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.
Cite this paper
Zhang, H. and Zhang, X. (2015). Stability and Regularization Method for Inverse Initial Value Problem of Biparabolic Equation. Open Access Library Journal, 2, e1542. doi: http://dx.doi.org/10.4236/oalib.1101542.
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
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
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
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
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
Atakhadzhaev, M.A. and Egamberdiev, O.M. (1990) The Cauchy Problem for the Abstract Bicaloric Equation. Sibirskii Matematicheskii Zhurnal, 31, 187-191.
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
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
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.
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
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
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
