Construction of solutions of quasilinear parabolic equations in parametric form. (English. Russian original) Zbl 1131.35313

Differ. Equ. 43, No. 4, 507-512 (2007); translation from Differ. Uravn. 43, No. 4, 492-497 (2007).
From the introduction: In the present paper, we suggest a new method for constructing closed formulas for exact solutions of quasilinear parabolic equations. The algorithm is stated in conditional form under the assumption that all functions used in it exist and have the desired smoothness. Once a solution formula has been constructed, one can study all properties of the solution: smoothness, domain, and so on. Here we do not consider the problem of qualitative study of a solution; instead, we deal with an effective construction of the solution in a new way.


35C05 Solutions to PDEs in closed form
35K55 Nonlinear parabolic equations
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[1] Oleinik, O.A., Kalashnikov, A.S., and Zhou Yuilin’, Izv. Akad. Nauk Ser. Mat., 1958, vol. 22, no. 5, pp. 667–704.
[2] Maslov, V.P., Danilov, V.G., and Volosov, K.A., Matematicheskoe modelirovanie tekhnologicheskikh protsessov izgotovleniya BIS (Mathematical Modeling of Technological VLSI Production Processes), Moscow, 1984.
[3] Maslov, V.P., Danilov, V.G., and Volosov, K.A., Matematicheskoe modelirovanie protsessov teplomassoperenosa [evolyutsiya dissipativnykh struktur] (Mathematical Modeling of Heat and Mass Transfer Processes [Evolution of Dissipative Structures]), Moscow: Nauka, 1987. · Zbl 0645.73049
[4] Danilov, V.G., Maslov, V.P., and Volosov, K.A., Mathematical Aspects of Computer Engineering, Moscow: Mir, 1988. · Zbl 0667.35044
[5] Danilov, V.G., Maslov, V.P., and Volosov, K.A., Mathematical Modeling of Heat and Mass Transfer Processes, Dordrecht: Kluwer Acad. Publ., 1995. · Zbl 0839.35001
[6] Engler, H.P., Proc. Acad. Math. Soc., 1985, vol. 93, no. 2, pp. 297–302. · doi:10.1090/S0002-9939-1985-0770540-6
[7] In Matematicheskoe modelirovanie. Protsessy v nelineinykh sredakh (Mathematical Modeling. Processes in Nonlinear Media), Moscow: Nauka, 1986, pp. 142–183, 279–309.
[8] Volosov, K.A., Danilov, V.G., and Maslov, V.P., Mat. Zametki, 1988, vol. 43, no. 6, pp. 829–839.
[9] Volosov, K.A. and Fedotov, I.A., Zh. Vychisl. Mat. Mat. Fiz., 1983, vol. 5, no. 1, pp. 93–101.
[10] Pukhnachev, V.V., Dokl. Akad. Nauk, 1987, vol. 294, no. 3, pp. 535–538.
[11] Galaktionova, V.A. and Posashkov, S.A., Zh. Vychisl. Mat. Mat. Fiz., 1994, vol. 34, no. 3, pp. 373–384.
[12] Volosov, K.A., Danilov, V.G., Kolobov, N.A., and Maslov, V.P., Dokl. Akad. Nauk, 1986, vol. 287, no. 6, pp. 535–538.
[13] Belyaev, N.M., Metody nestatsionarnoi teploprovodnosti (Nonstationary Thermal Conduction Methods), Moscow: Vysshaya Shkola, 1978.
[14] Crank, J., The Mathematics of Diffusion, Oxford: Clarendon Press, 1956. · Zbl 0071.41401
[15] Volosov, K.A., Pavlov, K.B., Romanov, A.S., and Fedotov, I.A., Zh. Prikl. Mekh. Tekh. Fiz., 1982, no. 5, pp. 89–92.
[16] Rudnykh, G.A. and Semenov, E.I., Sib. Mat. Zh., 2000, vol. 41, pp. 1141–1166. · Zbl 0971.46023 · doi:10.1023/A:1004880406077
[17] Galaktionov, V.A., Dorodnitsin, V.A., Elenin, G.E., et al., in Itogi nauki i tekhniki. Sovremennye problemy matematiki. Noveishie dostizheniya (Current Problems in Mathematics. Newest Results), Moscow: Akad. Nauk SSSR, 1987, vol. 28, pp. 95–205.
[18] Polyanin, A.D., Vyaz’min, A.V., Zhurov, A.I., and Kazenin, D.A., Spravochnik po tochnym resheniyam teplo-i massoperenosa (Handbook of Exact Solutions for Heat and Mass Transfer Equations), Moscow: Faktorial, 1998.
[19] Volosov, K.A., Mat. Zametki, 1994, vol. 56, no. 6, pp. 122–126.
[20] Zel’dovich, Ya.B. and Kompaneets, A.S., in K 70-letiyu A.F.Ioffe (Collection Dedicated to the 70th Birthday of Academician A.F. Ioffe), Moscow: Akad. Nauk SSSR, 1950, pp. 61–71.
[21] Kawahara, T. and Tanaka, M., Phys. Lett., 1983, vol. A 97, pp. 311–330. · doi:10.1016/0375-9601(83)90648-5
[22] Volosov, K.A., Differ. Uravn., 2005, vol. 41, no. 11, pp. 1572–1575.
[23] Nucci, M.C. and Clarkson, P.A., Phys. Lett. A, 1992, vol. 164, pp. 49–56. · doi:10.1016/0375-9601(92)90904-Z
[24] Polyanin, A.D. and Zaitsev, V.F., Handbook of Nonlinear Partial Differential Equations, Chapman and Hall, CPC, 2004. · Zbl 1053.35001
[25] Polyanin, A.D., Zaitsev, V.F., and Zhurov, A.I., Metody resheniya nelineinykh uravnenii matematicheskoi fiziki i mekhaniki (Solution Methods for Nonlinear Equations of Mathematical Physics and Mechanics), Moscow: Fizmatlit, 2005.
[26] Vorob’ev, E.V. and Olver, P.J., CRC Handbook of Lie Group Analysis of Differential Equations, CRC Press, 1996, vol. 3.
[27] Volosov, K.A., in Konf. ”Gertsenovskie chteniya,” 17–22 aprelya 2006 g. (Proc. Conf. ”Gertsen Readings,” April 17–22, 2006), St. Petersburg, 2006.
[28] Volosov, K.A., in The Inter. Conf. ”Tikhonov and Contemporary Mathematics,” June 19–24, 2006, M., 2006, pp. 133–134.
[29] Volosov, K.A., in Inter. Conf. of Differ. Equat. And Dynamical Systems, July 10–15, 2006, Suzdal’, 2006, pp. 56–60.
[30] Volosov, K.A., IUTAM Symp. Aug. 25–30, 2006, Steklov Math. Inst., pp. 147–149.
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