Solitary wave and other solutions for nonlinear heat equations
Anatoly Nikitin; Tetyana Barannyk
Open Mathematics (2004)
- Volume: 2, Issue: 5, page 840-858
- ISSN: 2391-5455
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topAnatoly Nikitin, and Tetyana Barannyk. "Solitary wave and other solutions for nonlinear heat equations." Open Mathematics 2.5 (2004): 840-858. <http://eudml.org/doc/268807>.
@article{AnatolyNikitin2004,
abstract = {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.},
author = {Anatoly Nikitin, Tetyana Barannyk},
journal = {Open Mathematics},
keywords = {35K55; 35Q51; 35Q80},
language = {eng},
number = {5},
pages = {840-858},
title = {Solitary wave and other solutions for nonlinear heat equations},
url = {http://eudml.org/doc/268807},
volume = {2},
year = {2004},
}
TY - JOUR
AU - Anatoly Nikitin
AU - Tetyana Barannyk
TI - Solitary wave and other solutions for nonlinear heat equations
JO - Open Mathematics
PY - 2004
VL - 2
IS - 5
SP - 840
EP - 858
AB - 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.
LA - eng
KW - 35K55; 35Q51; 35Q80
UR - http://eudml.org/doc/268807
ER -
References
top- [1] J.D. Murray: Mathematical Biology, Springer, 1991. Zbl0704.92001
- [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). Zbl0187.25102
- [5] V.A. Dorodnitsyn: “On invariant solutions of nonlinear heat equation with source”, Comp. Meth. Phys., Vol. 22(115), (1982). Zbl0535.35040
- [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). Zbl0187.03502
- [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). Zbl0694.35159
- [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). Zbl0727.35066
- [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). Zbl0788.35067
- [14] E. Fan: “Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method”, J. Phys. A, Vol. 35(6853), (2002). Zbl1039.35029
- [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). Zbl0423.35079
- [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). Zbl0850.35097
- [18] R.Z. Zhdanov and V.I. Lahno: “Conditional symmetry of a porous medium equation”, Physica D, Vol. 122(178), (1998). Zbl0952.76087
- [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).
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