Molhem, H., Pourgholi, R., Borghei, M. (2011). A numerical approach for solving a nonlinear inverse diusion problem by Tikhonov regularization. Theory of Approximation and Applications, 7(2), 39-54.

H. Molhem; R. Pourgholi; M. Borghei. "A numerical approach for solving a nonlinear inverse diusion problem by Tikhonov regularization". Theory of Approximation and Applications, 7, 2, 2011, 39-54.

Molhem, H., Pourgholi, R., Borghei, M. (2011). 'A numerical approach for solving a nonlinear inverse diusion problem by Tikhonov regularization', Theory of Approximation and Applications, 7(2), pp. 39-54.

Molhem, H., Pourgholi, R., Borghei, M. A numerical approach for solving a nonlinear inverse diusion problem by Tikhonov regularization. Theory of Approximation and Applications, 2011; 7(2): 39-54.

A numerical approach for solving a nonlinear inverse diusion problem by Tikhonov regularization

^{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.

Abstract

In this paper, we propose an algorithm for numerical solving an inverse non- linear diusion problem. In additional, the least-squares method is adopted to nd the solution. To regularize the resultant ill-conditioned linear system of equations, we apply the Tikhonov regularization method to obtain the stable numerical approximation to the solution. Some numerical experiments con- rm the utility of this algorithm as the results are in good agreement with the exact data.

Article Title [Persian]

A numerical approach for solving a
nonlinear inverse diusion problem by
Tikhonov regularization

Authors [Persian]

H. 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.

Abstract [Persian]

In this paper, we propose an algorithm for numerical solving an inverse non- linear diusion problem. In additional, the least-squares method is adopted to nd the solution. To regularize the resultant ill-conditioned linear system of equations, we apply the Tikhonov regularization method to obtain the stable numerical approximation to the solution. Some numerical experiments con- rm the utility of this algorithm as the results are in good agreement with the exact 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 diusion 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 Diusion Problem, Comput. Math. Appl. 52 (2006), 1021{1030. [10] A. Shidfar, R. Pourgholi, Application of nite dierence 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: ecient 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.