PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
2004 | 2 | 5 | 840-858
Tytuł artykułu

Solitary wave and other solutions for nonlinear heat equations

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
A number of explicit solutions for the heat equation with a polynomial non-linearity and for the Fisher equation is presented. An extended class of non-linear heat equations admitting solitary wave solutions is described. The generalization of the Fisher equation is proposed whose solutions propagate with arbitrary ad hoc fixed velocity.
Wydawca
Czasopismo
Rocznik
Tom
2
Numer
5
Strony
840-858
Opis fizyczny
Daty
wydano
2004-10-01
online
2004-10-01
Twórcy
  • National Academy of Sciences
Bibliografia
  • [1] J.D. Murray: Mathematical Biology, Springer, 1991.
  • [2] A.N. Kolmogorov, I.G. Petrovskii and N.S. Piskunov: “A study of the diffusion equation with increase in the quantity of matter and its application to a biological problem”, Bull. Moscow Univ. Sér. Int. A, Vol. 1(1), (1937).
  • [3] R. Fitzhugh: “Impulses and physiological states in models of nerve membrane”, Biophys. J., Vol. 1(445), 1961; J.S. Nagumo, S. Arimoto and S. Yoshizawa: “An active pulse transmission line simulating nerve axon”, Proc. IRE, Vol. 50(2061), (1962).
  • [4] A.C. Newell and J.A. Whitehead: “Finite bandwidth, finite amplitude convection”, J. Fluid Mech., Vol. 38(279), (1969).
  • [5] V.A. Dorodnitsyn: “On invariant solutions of nonlinear heat equation with source”, Comp. Meth. Phys., Vol. 22(115), (1982).
  • [6] S. Lie: Transformationgruppen, Leipzig, 1883.
  • [7] P. Olver: Application of Lie groups to differential equations, Springer, Berlin, 1986.
  • [8] A.G. Nikitin and R. Wiltshire: “Symmetries of Systems of Nonlinear Reaction-Diffusion Equations”, In: A.M. Samoilenko (Ed.): Symmetries in Nonlinear Mathematical Physics, Proc. of the Third Int. Conf., Kiev, July 12–18, 1999, Inst. of Mathematics of Nat. Acad. Sci. of Ukraine, Kiev, 2000; R. Cherniha and J. King: “Lie symmetries of nonlinear multidimensional reaction-diffusion systems: I”, J. Phys. A, Vol. 33(257), (2000); A.G. Nikitin and R. Wiltshire: “Systems of Reaction Diffusion Equations and their symmetry properties”, J. Math. Phys., Vol. 42(1666), (2001).
  • [9] G.W. Bluman and G.D. Cole: “The general similarity solution of the heat equation”, J. Math. Mech., Vol. 18(1025), (1969).
  • [10] W.I. Fushchych and A.G. Nikitin: Symmetries of Maxwell's equations, Reidel, Dordrecht, 1987; W.I. Fushchych: “Conditional symmetry of mathematical physics equations”, Ukr. Math. Zh., Vol. 43(1456), 1991.
  • [11] D. Levi and P. Winternitz: “Non-classical symmetry reduction: example of the Boussinesq equation”, J. Phys. A, Vol. 22(2915), (1989).
  • [12] W.I. Fushchich and M.I. Serov: “Conditional invariance and reduction of the nonlinear heat equation”, Dokl. Akad. Nauk Ukr. SSR, Ser. A, Vol. 4(24), (1990).
  • [13] P.A. Clarkson and E.L. Mansfield: “Symmetry reductions and exact solutions of a class of nonlinear heat equations”, Physica D, Vol. 70(250), (1993).
  • [14] E. Fan: “Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method”, J. Phys. A, Vol. 35(6853), (2002).
  • [15] R. Hirota and J. Satsuma: “Soliton solutions of a coupled Korteweg-de Vries equation”, Phys. Lett. A, Vol. 85(07), (1981).
  • [16] M.J. Ablowitz and A. Zeppetella: “Explicit solution of Fisher's equation for a special wave speed”, Bull. Math. Biol., Vol. 41(835), (1979).
  • [17] A.S. Fokas and Q.M. Liu: “Generalized Conditional Symmetries and Exact Solutions of Non Integrable Equations”, Theor. Math. Phys., Vol. 99(371), (1994).
  • [18] R.Z. Zhdanov and V.I. Lahno: “Conditional symmetry of a porous medium equation”, Physica D, Vol. 122(178), (1998).
  • [19] D.J. Needham and A.C. King: “The evolution of travelling waves in the weakly hyperbolic generalized Fisher model”, Proc. Roy. Soc. (London) Vol. 458(1055), (2002). P.S. Bindu, M. Santhivalavan and M. Lakshmanan: “Singularity structure, symmetries and integrability of generalized Fisher-type nonlinear diffusion equation”, J. Phys. A, Vol. 34(l689), (2001).
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_BF02475981
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.