Research Article
Solvability of an Inverse Problem for an Elliptic-Type Equation
Abstract
In this study, we consider an inverse problem of determining an unknown source function in the right-hand side of an elliptic equation which is ill-posed in the Hadamard sense. To investigate the solvability of the problem, we reduce it to a Dirichlet problem for a third-order partial differential equation with homogeneous boundary condition. Since the problem is linear, the proof of the uniqueness theorem is based on the Fredholm Alternative Theorem. We prove the existence of the solution to the problem by using the Galerkin method.
References
- [1] Payne L.E., Improperly Posed Problems in Partial Differential Equations. 1st ed. Philadelphia: Society for Industrial and Applied Mathematics, (1975) 19-42.
- [2] Isakov V., Inverse Problems for Partial Differential Equations. 2nd ed. New York: Springer, (2006) 89-295.
- [3] Kabanikhin S.I., Definitions and Examples of Inverse and Ill-Posed Problems, J. Inverse Ill-Pose. Probl., 16 (4) (2008) 317-357.
- [4] Klibanov M.V., Timonov A.A., Carleman Estimates for Coefficient Inverse Problems and Numerical Applications. 1st ed. The Netherlands: VSP, (2004) 79-145.
- [5] Lavrent’ev M.M., Romanov V.G., Shishatskii S.P., Ill-Posed Problems of Mathematical Physics and Analysis. 1st ed. Providence: American Mathematical Society, (1986) 7-261.
- [6] Amirov A., Gölgeleyen F., Solvability of an Inverse Problem for the Kinetic Equation and a Symbolic Algorithm, Comput. Model. Eng. Sci., 65 (2) (2010) 179-191.
- [7] Bellassoued M., Yamamoto M., Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems. 1st ed. Tokyo: Springer, (2017) 63-239.
- [8] Gölgeleyen F., Yamamoto M., Uniqueness of Solution of an Inverse Source Problem for Ultrahyperbolic Equations, Inverse Probl., 36 (3) (2020) 035008.
- [9] Gölgeleyen İ., Kaytmaz Ö., Uniqueness for a Cauchy Problem for the Generalized Schrödinger Equation, AIMS Math., 8 (3) (2023) 5703-5724.
- [10] Gölgeleyen İ., Yıldız M., On the Solution an Ill-Posed Boundary Value Problem for Second-Order Evolution Equations, Cumhuriyet Sci. J., 40 (1) (2019) 173-178.
Abstract
References
- [1] Payne L.E., Improperly Posed Problems in Partial Differential Equations. 1st ed. Philadelphia: Society for Industrial and Applied Mathematics, (1975) 19-42.
- [2] Isakov V., Inverse Problems for Partial Differential Equations. 2nd ed. New York: Springer, (2006) 89-295.
- [3] Kabanikhin S.I., Definitions and Examples of Inverse and Ill-Posed Problems, J. Inverse Ill-Pose. Probl., 16 (4) (2008) 317-357.
- [4] Klibanov M.V., Timonov A.A., Carleman Estimates for Coefficient Inverse Problems and Numerical Applications. 1st ed. The Netherlands: VSP, (2004) 79-145.
- [5] Lavrent’ev M.M., Romanov V.G., Shishatskii S.P., Ill-Posed Problems of Mathematical Physics and Analysis. 1st ed. Providence: American Mathematical Society, (1986) 7-261.
- [6] Amirov A., Gölgeleyen F., Solvability of an Inverse Problem for the Kinetic Equation and a Symbolic Algorithm, Comput. Model. Eng. Sci., 65 (2) (2010) 179-191.
- [7] Bellassoued M., Yamamoto M., Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems. 1st ed. Tokyo: Springer, (2017) 63-239.
- [8] Gölgeleyen F., Yamamoto M., Uniqueness of Solution of an Inverse Source Problem for Ultrahyperbolic Equations, Inverse Probl., 36 (3) (2020) 035008.
- [9] Gölgeleyen İ., Kaytmaz Ö., Uniqueness for a Cauchy Problem for the Generalized Schrödinger Equation, AIMS Math., 8 (3) (2023) 5703-5724.
- [10] Gölgeleyen İ., Yıldız M., On the Solution an Ill-Posed Boundary Value Problem for Second-Order Evolution Equations, Cumhuriyet Sci. J., 40 (1) (2019) 173-178.
There are 10 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Applied Mathematics (Other) |
| Journal Section | Natural Sciences |
| Authors | |
| Publication Date | March 28, 2024 |
| Submission Date | September 13, 2023 |
| Acceptance Date | February 26, 2024 |
| Published in Issue | Year 2024 Volume: 45 Issue: 1 |
