Geodesic
Lambert function and exact solutions of nonlinear parabolic equations
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2019), pp. 13-20.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the diffusion equations with degree type coefficient of diffusion and a nonlinear source. The main attention is paid to the construction of exact solutions expressed via the Lambert function. We prove a series of statements that determine the conditions for the source function that guarantee the existence of exact solutions of a certain type. We give examples of exact solutions of nonlinear diffusion equations (including those equations with polynomial and fractional-rational source functions) to illustrate the obtained results.
Keywords: equation of nonlinear diffusion, Lambert's function
Mots-clés : exact solutions. 
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}
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A. A. Kosov; E. I. Semenov. Lambert function and exact solutions of nonlinear parabolic equations. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2019), pp. 13-20. http://geodesic.mathdoc.fr/item/IVM_2019_8_a1/

[1] Wright E. M., “Solution of the equation $ze^{z}=a$”, Proceedings of the Royal Society Edinburgh, 65 (1959), 193–203 | MR | Zbl

[2] Corless R. M., Gonnet G. H., Hare D. E., Jeffrey D. J., Knuth D. E., “On the Lambert $W$-function”, Advances in Comput. Math., 5 (1996), 329–359 | DOI | MR | Zbl

[3] Valluri S. R., Jeffrey D. J., Corless R. M., “Some applications of the Lambert $W$-function to physics”, Canadian J. Phys., 78 (2000), 823–831 | MR

[4] Dubinova I. D., “Primenenie $W$-funktsii Lamberta v matematicheskikh zadachakh fiziki plazmy”, Fizika plazmy, 30:10 (2004), 937–943

[5] Dubinov A. E., Dubinova I. D., Saikov S. K., $W$-funktsiya Lamberta i ee primenenie v matematicheskikh zadachakh fiziki, RFYaTs VNIIEF, Sarov, 2006

[6] Samarskii A. A., Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P., Rezhimy s obostreniem v zadachakh dlya kvazilineinykh parabolicheskikh uravnenii, Nauka, M., 1987 | MR

[7] Rudykh G. A., Semenov E. I., “Novye tochnye resheniya odnomernogo uravneniya nelineinoi diffuzii”, Sib. matem. zhurn., 38:5 (1997), 1130–1139 | MR | Zbl

[8] Rudykh G. A., Semenov E. I., “O novykh tochnykh resheniyakh odnomernogo uravneniya nelineinoi diffuzii s istochnikom (stokom)”, Zhurn. vychisl. matem. i matem. fiziki, 38:6 (1998), 971–977 | MR | Zbl

[9] Polyanin A. D., Zaitsev V. F., Spravochnik po nelineinym uravneniyam matematicheskoi fiziki: Tochnye resheniya, Fizmatlit, M., 2002

[10] Polyanin A. D., Zaitsev V. F., Nelineinye uravneniya matematicheskoi fiziki, Ucheb. posobie, v. 1, YuRAIT, M., 2017

[11] Polyanin A. D., Zaitsev V. F., Nelineinye uravneniya matematicheskoi fiziki, Ucheb. posobie, v. 2, YuRAIT, M., 2017

[12] Zhuravlev V. M., “Printsip superpozitsii i tochnye resheniya uravneniya nelineinoi diffuzii”, TMF, 183:1 (2015), 36–50 | DOI | MR | Zbl

[13] Barenblatt G. I., “O predelnykh avtomodelnykh dvizheniyakh v teorii nestatsionarnoi filtratsii gaza v poristoi srede i teorii pogranichnogo sloya”, PMM, 18:4 (1954), 409–414 | Zbl

[14] Barenblatt G. I., Podobie, avtomodelnost, promezhutochnaya asimptotika, Gidrometeoizdat, L., 1978 | MR

[15] Zaitsev V. F., Polyanin A. D., Spravochnik po obyknovennym differentsialnym uravneniyam, Fizmatlit, M., 2001 | MR

[16] Ibragimov N. Kh., Gruppy preobrazovanii v matematicheskii fizike, Nauka, M., 1983 | MR