A numerical method based on the polynomial regression for the inverse diffusion problem

References

• M. Alifanov, Inverse Heat Transfer Problems, Springer-Verlag, New York, 1994.
   Google Scholar
• A.V. Avdeev, E.V. Goriounov, M.M. LavrentievJr., and R. Spigler, A behavior-oriented model for long-term coastal profile evolution: Validation, identification, and prediction, Appl. Math. Model. 33 (2009), pp. 3981–3996. doi: 10.1016/j.apm.2009.01.010
   Web of Science ®Google Scholar
• H. Azari and S. Zhang, A numerical approximation to the solution of an inverse heat conduction problem, Num. Methods Partial Differ. Equ. 26(1) (2010), pp. 95–106. doi: 10.1002/num.20418
   Web of Science ®Google Scholar
• A. Benabdallah, Y. Dermenjian, and J. Le Rousseau, Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem, J. Math. Anal. Appl. 336 (2007), pp. 865–887. doi: 10.1016/j.jmaa.2007.03.024
   Web of Science ®Google Scholar
• P. DuChateau, Monotonicity and invertibility of coefficient-to-data mappings for parabolic inverse problems, SIAM J. Math. Anal. 26(6) (1995), pp. 1473–1487. doi: 10.1137/S0036141093259257
   Web of Science ®Google Scholar
• P. DuChateau, R. Thelwell, and G. Butters, Analysis of an adjoint problem approach to the identification of an unknown diffusion coefficient, Inverse Probl. 20 (2004), pp. 601–625. doi: 10.1088/0266-5611/20/2/019
   Web of Science ®Google Scholar
• A.G. Fatullayev and S. Cula, An iterative procedure for determining an unknown spacewise-dependent coefficient in a parabolic equation, Appl. Math. Lett. 22(7) (2009), pp. 1033–1037. doi: 10.1016/j.aml.2009.01.022
   Web of Science ®Google Scholar
• R. Fletcher and C.M. Reeves, Function minimization by conjugate gradients, Computer J. 7 (1964), pp. 149–154. doi: 10.1093/comjnl/7.2.149
   Web of Science ®Google Scholar
• D.N. Hào, A non-characteristic cauchy problem for linear parabolic equations and related inverse problems: I. Solvability, Inverse Probl. 10 (1994), pp. 295–315. doi: 10.1088/0266-5611/10/2/007
   Web of Science ®Google Scholar
• A. Hasanov, A. Demir, and A. Erdem, Monotonicity of input-output mappings in inverse coefficient and source problems for parabolic equations, J. Math. Anal. Appl. 335 (2007), pp. 1434–1451. doi: 10.1016/j.jmaa.2007.01.097
   Web of Science ®Google Scholar
• A. Hasanov, B. Pektas, and A. Erdem, Comparative analysis of inverse coefficient problems for parabolic equations. Part I: Adjoint problem approach, Inverse Probl. Sci. Eng. 19(5) (2011), pp. 599–615. doi: 10.1080/17415977.2011.579605
   Web of Science ®Google Scholar
• Y. Jarny, M.N. Ozisik, and J.P. Bardon, A general optimization method using adjoint equation for solving multidimensional inverse heat conduction, Int. J. Heat Mass Transfer 34(11) (1991), pp. 2911–2919. doi: 10.1016/0017-9310(91)90251-9
   Web of Science ®Google Scholar
• S. Kabanikhin and A. Penenko, Gradient-type methods in inverse parabolic problems, J. Phys: Conf. Ser. 135 (2008), pp. 012054.
   Google Scholar
• Z. Liu and B. Wang, Coefficient identification in parabolic equations, Appl. Math. Comput. 209 (2009), pp. 379–390. doi: 10.1016/j.amc.2008.12.062
   Web of Science ®Google Scholar
• D.W. Marquardt, An algorithm for least squares estimation of nonlinear parameters, J. Soc. Ind. Appl. Math ll (1963), pp. 431–441. doi: 10.1137/0111030
   Google Scholar
• B. Pektaş, Determination of unknown coefficients in a nonlocal parabolic problem from Neumann type boundary measured data, Adv. Dyn. Syst. Appl. 5(2) (2010), pp. 215–230.
   Google Scholar
• A. Shidfar, M. Fakhraie, R. Pourgholi, and M. Ebrahimi, A numerical solution technique for a one-dimensional inverse nonlinear parabolic problem, Appl. Math. Comput. 184(2) (2007), pp. 308–315. doi: 10.1016/j.amc.2006.05.183
   Web of Science ®Google Scholar
• M. Tadi, M.V. Klibanov, and W. Cai, An inversion method for parabolic equations based on quasireversibility, Comput. Math. Appl. 43 (2002), pp. 927–941. doi: 10.1016/S0898-1221(02)80003-7
   Web of Science ®Google Scholar