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2013 | 11 | 8 | 1375-1391
Tytuł artykułu

Numerical investigation of a new class of waves in an open nonlinear heat-conducting medium

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Języki publikacji
EN
Abstrakty
EN
The paper contributes to the problem of finding all possible structures and waves, which may arise and preserve themselves in the open nonlinear medium, described by the mathematical model of heat structures. A new class of self-similar blow-up solutions of this model is constructed numerically and their stability is investigated. An effective and reliable numerical approach is developed and implemented for solving the nonlinear elliptic self-similar problem and the parabolic problem. This approach is consistent with the peculiarities of the problems - multiple solutions of the elliptic problem and blow-up solutions of the parabolic one.
Twórcy
Bibliografia
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  • [2] Akhromeeva T.S., Kurdyumov S.P., Malinetskii G.G., Samarskii A.A., Nonstationary Structures and Diffusion Chaos, Nauka, Moscow, 1992 (in Russian)
  • [3] Bakirova M.I., Dimova S.N., Dorodnitsyn V.A., Kurdyumov S.P., Samarskii A.A., Svirshchevskii S.R., Invariant solutions of the heat equation that describe the directed propagation of combustion and spiral waves in a nonlinear medium, Soviet Phys. Dokl., 1988, 33(3), 187–189
  • [4] Cho C.-H., On the finite difference approximation for blow-up solutions of the porous medium equation with a source, Appl. Numer. Math., 2013, 65, 1–26 http://dx.doi.org/10.1016/j.apnum.2012.11.001
  • [5] Dimova M., Dimova S., Numerical investigation of spiral structure solutions of a nonlinear elliptic problem, In: Numerical Methods and Applications, Borovets, August 20–24, 2010, Lecture Notes in Comput. Sci., 6046, Springer, Berlin-Heidelberg, 2011, 395–403
  • [6] Dimova S.N., Kaschiev M.S., Koleva M.G., Investigation of eigenfunctions for combustion of nonlinear medium in polar coordinates with finite elements method, Mat. Model., 1992, 4(3), 76–83 (in Russian)
  • [7] Dimova S., Kaschiev M., Koleva M., Vasileva D., Numerical analysis of radially nonsymmetric blow-up solutions of a nonlinear parabolic problem, J. Comput. Appl. Math., 1998, 97(1–2), 81–97 http://dx.doi.org/10.1016/S0377-0427(98)00103-4
  • [8] Dimova S.N., Kaschiev M.S., Kurdyumov S.P., Numerical analysis of the eigenfunctions of the burning of a nonlinear medium in the radially symmetric case, U.S.S.R. Comput. Math. and Math. Phys., 1991, 29(6), 61–73 http://dx.doi.org/10.1016/S0041-5553(89)80008-4
  • [9] Dimova S.N., Vasileva D.P., Numerical realization of blow-up spiral wave solutions of a nonlinear heat-transfer equation, Internat. J. Numer. Methods Heat Fluid Flow, 1994, 4(6), 497–511 http://dx.doi.org/10.1108/EUM0000000004052
  • [10] Elenin G.G., Kurdyumov S.P., Samarskii A.A., Nonstationary dissipative structures in a nonlinear heat-conducting medium, Zh. Vychisl. Mat. i Mat. Fiz., 1983, 23(2), 380–390 (in Russian)
  • [11] Ferreira R., Groisman P., Rossi J.D., Numerical blow-up for the porous medium equation with a source, Numer. Methods Partial Differential Equations, 2004, 20(4), 552–575 http://dx.doi.org/10.1002/num.10103
  • [12] Galaktionov V.A., Dorodnitsyn V.A., Elenin G.G., Kurdyumov S.P., Samarskii A.A., A quasilinear equation of heat conduction with a source: peaking, localization, symmetry, exact solutions, asymptotic behavior, structures, J. Soviet Math., 1988, 41(5), 1222–1292 http://dx.doi.org/10.1007/BF01098785
  • [13] Gavurin M.K., Nonlinear functional equations and continuous analogues of iteration methods, Izv. Vysš. Učebn. Zaved. Matematika, 1958, 5(6), 18–31 (in Russian)
  • [14] Hajipour M., Malek A., High accurate NRK and MWENO scheme for nonlinear degenerate parabolic PDEs, Appl. Math. Model., 2012, 36(9), 4439–4451 http://dx.doi.org/10.1016/j.apm.2011.11.069
  • [15] Kurdyumov S.P., Kurkina E.S., Potapov A.B., Samarskii A.A., The architecture of multidimensional thermal structures, Soviet Phys. Dokl., 1984, 29, 106–108
  • [16] Kurkina E.S., Nikol’skii I.M., Bifurcation analysis of the spectrum of two-dimensional thermal structures evolving with blow-up, Comput. Math. Model., 2006, 17(4), 320–340 http://dx.doi.org/10.1007/s10598-006-0027-z
  • [17] Kurkina E.S., Nikol’skii I.M., Stability and localization of unbounded solutions of a nonlinear heat equation in a plane, Comput. Math. Model., 2009, 20(4), 348–366 http://dx.doi.org/10.1007/s10598-009-9042-1
  • [18] Le Roux A.-Y., Le Roux M.-N., Numerical solution of a Cauchy problem for nonlinear reaction diffusion processes, J. Comput. Appl. Math., 2008, 214(1), 90–110 http://dx.doi.org/10.1016/j.cam.2007.02.035
  • [19] Novikov V.A., Novikov E.A., Control of the stability of explicit one-step methods of integration of ordinary differential equations, Dokl. Akad. Nauk SSSR, 1984, 277(5), 1058–1062 (in Russian)
  • [20] Puzynin I.V., Boyadzhiev T.L., Vinitskii S.I., Zemlyanaya E.V., Puzynina T.P., Chuluunbaatar O., Methods of computational physics for investigation of models of complex physical systems, Physics of Particles and Nuclei, 2007, 38(1), 70–116 http://dx.doi.org/10.1134/S1063779607010030
  • [21] Puzynin I.V., Puzynina T.P., SLIP1 - Program for the numerical solution of the Sturm-Liouville problem basing on the continuous analog of the Newton method, In: Algorithms and Programs for Solution of Some Problems in Physics, KFKI-74-34, Central Research Institute for Physics, Hungarian Academy of Sciences, Budapest, 1974, 93–112
  • [22] Samarskii A.A., Galaktionov V.A., Kurdyumov S.P., Mikhailov A.P., Blow-Up in Quasilinear Parabolic Equations, de Gruyter Exp. Math., 19, Walter de Gruyter, Berlin, 1995 http://dx.doi.org/10.1515/9783110889864
  • [23] Samarskii A.A., Zmitrenko N.V., Kurdyumov S.P., Mikhailov A.P., Thermal structures and fundamental length in a medium with non-linear heat conduction and volumetric heat sources, Soviet Phys. Dokl., 1976, 21, 141–143
  • [24] Thomée V., Galerkin Finite Element Methods for Parabolic Problems, Lecture Notes in Math., 1054, Springer, Berlin, 1984
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-013-0253-5
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