49
Views
0
CrossRef citations to date
0
Altmetric
Research Articles

A Source Identification Problem in a Bi-parabolic Equation: Convergence Rates and Some Optimal Results

ORCID Icon &
Pages 189-215 | Received 22 May 2023, Accepted 05 Feb 2024, Published online: 21 Feb 2024

References

  • Payne, L. E., Song, J. C. (2006). On a proposed model for heat conduction. IMA J. Appl. Math. 71(4):590–599. DOI: 10.1093/imamat/hxh112.
  • Fichera, G. (1992). Is the Fourier theory of heat propagation paradoxical? Rend. Circ. Mat. Palermo (2) 41(1):5–28. DOI: 10.1007/BF02844459.
  • Joseph, D. D., Preziosi, L. (1989). Heat waves. Rev. Mod. Phys. 61(1):41–73. DOI: 10.1103/RevModPhys.61.41.
  • Fushchich, V. I., Galitsyn, A. S., Polubinskii, A. S. (1990). A new mathematical model of heat conduction processes. Ukrainian Math. J. 42(2):210–216. DOI: 10.1007/BF01071016.
  • Kalantarov, V., Zelik, S. (2009). Finite-dimensional attractors for the quasi-linear strongly-damped wave equation. J. Differ. Equ. 247(4):1120–1155. DOI: 10.1016/j.jde.2009.04.010.
  • Greer, B. J., Bertozzi, A. L., Sapiro, G. (2006). Fourth order partial differential equations on general geometries. J. Comput. Phys. 216(1):216–246. DOI: 10.1016/j.jcp.2005.11.031.
  • Isakov, V. (2006). Inverse Problems for Partial Differential Equations. 2nd ed. Applied Mathematical Sciences, 127. New York: Springer, xiv + 344 pp.
  • Andrle, M., El Badia, A. (2012). Identification of multiple moving pollution sources in surface waters or atmospheric media with boundary observations. Inverse Probl. 28(7):075009, 22 pp. DOI: 10.1088/0266-5611/28/7/075009.
  • Magnoli, N., Viano, G. A. (1997). The source identification problem in electromagnetic theory. J. Math. Phys. 38(5):2366–2388. DOI: 10.1063/1.531978.
  • Quoc Nam, D. H., Long, L. D., O’Regan, D., Ngoc, T. B., Tuan, N. H. (2022). Identification of the right-hand side in a bi-parabolic equation with final data. Appl. Anal. 101(4):1157–1175. DOI: 10.1080/00036811.2020.1775817.
  • Engl, H. W., Hanke, M., Neubauer, A. (1996). Regularization of inverse problems. Dordrecht: Kluwer.
  • Nair, M. T. (2009). Linear Operator Equations: Approximation and Regularization. Hackensack: World Scientific.
  • Zouyed, F., Djemoui, S. (2015). An iterative regularization method for identifying the source term in a second order differential equation. Math. Probl. Eng. 2015:Art. ID 713403, 9 pp.
  • Tuan, N. H. (2022). On some inverse problem for bi-parabolic equation with observed data in Lp spaces. Opuscula Math. 42(2):305–335. DOI: 10.7494/OpMath.2022.42.2.305.
  • Phuong, N. D., Luc, N. H., Long, L. D. (2020). Modified quasi boundary value method for inverse source problem of the bi-parabolic equation. Adv. Theory Nonlinear Anal. Appl. 4(3):132–142.
  • Lattès, R., Lions, J.-L. (1967). Méthode de quasi-réversibilité et applications. (French) Travaux et Recherches Mathématiques, No. 15. Paris: Dunod, xii + 368 pp.
  • Bourgeois, L. (2006). Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace’s equation. Inverse Probl. 22(2):413–430. DOI: 10.1088/0266-5611/22/2/002.
  • Dorroh, J. R., Ru, X. (1999). The application of the method of quasi-reversibility to the sideways heat equation. J. Math. Anal. Appl. 236(2):503–519. DOI: 10.1006/jmaa.1999.6462.
  • Mophou, G., Warma, M. (2023). Quasi-reversibility methods of optimal control for ill-posed final value diffusion equations. J. Math. Anal. Appl. 517(2):Paper No. 126618, 35 pp. DOI: 10.1016/j.jmaa.2022.126618.
  • Nguyen, H. T., Khoa, V. A., Vo, V. A. (2019). Analysis of a quasi-reversibility method for a terminal value quasi-linear parabolic problem with measurements. SIAM J. Math. Anal. 51(1):60–85. DOI: 10.1137/18M1174064.
  • Showalter, R. E. (1975). Quasi-reversibility of first and second order parabolic evolution equations. Improperly posed boundary value problems. In: Conf. Univ. New Mexico, Albuquerque, N.M., 1974. Research Notes in Mathematics, No. 1. London: Pitman, pp. 76–84.
  • Tuan, N. H., Nane, E., Trong, D. D. (2021). Analysis of a quasi-reversibility method for nonlinear parabolic equations with uncertainty data. Illinois J. Math. 65(4):793–845. DOI: 10.1215/00192082-9501497.
  • Le, T. T., Nguyen, L. H., Nguyen, T. P., Powell, W. (2021). The quasi-reversibility method to numerically solve an inverse source problem for hyperbolic equations. J. Sci. Comput. 87(3):Paper No. 90, 23 pp. DOI: 10.1007/s10915-021-01501-3.
  • Nguyen, L. H. (2019). An inverse space-dependent source problem for hyperbolic equations and the Lipschitz-like convergence of the quasi-reversibility method. Inverse Probl. 35(3):035007, 28 pp. DOI: 10.1088/1361-6420/aafe8f.
  • Duc, N. V., Thang, N. V., Thành, N. T. (2023). The quasi-reversibility method for an inverse source problem for time-space fractional parabolic equations. J. Differ. Equ. 344:102–130. DOI: 10.1016/j.jde.2022.10.029.
  • Yang, F., Ren, Y. P., Li, X. X. (2018). The quasi-reversibility method for a final value problem of the time-fractional diffusion equation with inhomogeneous source. Math. Methods Appl. Sci. 41(5):1774–1795. DOI: 10.1002/mma.4705.
  • Ames, K. A., Clark, G. W., Epperson, J. F., Oppenheimer, S. F. (1998). A comparison of regularizations for an ill-posed problem. Math. Comp. 67(224):1451–1471. DOI: 10.1090/S0025-5718-98-01014-X.
  • Clark, G. W., Oppenheimer, S. F. (1994). Quasireversibility methods for non-well-posed problems. Electron. J. Differ. Equ. 1994(08):approx. 9 pp.
  • Clason, C., Klibanov, M. V. (2007/08). The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium. SIAM J. Sci. Comput. 30(1):1–23. DOI: 10.1137/06066970X.
  • Huang, Y. (2008). Modified quasi-reversibility method for final value problems in Banach spaces. J. Math. Anal. Appl. 340(2):757–769. DOI: 10.1016/j.jmaa.2007.09.019.
  • Showalter, R. E. (1974). The final value problem for evolution equations. J. Math. Anal. Appl. 47:563–572. DOI: 10.1016/0022-247X(74)90008-0.
  • Háo, D. N., Liu, J., Duc, N. V., Thang, N. V. (2019). Stability results for backward time-fractional parabolic equations. Inverse Probl. 35(12):125006, 25 pp. DOI: 10.1088/1361-6420/ab45d3.
  • Evans, L. C. (2010). Partial Differential Equations, 2nd ed. Graduate Studies in Mathematics, 19. Providence, RI: American Mathematical Society, xxii + 749 pp.
  • Háo, D. N., Duc, N. V., Thang, N. V., Thánh, N. T. (2020). Regularization of backward time-fractional parabolic equations by Sobolev-type equations. J. Inverse Ill-Posed Probl. 28(5):659–676. DOI: 10.1515/jiip-2020-0062.
  • Hofmann, B., Mathé, P., Schieck, M. (2008). Modulus of continuity for conditionally stable ill-posed problems in Hilbert space. J. Inverse Ill-Posed Probl. 16(6):567–585.
  • Trong, D. D., Hai, D. N. D. (2021). Backward problem for time-space fractional diffusion equations in Hilbert scales. Comput. Math. Appl. 93:253–264. DOI: 10.1016/j.camwa.2021.04.018.
  • Tautenhahn, U. (1998). Optimality for ill-posed problems under general source conditions. Numer. Funct. Anal. Optim. 19(3–4):377–398. DOI: 10.1080/01630569808816834.
  • Courant, R., Hilbert, D. (1953). Methods of Mathematical Physics, Vol. I. New York, NY: Interscience Publishers, Inc., xv + 561 pp.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.