In this work, we investigate an abstract parabolic problem associated with the final data, with one specific case of the abstract parabolic problem being considered in polar coordinates. Specially, the present paper gives the first treatment which is not only more general but also more applicable in the case the backward heat problem (BHP) is not symmetric and not axisymmetric in polar coordinates. In general, the above problem is severely ill-posed in the sense of Hadamard. Here, we propose the modified quasi-boundary value (MQBV) method in order to obtain the stability of the regularized solution for the problem. To our knowledge, our results are new and they improve the results of two previous papers (see Cheng and Fu (Inverse Probl. Sci. Eng. 17(8), 1085–1093 2009), (Acta Math. Sinica, English Series 26(11), 2157–2164 2010)) while the authors only considered axisymmetric or radially symmetric data. Finally, a numerical example is given to demonstrate the feasibility and efficiency of our method.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Cheng, W., Fu, C.L.: A spectral method for an axisymmetric backward heat equation. Inverse Probl. Sci. Eng. 17(8), 1085–1093 (2009)
Cheng, W., Fu, C.L.: A modified Tikhonov regularization method for an axisymmetric backward heat equation. Acta Math. Sinica, English Series 26(11), 2157–2164 (2010)
Denche, M., Djezzar, S.: A modified quasi-boundary value method for a class of abstract parabolic ill-posed problems. Hindawi Publ. Corp. Bound. Value Probl. 2006, 1–8 (2006)
Fu, C. L., Xiong, X. T., Qian, Z.: Fourier regularization for a backward heat equation. J. Math. Anal. Appl. 331, 472–480 (2007)
Nakhlé, H.A.: Partial Differential Equations with Fourier Series and Boundary Value Problems. Upper Saddle River, p. 07458
Quan, P. H., Trong, D. D., Triet, L. M., Tuan, N. H.: A modified quasi-boundary value method for regularizing of a backward problem with time-dependent coefficient. Inverse Probl. Sci. Eng. 19(3), 409–423 (2011)
Trong, D. D., Tuan, N. H.: Stabilized quasi-reversibility method for a class of nonlinear ill-posed problems. Electron. J. Differ. Equ. 2008(84), 1–12 (2008)
Trong, D. D., Quan, P. H., Tuan, N. H.: A quasi-boundary value method for regularizing nonlinear ill-posed problems. Electron. J. Differ. Equ. 2009(109), 1–16 (2009)
Trong, D. D., Tuan, N. H.: A nonhomogeneous backward heat problem: regularization and error estimates. Electron. J. Differ. Equ. 2008(33), 1–14 (2008)
Tuan, N. H., Trong, D. D.: A note on a nonlinear backward heat equation stability and error estimates. Acta Universitatis Apulensis 28, 279–292 (2011)
Showalter, R.E.: Cauchy problem for hyper-parabolic partial differential equations. In: Trends in the Theory and Practice of Non-Linear Analysis. Elsevier (1983)
This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2015.23.
About this article
Cite this article
Triet, L.M., Phong, L.H. & Quan, P.H. On a Backward Heat Conduction Problem Associated with Asymmetric Final Data. Acta Math Vietnam 43, 341–356 (2018). https://doi.org/10.1007/s40306-017-0238-8
- Backward heat problem
- Quasi-boundary value method
- Polar coordinates
- Ill-posed problem
Mathematics Subject Classification (2010)