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

Hongwu Zhang, Xiaoju Zhang

School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan, China.

DOI: 10.4236/oalib.1101542                

Abstract

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.

Keywords

Inverse Initial Value Problem, Biparabolic Equation, Conditional Stability, Regularization Method, Convergence Estimate

Conflicts of Interest

The authors declare no conflicts of interest.

References

[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