A numerical approach for solving a nonlinear inverse di usion problem by Tikhonov regularization
محورهای موضوعی : Applied MathematicsH. Molhem 1 , R. Pourgholi 2 , M. Borghei 3
1 - Department of Physics , Faculty of Science, Islamic Azad University, Karaj
Branch, Karaj, Iran
2 - School of Mathematics and Computer Sciences,
Damghan University, P.O.Box 36715-364, Damghan, Iran.
3 - Department of Physics , Faculty of Science, Islamic Azad University, Karaj
Branch, Karaj, Iran.
کلید واژه: Laplace transform, Inverse nonlinear di usion problem, Finite di erence method, Least-squares method, Regularization method, SVD Method,
چکیده مقاله :
In this paper, we propose an algorithm for numerical solving an inverse non-linear di usion problem. In additional, the least-squares method is adopted to nd the solution. To regularize the resultant ill-conditioned linear system ofequations, we apply the Tikhonov regularization method to obtain the stablenumerical approximation to the solution. Some numerical experiments con- rm the utility of this algorithm as the results are in good agreement with theexact data.
In this paper, we propose an algorithm for numerical solving an inverse non-linear di usion problem. In additional, the least-squares method is adopted to nd the solution. To regularize the resultant ill-conditioned linear system ofequations, we apply the Tikhonov regularization method to obtain the stablenumerical approximation to the solution. Some numerical experiments con- rm the utility of this algorithm as the results are in good agreement with theexact data.
[1] J. Bear , Dynamics of Fluids in Porous Media, 2nd edn. Elsevier, New
York, 1975.
[2] J. R. Cannon, The One-Dimensional Heat Equation. Addison-Wesley,
Menlo Park, California, 1984.
[3] P. C. Hansen, Analysis of discrete ill-posed problems by means of the
L-curve, SIAM Rev. 34 (1992), 561{80.
[4] A. N. Tikhonov, V. Y. Arsenin, On the solution of ill-posed problems, New
York, Wiley, 1977.
[5] C. L. Lawson, R. J. Hanson, Solving least squares problems, Philadelphia,
PA,SIAM, 1995, First published by Prentice-Hall, 1974.
[6] J. R. Cannon, P. Duchateau, An inverse problem for a nonlinear di usion
equation, SIAM J. appl. Math. 39 (1980), 272{289.
[7] O. A. Ladyzhenskaya, V. A. Sollonikov, N. N. Uralceva, Linear and
Quasilinear Equations Parabolic Type,Amer. Math. Soc. Providence, RI
(1967).
[8] H. T. Chen, S. M. Chang, Application of the hybrid method to inverse heat
conduction problems, Int. J. Heat Mass Transfer, 33 (1990), 621{628.
[9] A. Shidfar, R. Pourgholi, M. Ebrahimi, A Numerical Method for Solving
of a Nonlinear Inverse Di usion Problem, Comput. Math. Appl. 52 (2006),
1021{1030.
[10] A. Shidfar, R. Pourgholi, Application of nite di erence method to
analysis an ill-posed problem, Appl. Math. Comput. 168 (2005), 1400{
1408 .
[11] R. Pourgholi, N. Azizi, Y. S. Gasimov, F. Aliev, H. K. Khala ,
Removal of Numerical Instability in the Solution of an Inverse
Heat Conduction Problem, Communications in Nonlinear Science and
Numerical Simulation, (2008) In Press.
[12] F. Durbin, Numerical inversion of Laplace transforms: e cient
improvement to Dubner and Abates method, Comp. J. 17 (1973), 371{
376.
[13] G. Honig and U. Hirdes, A method for the numerical inversion of Laplace
transforms, J. Comp. Appl. Math. 9 (1984), 113-132.
[14] G. H. Golub, C. F. Van Loan, Matrix computations, John Hopkins
university press, Baltimore, MD, 1983.
