Research Article
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Year 2024, , 130 - 134, 28.03.2024
https://doi.org/10.17776/csj.1359651

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.

Solvability of an Inverse Problem for an Elliptic-Type Equation

Year 2024, , 130 - 134, 28.03.2024
https://doi.org/10.17776/csj.1359651

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.
There are 10 citations in total.

Details

Primary Language English
Subjects Applied Mathematics (Other)
Journal Section Natural Sciences
Authors

Özlem Kaytmaz 0000-0003-0420-007X

Mustafa Yıldız 0000-0003-3367-7176

Publication Date March 28, 2024
Submission Date September 13, 2023
Acceptance Date February 26, 2024
Published in Issue Year 2024

Cite

APA Kaytmaz, Ö., & Yıldız, M. (2024). Solvability of an Inverse Problem for an Elliptic-Type Equation. Cumhuriyet Science Journal, 45(1), 130-134. https://doi.org/10.17776/csj.1359651