Abstract
A reduction aproach is developed for determining exact solutions of anonlinear second order parabolic PDE. The method in point makes acomplementary use of the leading ideas of the theory of quasilinearhyperbolic systems of first order endowed by differential constraintsand of the techniques providing multiple wave-like solutions ofnonlinear PDEs. The searched solutions exhibit a inherent wave featuresand they are obtained by solving a consistent overdetermined system ofPDEs. Remarkably, in the process it is possible to define nonlinearmodel equations which allow special classes of initial or boundary valueproblems to be solved in a closed form. Within the present reductionapproach exact solutions and model material response functions areobtained for an equation of widespread application in many fields ofinterest.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Yanenko, N. N., 'Compatibility theory and methods of integration of systems of nonlinear partial differential equations', in Proceedings of the Fourth All-Union Mathematics Congress, Nauka, Leningrad, 1964, p. 613.
Sidorov, A. F., Shapeev, V. P., and Yanenko, N. N., The Method of Differential Constraints and Its Applications in Gas Dynamics, Nauka, Novosibirsk, 1984.
Shapeev, V. P., 'A problem of continuous adjoint of DP-solutions of one-dimensional inelastic continuum media', Chisl. metody mech. splosh. sredy 5(4), 1974, 116.
Kaptsov, O. V., 'B-determining equations: Applications to nonlinear partial differential equations', European Journal of Applied Mathematics 6, 1995, 265.
Boillat, G., 'Involutions des systèmes conservatifs', Comptes Rendus Académie des Sciences, Paris I 307, 1988, 891.
Fusco, D. and Manganaro, N., 'A method for finding exact solutions to hyperbolic systems of first-order', IMA Journal of Applied Mathematics 57, 1996, 223.
Fusco, D., Manganaro, N., and Migliardo, M., 'Pulselike exact solutions for a model describing nerve fibers', Studies in Applied Mathematics 103, 1999, 25.
Manganaro, N. and Meleshko, S., 'Reduction procedure and generalized simple waves for systems written in Riemann variables', Nonlinear Dynamics 30, 2002, 87–102.
Ibragimov, N., Transformation Groups Applied to Mathematical Physics, Nauka, Moscow, 1983 [English translation by Reidel, Dordrecht, 1989].
Ovsiannikov, L. V., Group Analysis of Differential Equations, Nauka, Moscow, 1978 [English translation, Ames, W. F., ed., Academic Press, New York, 1982].
Meleshko, S., On classification of plane isentropic double wave type gas flow, Prikladnaya Matematika i Mekhanika 49, 1982.
Yanenko, N. N., 'Travelling waves of a system of quasilinear equation', Doklady Akademii Nauk S.S.S.R. 109(1), 1956, 44.
Currò, C. and Fusco, D., 'A reduction method for quasilinear hyperbolic systems of multicomponent field PDEs with application to wave interaction', International Journal of Non-Linear Mechanics 37, 2002, 281.
Fusco, D. and Manganaro, N., 'Generalized simple wave solutions to a class of hyperbolic dissipative models', preprint.
Ames, W. F., Nonlinear Partial Differential Equations in Engineering, Academic Press, New York, 1972.
Aris, R., The Mathematical Theory of Diffusion and Reaction Impermeable Catalysts, Clarendon Press, Oxford, 1975.
Murray, J. D., Nonlinear Differential Equation Models in Biology, Clarendon Press, Oxford, 1977.
Cherniha, R. and Serov, M., 'Symmetries, ansatz and exact solutions of nonlinear second-order evolution equations with convection terms', European Journal of Applied Mathematics 9, 1998, 527.
Gilding, B. H. and Kersner, R., 'The characterization of reaction-convection-diffusion processes by travelling waves', Journal of Differential Equations 124, 1996, 27.
Bluman, G.W. and Cole, J. D., 'The general similarity solution of the heat equation', Journal of Mathematics and Mechanics 18, 1969, 1025.
Meleshko, S., 'DP-conditions and the adjoin problem of different DP-solutions', Chisl. Metody Mech. Splosh. Sredy 11(6), 1980, 96.
Gzigzin, A. E. and Shapeev, V. P., 'To the problem about continuous adjoining of particular solutions of systems of partial differential equations', Chisl. Metody Mech. Splosh. Sredy 6(7), 1975, 44.
Gzigzin, A. E., 'On integrability of some nonanalytic involutive systems', Doklady AS USSR 238(1), 1978, 15.
Rozhdestvenskii, B. L. and Yanenko, N. N., Systems of Quasilinear Equations and Their Applications to Gas Dynamics, 2nd edn., Nauka, Moscow, 1978 [English translation published by American Mathematical Society, Providence, RI, 1983].
Meleshko, S. V., 'One class of solutions for the systems of quasilinear differential equations with many independent variables', Chisl. Metody Mech. Splosh. Sredy 12(4), 1981, 87.
Meleshko, S. V., 'A solution of wave problem for dynamics of nonelastic continuous media with method of differential constraints', Ph.D. Thesis, Institute of Hydrodynamics, Novosibirsk, 1981.
Smoller, J., Shock Waves and Reaction-Diffusion Equation, Springer, Berlin, 1983.
Murray, J. D., Mathematical Biology, Springer, Berlin, 1989.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Carini, M., Fusco, D. & Manganaro, N. Wave-Like Solutions for a Class of Parabolic Models. Nonlinear Dynamics 32, 211–222 (2003). https://doi.org/10.1023/A:1024408401310
Issue Date:
DOI: https://doi.org/10.1023/A:1024408401310