Theoretical Foundations of Chemical Engineering

, Volume 49, Issue 5, pp 622–635 | Cite as

Exact solutions and qualitative features of nonlinear hyperbolic reaction—diffusion equations with delay

Article

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.

Keywords

nonlinear reaction–diffusion equations with delay exact solutions generalized separable solutions functional separable solutions delay differential equations global instability of solutions generalized Stokes problem 

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References

  1. 1.
    Carslaw, H.S. and Jaeger, J.C., Conduction of Heat in Solids, Oxford: Oxford Univ. Press, 1959, 2nd ed.Google Scholar
  2. 2.
    Lykov, A.V., Teoriya teploprovodnosti (Heat Conduction Theory), Moscow: Vysshaya Shkola, 1967.Google Scholar
  3. 3.
    Kutateladze, S.S., Osnovy teorii teploobmena (Fundamentals of Heat Transfer Theory), Moscow: Atomizdat, 1979.Google Scholar
  4. 4.
    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.Google Scholar
  5. 5.
    Kutepov, A.M., Polyanin, A.D., Zapryanov, Z.D., Vyazmin, A.V., and Kazenin, D.A., Khimicheskaya gidrodinamika (Chemical Hydrodynamics), Moscow: Byuro Kvantum, 1996.Google Scholar
  6. 6.
    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.Google Scholar
  7. 7.
    Cattaneo, C., A form of heat conduction equation which eliminates the paradox of instantaneous propagation, Comptes Rendus, 1958, vol. 247, p. 431.Google Scholar
  8. 8.
    Vernotte, P., Some possible complications in the phenomena of thermal conduction, Comptes Rendus, 1961, vol. 252, p. 2190.Google Scholar
  9. 9.
    Lykov, A.V., Teplomassoobmen: spravochnik (Heat and Mass Transfer: A Handbook), Moscow: Energiya, 1978.Google Scholar
  10. 10.
    Taganov, I.N., Modelirovanie protsessov massoi energoperenosa (Modeling of Mass and Energy Transfer Processes), Leningrad: Khimiya, 1979.Google Scholar
  11. 11.
    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.Google Scholar
  12. 12.
    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.CrossRefGoogle Scholar
  13. 13.
    Demirel, Y., Nonequilibrium Thermodynamics: Transport and Rate Processes in Physical, Chemical and Biological Systems, Amsterdam: Elsevier, 2007, 2nd ed.Google Scholar
  14. 14.
    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.CrossRefGoogle Scholar
  15. 15.
    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.CrossRefGoogle Scholar
  16. 16.
    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.CrossRefGoogle Scholar
  17. 17.
    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.CrossRefGoogle Scholar
  18. 18.
    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.CrossRefGoogle Scholar
  19. 19.
    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.CrossRefGoogle Scholar
  20. 20.
    Jou, D., Casas-Vazquez, J., and Lebon, G., Extended Irreversible Thermodynamics, New York: Springer, 2010, 4th ed.CrossRefGoogle Scholar
  21. 21.
    Wu, J., Theory and Applications of Partial Functional Differential Equations, New York: Springer-Verlag, 1996.CrossRefGoogle Scholar
  22. 22.
    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.CrossRefGoogle Scholar
  23. 23.
    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.CrossRefGoogle Scholar
  24. 24.
    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.CrossRefGoogle Scholar
  25. 25.
    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.CrossRefGoogle Scholar
  26. 26.
    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.CrossRefGoogle Scholar
  27. 27.
    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.CrossRefGoogle Scholar
  28. 28.
    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.CrossRefGoogle Scholar
  29. 29.
    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.CrossRefGoogle Scholar
  30. 30.
    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.CrossRefGoogle Scholar
  31. 31.
    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.CrossRefGoogle Scholar
  32. 32.
    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.CrossRefGoogle Scholar
  33. 33.
    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.CrossRefGoogle Scholar
  34. 34.
    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.CrossRefGoogle Scholar
  35. 35.
    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.CrossRefGoogle Scholar
  36. 36.
    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.CrossRefGoogle Scholar
  37. 37.
    Bellman, R. and Cooke, K.L., Differential-Difference Equations, New York: Academic, 1963.Google Scholar
  38. 38.
    Driver, R.D., Ordinary and Delay Differential Equations, New York: Springer, 1977.CrossRefGoogle Scholar
  39. 39.
    Kuang, Y., Delay Differential Equations with Applications in Population Dynamics, Boston: Academic, 1993.Google Scholar
  40. 40.
    Smith, H.L., An Introduction to Delay Differential Equations with Applications to the Life Sciences, New York: Springer, 2010.Google Scholar
  41. 41.
    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.CrossRefGoogle Scholar
  42. 42.
    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.CrossRefGoogle Scholar
  43. 43.
    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.CrossRefGoogle Scholar
  44. 44.
    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.CrossRefGoogle Scholar
  45. 45.
    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.CrossRefGoogle Scholar
  46. 46.
    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.CrossRefGoogle Scholar
  47. 47.
    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.Google Scholar
  48. 48.
    Polyanin, A.D. and Manzhirov, A.V., Handbook of Mathematics for Engineers and Scientists, Boca Raton, Fla.: Chapman & Hall/CRC, 2007.Google Scholar
  49. 49.
    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.Google Scholar
  50. 50.
    Polyanin, A.D. and Zaitsev, V.F., Handbook of Nonlinear Partial Differential Equations, Boca Raton, Fla.: Chapman & Hall/CRC, 2012, 2nd ed.Google Scholar
  51. 51.
    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.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  • A. D. Polyanin
    • 1
    • 2
  • V. G. Sorokin
    • 3
  • A. V. Vyazmin
    • 4
  1. 1.Ishlinskii Institute for Problems in MechanicsRussian Academy of SciencesMoscowRussia
  2. 2.National Research Nuclear University MEPhIMoscowRussia
  3. 3.Bauman Moscow State Technical UniversityMoscowRussia
  4. 4.Environmental and Chemical Engineering InstituteMoscow State University of Mechanical EngineeringMoscowRussia

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