Geodesic
The solution of the inverse problem for the diffusion equation based on Laguerre transformation
Matematičeskoe modelirovanie, Tome 19 (2007) no. 9, pp. 15-26 
A. F. Mastryukov. The solution of the inverse problem for the diffusion equation based on Laguerre transformation. Matematičeskoe modelirovanie, Tome 19 (2007) no. 9, pp. 15-26. http://geodesic.mathdoc.fr/item/MM_2007_19_9_a1/
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     author = {A. F. Mastryukov},
     title = {The solution of the inverse problem for the diffusion equation based on {Laguerre} transformation},
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this work a method of solving the inverse problem for the diffusion equations, based on Laguerre spectral transformation, is suggested. The problem is considered in 1D space. Diffusion equation is obtained from Maxwell's equations in the low-frequency limit. By the given solution in a certain point of space a distribution of the conductivity in the media is found. The optimization method of solution is used. The Laguerre's harmonics function is minimized. The minimization is made using the conjugate gradient method or the Newton method. The results of defining the conductivity in the horizontally layered media are presented. An influence of the accuracy of the edge problem approximation upon that of the inverse problem solution, is analyzed. The accuracies of the inverse problem solution method, based on Laguerre transformation, and the method using Fourier transformation, are compared.

[1] Landau L. D., Lifshits E. M., Elektrodinamika sploshnykh sred, Nauka, M., 1982 | MR

[2] Tarkhov A. G. (red.), Elektrorazvedka. Spravochnik geofizika, Nedra, M., 1980

[3] Kashkin V. B., Sukhinin A. I., Distantsionnoe zondirovanie Zemli iz kosmosa, Logos, M., 2001

[4] Mastryukov A. F., “Opredelenie plotnosti, skorosti i koeffitsienta pogloscheniya akusticheskikh voln”, Matem. modelirovanie, 11:10 (1999), 62–76 | MR

[5] Farquharson C. G., Oldenburg D. W., “Inversion of time electromagnetic data for horizontally layered Earth”, Geophysical Journal International, 114:3, 433–442 | DOI

[6] Konyukh G. V., Mikhailenko B. G., “Primenenie integralnogo preobrazovaniya Lagerra pri reshenii dinamicheskikh zadach seismiki.”, Mat. modelirovanie v geofizike, Trudy IVMiMG, 5, Novosibirsk, 1998, 107–112

[7] Mikhailenko B. G., “Spectral Laguerre method for the approximate solution of time dependent problems”, Applied Mathematics Letters, 12 (1999), 105–110 | DOI | MR | Zbl

[8] Mastryukov A. F., Mikhailenko B. G., “Chislennoe modelirovanie rasprostraneniya elektromagnitnykh voln v neodnorodnykh sredakh s zatukhaniem na osnove spektralnogo preobrazovaniya Lagerra”, Geologiya i geofizika, 44:10 (2003), 1060–1069

[9] M. Abramovits, I. Stigun (red.), Spravochnik po spetsialnym funktsiyam, Nauka, M., 1979

[10] Faddeev D. K., Faddeeva V. N., Vychislitelnye metody lineinoi algebry, Nauka, M., 1963 | MR

[11] Fedorenko R. P., Priblizhennoe reshenie zadach optimalnogo upravleniya, Nauka, M., 1978 | MR | Zbl

[12] Vasilev F. P., Chislennye metody resheniya ekstremalnykh zadach, Nauka, M., 1980 | MR

[13] Fornberg B., Ghrist M., “Spatial finite difference approximation for wave-type equations”, SIAM J. Numer.Anal., 37, 105–130 | DOI | MR | Zbl

[14] Ursin B., “Review of elastic and electromagnetic wave propagation in horizontally layered media”, Geophysics, 48 (1983), 1063–1081 | DOI