Abstract
In this paper, we consider a time-inverse problem for a nonlinear spherically symmetric backward heat equation which is a severely ill-posed problem. Using a modified integral equation method with two regularization parameters: one related to the error in a measurement process and the other is related to the regularity of solution, we regularize this problem and obtain the Hölder-type estimation error for the whole time interval. Numerical results are presented to illustrate the accuracy and efficiency of the method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions: with formulas, graphs, and mathematical tables. Dover Publications, New york (1965)
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover Publications, New York (1972)
Cheng, W., Fu, C.L.: A spectral method for an axisymmetric backward heat equation. Inverse. Probl. Sci. Eng. 17, 1081–1093 (2009)
Cheng, W., Fu, C.L., Qin, F.J.: Regularization and error estimate for a spherically symmetric backward heat equation. J. Inverse. Ill-posed Prob. 19, 369–377 (2011)
Cheng, W., Fu, C.L.: Identifying an unknown source term in a spherically symmetric parabolic equation. Appl. Math. Lett. 26, 387–391 (2013)
Cheng, W., Ma, Y.J., Fu, C.L.: A regularization method for solving the radially symmetric backward heat conduction problem. Appl. Math. Lett. 30, 38–43 (2014)
Clark, G.W., Oppenheimer, S.F.: Quasireversibility methods for non-well posed problems. Electr. J. Differ. Eqs. 8, 1–9 (1994)
Daido, Y., Kang, H., Nakamura, G.: A probe method for the inverse boundary value problem of nonstationary heat equations. Inverse Prob. 23, 1787–1800 (2007)
Denche, M., Bessila, K.: A modified quasi-boundary value method for ill-posed problems. J. Math. Anal. Appl. 301, 419–426 (2005)
Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence, RI (1998)
Fu, C.L., Qian, Z., Shi, R.: A modified method for a backward heat conduction problem. Appl. Math. Comput. 185, 564–573 (2007)
Glasko, V.B.: Inverse Problems of Mathematical Physics. American Institute of Physics, New York (1984)
Hao, D.N., Duc, N.V.: Stability results for the heat equation backward in time. J. Math. Anal. Appl. 353, 627–641 (2009)
Hao, D.N., Duc, N.V., Lesnic, D.: Regularization of parabolic equations backward in time by a non-local boundary value problem method. IMA J. Appl. Math. 75, 291–315 (2010)
Hao, D.N., Duc, N.V.: Regularization of backward parabolic equations in Banach spaces. J. Inverse. Ill-posed Prob. 20, 745–763 (2012)
Kaltenbacher, B.: On convergence rates of some iterative regularization methods for an inverse problem for a nonlinear parabolic equation connected with continuous casting of steel. J. Inverse. Ill-posed Prob. 7, 145–164 (1999)
Kaltenbacher, B., Klibanov, M.V.: An inverse problem for a nonlinear parabolic equation with applications in population dynamics and magnetics. SIAM J. Math. Anal. 39, 1863–1889 (2008)
Lattes, R., Lions, J.L.: Méthode de quasi-réversibilité et applications. Dunod, Paris (1967)
Muniz, B.W.: A comparison of some inverse methods for estimating the initial condition of the heat equation. J. Comput. Appl. Math. 103, 145–163 (1999)
Qian, Z., Hon, B.Y.C., Xiong, X.T.: Numerical solution of two-dimensional radially symmetric inverse heat conduction problem. J. Inverse. Ill-posed Prob. 23, 121–134 (2015)
Showalter, R.E.: The final value problem for evolution equations. J. Math. Anal. Appl. 47, 563–572 (1974)
Showalter, R.E.: Quasi-reversibility of first and second order parabolic evolution equations, Improperly posed boundary value problems (Conf., Univ. New Mexico, Albuquerque), Pitman, London, pp. 76–84 (1975)
Tautenhahn, U.: Optimality for ill-posed problems under general source conditions. Numer. Funct. Anal. Opt. 19, 377–398 (1998)
Tikhonov, A.N., Arsenin, V.Y.: Solutions of illposed problems. V.H Winston and Sons. Washington DC (1977)
Trong, D.D., Tuan, N.H.: Regularization and error estimate for the nonlinear backward heat problem using a method of integral equation. Nonlinear Anal. Theory Methods Appl. 71, 4167–4176 (2009)
Wei, T., Wang, J.: Modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation. Appl. Numer. Math. 78, 95–111 (2014)
Xiong, X.T., Fu, C.L.: Error estimates on a backward heat equation by a wavelet dual least squares method. Int. J. Wavelets. Multiresolut. Inf. Process. 5, 389–397 (2007)
Xiong, X.T.: On a radially symmetric inverse heat conduction problem. Appl. Math. Model. 34, 520–529 (2010)
Yamamoto, M.: Carleman estimates for parabolic equations and applications. Inverse Prob. 25, 123013 (2009)
Acknowledgements
The authors would like to express their gratitude to the anonymous referees for their helpful comments and suggestions, which have greatly improved the paper. The authors would like to thank Prof. Dang Duc Trong, Prof. Nguyen Huy Tuan and Dr. Vo Hoang Hung for the great support during their undergraduate period. This work is supported by the Vietnam National Foundation of Scientific and Technology Development (NAFOSTED)-Project 101.02-2016.26.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Khanh, T.Q., Khieu, T.T. & Hoa, N.V. Stabilizing the nonlinear spherically symmetric backward heat equation via two-parameter regularization method. J. Fixed Point Theory Appl. 19, 2461–2481 (2017). https://doi.org/10.1007/s11784-017-0429-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11784-017-0429-x