Abstract
New classes of exact solutions to nonlinear hyperbolic reaction—diffusion equations with delay are described. All of the equations under consideration depend on one or two arbitrary functions of one argument, and the derived solutions contain free parameters (in certain cases, there can be any number of these parameters). The following solutions are found: periodic solutions with respect to time and space variable, solutions that describe the nonlinear interaction between a standing wave and a traveling wave, and certain other solutions. Exact solutions are also presented for more complex nonlinear equations in which delay arbitrarily depends on time. Conditions for the global instability of solutions to a number of reaction—diffusion systems with delay are derived. The generalized Stokes problem subject to the periodic boundary condition, which is described by a linear diffusion equation with delay, is solved.
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References
Carslaw, H.S. and Jaeger, J.C., Conduction of Heat in Solids, Oxford: Oxford Univ. Press, 1959, 2nd ed.
Lykov, A.V., Teoriya teploprovodnosti (Heat Conduction Theory), Moscow: Vysshaya Shkola, 1967.
Kutateladze, S.S., Osnovy teorii teploobmena (Fundamentals of Heat Transfer Theory), Moscow: Atomizdat, 1979.
Planovskii, A.N. and Nikolaev, P.I., Protsessy i apparaty khimicheskoi i neftekhimicheskoi tekhnologii (Processes and Apparatuses in Chemical and Petrochemical Technology), Moscow: Khimiya, 1987, 3rd ed.
Kutepov, A.M., Polyanin, A.D., Zapryanov, Z.D., Vyazmin, A.V., and Kazenin, D.A., Khimicheskaya gidrodinamika (Chemical Hydrodynamics), Moscow: Byuro Kvantum, 1996.
Polyanin, A.D., Kutepov, A.M., Vyazmin, A.V., and Kazenin, D.A., Hydrodynamics, Mass and Heat Transfer in Chemical Engineering, London: Taylor & Francis, 2002.
Cattaneo, C., A form of heat conduction equation which eliminates the paradox of instantaneous propagation, Comptes Rendus, 1958, vol. 247, p. 431.
Vernotte, P., Some possible complications in the phenomena of thermal conduction, Comptes Rendus, 1961, vol. 252, p. 2190.
Lykov, A.V., Teplomassoobmen: spravochnik (Heat and Mass Transfer: A Handbook), Moscow: Energiya, 1978.
Taganov, I.N., Modelirovanie protsessov massoi energoperenosa (Modeling of Mass and Energy Transfer Processes), Leningrad: Khimiya, 1979.
Shashkov, A.G., Bubnov, V.A., and Yanovskii, S.Yu., Volnovye yavleniya teploprovodnosti: sistemno-strukturnyi podkhod (Wave Phenomena in Heat Conduction: A Systems Approach), Moscow: Editorial URSS, 2004.
Mitra, K., Kumar, S., Vedavarz, A., and Moallemi, M.K., Experimental evidence of hyperbolic heat conduction in processed meat, J. Heat Transfer, 1995, vol. 117, no. 3, p. 568.
Demirel, Y., Nonequilibrium Thermodynamics: Transport and Rate Processes in Physical, Chemical and Biological Systems, Amsterdam: Elsevier, 2007, 2nd ed.
Ordonez-Miranda, J. and Alvarado-Gil, J.J., Thermal wave oscillations and thermal relaxation time determination in a hyperbolic heat transport model, Int. J. Therm. Sci., 2009, vol. 48, p. 2053.
Roetzel, W., Putra, N., and Saritdas, K., Experiment and analysis for non-Fourier conduction in materials with nonhomogeneous inner structure, Int. J. Therm. Sci., 2003, vol. 42, no. 6, p. 541.
Kalospiros, N.S., Edwards, B.J., and Beris, A.N., Internal variables for relaxation phenomena in heat and mass transfer, Int. J. Heat Mass Transfer, 1993, vol. 36, p. 1191.
Polyanin, A.D. and Vyazmin, A.V., Differential-difference heat-conduction and diffusion models and equations with a finite relaxation time, Theor. Found. Chem. Eng., 2013, vol. 47, no. 3, p. 217.
Polyanin, A.D. and Zhurov, A.I., Exact solutions of linear and non-linear differential-difference heat and diffusion equations with finite relaxation time, Int. J. NonLinear Mech., 2013, vol. 54, pp. 115–126.
Polyanin, A.D., Exact solutions to differential-difference heatand mass-transfer equations with a finite relaxation time, Theor. Found. Chem. Eng., 2014, vol. 48, no. 2, pp. 167–174.
Jou, D., Casas-Vazquez, J., and Lebon, G., Extended Irreversible Thermodynamics, New York: Springer, 2010, 4th ed.
Wu, J., Theory and Applications of Partial Functional Differential Equations, New York: Springer-Verlag, 1996.
Smith, H.L. and Zhao, X.-Q., Global asymptotic stability of travelling waves in delayed reaction–diffusion equations, SIAM J. Math. Anal., 2000, vol. 31, pp. 514–534.
Wu, J. and Zou, X., Traveling wave fronts of reaction–diffusion systems with delay, J. Dyn. Differ. Equations, 2001, vol. 13, no. 3, pp. 651–687.
Huang, J. and Zou, X., Traveling wavefronts in diffusive and cooperative Lotka–Volterra system with delays, J. Math. Anal. Appl., 2002, vol. 271, pp. 455–466.
Faria, T. and Trofimchuk, S., Nonmonotone travelling waves in a single species reaction–diffusion equation with delay, J. Differ. Equations, 2006, vol. 228, pp. 357–376.
Meleshko, S.V. and Moyo, S., On the complete group classification of the reaction–diffusion equation with a delay, J. Math. Anal. Appl., 2008, vol. 338, pp. 448–466.
Polyanin, A.D. and Zhurov, A.I., Exact separable solutions of delay reaction–diffusion equations and other nonlinear partial functional-differential equations, Commun. Nonlinear Sci. Numer. Simul., 2014, vol. 19, no. 3, pp. 409–416.
Polyanin, A.D. and Zhurov, A.I., Functional constraints method for constructing exact solutions to delay reaction–diffusion equations and more complex nonlinear equations, Commun. Nonlinear Sci. Numer. Simul., 2014, vol. 19, no. 3, pp. 417–430.
Polyanin, A.D. and Zhurov, A.I., New generalized and functional separable solutions to non-linear delay reaction–diffusion equations, Int. J. Non-Linear Mech., 2014, vol. 59, pp. 16–22.
Polyanin, A.D. and Zhurov, A.I., Non-linear instability and exact solutions to some delay reaction–diffusion systems, Int. J. Non-Linear Mech., 2014, vol. 62, pp. 33–40.
Polyanin, A.D., Exact generalized separable solutions to nonlinear delay reaction–diffusion equations, Theor. Found. Chem. Eng., 2015, vol. 49, no. 1, pp. 107–114.
Polyanin, A.D. and Zhurov, A.I., Nonlinear delay reaction–diffusion equations with varying transfer coefficients: exact methods and new solutions, Appl. Math. Lett., 2014, vol. 37, pp. 43–48.
Polyanin, A.D. and Zhurov, A.I., The functional constraints method: application to non-linear delay reaction–diffusion equations with varying transfer coefficients, Int. J. Non-Linear Mech., 2014, vol. 67, pp. 267–277.
Polyanin, A.D., Exact solutions to new classes of reaction–diffusion equations containing delay and arbitrary functions, Theor. Found. Chem. Eng., 2015, vol. 49, no. 2, pp. 169–175.
Jordan, P.M., Dai, W., and Mickens, R.E., A note on the delayed heat equation: instability with respect to initial data, Mech. Res. Commun., 2008, vol. 35, no. 6, p. 414.
Polyanin, A.D. and Zhurov, A.I., Exact solutions of non-linear differential-difference equations of a viscous fluid with finite relaxation time, Int. J. Non-Linear Mech., 2013, vol. 57, no. 5, pp. 116–122.
Bellman, R. and Cooke, K.L., Differential-Difference Equations, New York: Academic, 1963.
Driver, R.D., Ordinary and Delay Differential Equations, New York: Springer, 1977.
Kuang, Y., Delay Differential Equations with Applications in Population Dynamics, Boston: Academic, 1993.
Smith, H.L., An Introduction to Delay Differential Equations with Applications to the Life Sciences, New York: Springer, 2010.
Tanthanuch, J., Symmetry analysis of the nonhomogeneous inviscid Burgers equation with delay, Commun. Nonlinear Sci. Numer. Simul., 2012, vol. 17, no. 12, pp. 4978–4987.
Polyanin, A.D. and Zhurov, A.I., Generalized and functional separable solutions to nonlinear delay Klein–Gordon equations, Commun. Nonlinear Sci. Numer. Simul., 2014, vol. 19, no. 8, pp. 2676–2689.
He, Q., Kang, L., and Evans, D.J., Convergence and stability of the finite difference scheme for nonlinear parabolic systems with time delay, Numer. Algorithms, 1997, vol. 16, no. 2, p. 129.
Pao, C.V., Numerical methods for systems of nonlinear parabolic equations with time delays, J. Math. Anal. Appl., 1999, vol. 240, no. 1, p. 249.
Jackiewicza, Z. and Zubik-Kowal, B., Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations, Appl. Numer. Math., 2006, vol. 56, nos. 3–4, p. 433.
Zhang, Q. and Zhang, C., A new linearized compact multisplitting scheme for the nonlinear convection–reaction–diffusion equations with delay, Commun. Nonlinear Sci. Numer. Simul., 2013, vol. 18, no. 12, p. 3278.
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: Gos. Izd. Fiz.-Mat. Literatury, 2005.
Polyanin, A.D. and Manzhirov, A.V., Handbook of Mathematics for Engineers and Scientists, Boca Raton, Fla.: Chapman & Hall/CRC, 2007.
Galaktionov, V.A. and Svirshchevskii, S.R., Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, Boca Raton, Fla.: Chapman & Hall/CRC, 2007.
Polyanin, A.D. and Zaitsev, V.F., Handbook of Nonlinear Partial Differential Equations, Boca Raton, Fla.: Chapman & Hall/CRC, 2012, 2nd ed.
Polyanin, A.D. and Zhurov, A.I., The functional constraints method: exact solutions to nonlinear reaction–diffusion equations with delay, Vestn. Nats. Issled. Yadern. Univ. MIFI, 2013, vol. 2, no. 4, p. 425.
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Original Russian Text © A.D. Polyanin, V.G. Sorokin, A.V. Vyazmin, 2015, published in Teoreticheskie Osnovy Khimicheskoi Tekhnologii, 2015, Vol. 49, No. 5, pp. 527–541.
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Polyanin, A.D., Sorokin, V.G. & Vyazmin, A.V. Exact solutions and qualitative features of nonlinear hyperbolic reaction—diffusion equations with delay. Theor Found Chem Eng 49, 622–635 (2015). https://doi.org/10.1134/S0040579515050243
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DOI: https://doi.org/10.1134/S0040579515050243