Stability and Regularization Method for Inverse Initial Value Problem of Biparabolic Equation

We consider an inverse initial value problem of the biparabolic equation; this problem is ill-posed and the regularization methods are needed to stabilize the numerical computations. This paper firstly establishes a conditional stability of Holder type, then uses a modified regularization method to overcome its ill-posedness and gives the convergence estimate under an a-priori assumption for the exact solution. Finally, a numerical example is presented to show that this method works well.

Share and Cite:

Zhang, H. and Zhang, X. (2015) Stability and Regularization Method for Inverse Initial Value Problem of Biparabolic Equation. Open Access Library Journal, 2, 1-7. doi: 10.4236/oalib.1101542.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] Cheng, J. and Liu, J.J. (2008) A Quasi Tikhonov Regularization for a Two-Dimensional Backward Heat Problem by a Fundamental Solution. Inverse Problems, 24, Article ID: 065012. http://dx.doi.org/10.1088/0266-5611/24/6/065012 [2] Feng, X.L., Qian, Z. and Fu, C.L. (2008) Numerical Approximation of Solution of Nonhomogeneous Backward Heat Conduction Problem in Bounded Region. Mathematics and Computers in Simulation, 79, 177-188. http://dx.doi.org/10.1016/j.matcom.2007.11.005 [3] Liu, J.J. (2002) Numerical Solution of Forward and Backward Problem for 2-d Heat Conduction Equation. Journal of Computational and Applied Mathematics, 145, 459-482. http://dx.doi.org/10.1016/S0377-0427(01)00595-7 [4] Qian, Z., Fu, C.L. and Shi, R. (2007) A Modified Method for a Backward Heat Conduction Problem. Applied Mathematics and Computation, 185, 564-573. http://dx.doi.org/10.1016/j.amc.2006.07.055 [5] Fichera, G. (1992) Is The Fourier Theory of Heat Propagation Paradoxical? Rendiconti del Circolo Matematico di Palermo, 41, 5-28. [6] Joseph, L. and Preziosi, D.D. (1989) Heat Waves. Reviews of Modern Physics, 61, 41. http://dx.doi.org/10.1103/revmodphys.61.41 [7] Fushchich, V.L., Galitsyn, A.S. and Polubinskii, A.S. (1990) A New Mathematical Model of Heat Conduction Processes. Ukrainian Mathematical Journal, 42, 210-216. http://dx.doi.org/10.1007/BF01071016 [8] Atakhadzhaev, M.A. and Egamberdiev, O.M. (1990) The Cauchy Problem for the Abstract Bicaloric Equation. Sibirskii Matematicheskii Zhurnal, 31, 187-191. [9] Lakhdari, A. and Boussetila, N. (2015) An Iterative Regularization Method for an Abstract Ill-Posed Biparabolic Problem. Boundary Value Problems, 55, 1-17. http://dx.doi.org/10.1186/s13661-015-0318-4 [10] Ames, K.A. and Straughan, B. (1997) Non-Standard and Improperly Posed Problems. Academic Press, New York. [11] Carasso, A.S. (2010) Bochner Subordination, Logarithmic Diffusion Equations, and Blind Deconvolution of Hubble Space Telescope Imagery and Other Scientifc Data. SIAM Journal on Imaging Sciences, 3, 954-980. http://dx.doi.org/10.1137/090780225 [12] Payne, L.E. (2006) On a Proposed Model for Heat Conduction. IMA Journal of Applied Mathematics, 71, 590-599.http://dx.doi.org/10.1093/imamat/hxh112 [13] Wang, L., Zhou, X. and Wei, X. (2008) Heat Conduction: Mathematical Models and Analytical Solutions. Springer-Verlag, Berlin. [14] Engl, H.W., Hanke, M. and Neubauer, A. (1996) Regularization of Inverse Problems, Volume 375 of Mathematics and Its Applications. Kluwer Academic Publishers Group, Dordrecht. [15] Kirsch, A. (1996) An Introduction to the Mathematical Theory of Inverse Problems. Volume 120 of Applied Mathematical Sciences. Springer-Verlag, New York. http://dx.doi.org/10.1007/978-1-4612-5338-9 [16] Ames, K.A., Clark, G.W., Epperson, J.F. and Oppenheimer, S.F. (1998) A Comparison of Regularizations for an Ill-Posed Problem. Mathematics of Computation, 67, 1451-1472. http://dx.doi.org/10.1090/S0025-5718-98-01014-X [17] Clark, G.W. and Oppenheimer, S.F. (1994) Quasireversibility Methods for Non-Well-Posed Problems. Electronic Journal of Differential Equations, 1-9. [18] Denche, M. and Bessila, K. (2005) A Modified Quasi-Boundary Value Method for Ill-Posed Problems. Journal of Mathematical Analysis and Applications, 301, 419-426.http://dx.doi.org/10.1016/j.jmaa.2004.08.001