamed M., “Generalized Exponents and Their Applications to the Theory of Differential Equations”, Computer Mathematics Systems and Their Applications, 21, Smolensk State University Publ., Smolensk, 2020, 345–353 (in Russian)

[16] Tikhonov I. V., Almohamed M., “Inverse Problem with Overdetermination of the Third Kind for an Abstract Second-Order Differential Equation”, Differ. Equ., 58:7 (2022), 877–898 | DOI | DOI | MR | Zbl

[17] Tikhonov I. V., Eidelman Yu. S., “An Inverse Problem for a Differential Equation in Banach Space and the Distribution of Zeros of an Entire Mittag-Leffler Function”, Differ. Equ., 38:5 (2002), 669–677 | DOI | MR | Zbl

[18] Trenogin V. A., Functional Analysis, 3-rd ed., M., 2002, 488 pp. (in Russian) | MR

[19] Eidelman Yu. S., “Uniqueness of the Solution of the Inverse Problem for a Differential Equation in a Banach Space”, Differ. Uravn., 23:9 (1987), 1647–1649 (in Russian) | MR | Zbl

[20] Khaidarov A., “A Class of Inverse Problems for Elliptic Equations”, Dokl. Akad. Nauk SSSR, 277:6 (1984), 1335–1337 (in Russian) | MR

[21] Khaidarov A., “A Class of Inverse Problems for Elliptic Equations”, Differ. Uravn., 23:8 (1987), 1376–1383 (in Russian) | MR

[22] Almohamed M., Tikhonov I. V., “Specific Cases of One General Inverse Problem for Abstract Differential Equations”, Lobachevskii Journal of Mathematics, 44:2 (2023), 502–509 | DOI | MR | Zbl

[23] Prilepko A. I., Orlovsky D. G., Vasin I. A., Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker Inc, New York–Basel, 2000, 744 pp. | DOI | MR | Zbl