We present a number of new simple separable, generalized separable, and functional separable solutions to one-dimensional nonlinear delay reaction-diffusion equations with varying transfer coefficients of the formut = [G(u)ux ]x F(u,w),where w = u(x,t) and w = u(x,t ), with denoting the delay time. All of the equations considered contain one, two, or three arbitrary functions of a single argument. The generalized separable solutions are sought in the form =1 = () () N
n n n u x t , withn (x) and n (t) to be determined in the analysis using a new modification of the functional constraints method. Some of the results are extended to nonlinear delay reaction-diffusion equations with time-varying delay = (t). We also present exact solutions to more complex, three-dimensional delay reactiondiffusion equations of the formut = div[G(u)u] F(u,w).Most of the solutions obtained involve free parameters, so they may be suitable for solving certain problems as well as testing approximate analytical and numerical methods for non-linear delay PDEs.
 Wu J. Theory and applications of partial functional differential equations. New York, Springer-Verlag, 1996.
 Smith H.L., Zhao X.-Q. Global asymptotic stability of travelling waves in delayed reaction-diffusion equations. SIAM J. Math. Anal., 2000, no. 31, pp. 514–534.
 Wu J., Zou X. Traveling wave fronts of reaction-diffusion systems with delay. J. Dynamics and Differential Equations, 2001, vol. 13, no. 3, pp. 651–687.
 Huang J., Zou X. Traveling wave fronts in diffusive and cooperative Lotka — Volterra system with delays. J. Math. Anal. Appl., 2002, vol. 271, pp. 455–466.
 Faria T., Trofimchuk S. Nonmonotone travelling waves in a single species reaction - diffusion equation with delay. J. Differential Equations, 2006, vol. 228, pp. 357–376.
 Trofimchuk E., Tkachenko V., Trofimchuk S. Slowly oscillating wave solutions of a single species reaction–diffusion equation with delay. J. Differential Equations, 2008, vol. 245, pp. 2307–2332.
 Mei M., So J., Li M., Shen S. Asymptotic stability of travelling waves for Nicholson's blowflies equation with diffusion. Proc. Roy. Soc. Edinburgh Sect. A, 2004, vol. 134, pp. 579–594.
 Gourley S.A., Kuan Y. Wavefronts and global stability in time-delayed population model with stage structure. Proc. Roy. Soc. London A, 2003, vol. 459, pp. 1563–1579.
 Pao С. Global asymptotic stability of Lotka — Volterra competition systems with diffusion and time delays. Nonlinear Anal.: Real World Appl., 2004, vol. 5, no. 1, pp. 91–104.
 Liz Е., Pinto М., Tkachenko V., Trofimchuk S. A global stability criterion for a family of delayed population models. Quart. Appl. Math., 2005, vol. 63, pp. 56–70.
 Meleshko S.V., Moyo S. On the complete group classification of the reactiondiffusion equation with a delay. J. Math. Anal. Appl., 2008, vol. 338, pp. 448–466.
 Polyanin A.D., Zhurov A.I. Exact solutions of linear and non-linear differential- difference heat and diffusion equations with finite relaxation time. International J. of Non-Linear Mechanics, 2013, vol. 54, pp. 115–126.
 Arik S. Global asymptotic stability of a larger class of neural networks with constant time delay. Phys. Lett. A, 2003, vol. 311, pp. 504–511.
 Cao J. New results concerning exponential stability and periodic solutions of delayed cellular neural networks. Phys. Lett. A, 2003, vol. 307, pp. 136–147.
 Cao J., Liang J., Lam J. Exponential stability of high-order bidirectional associative memory neural networks with time delays. Physica D: Nonlinear Phenomena, 2004, vol. 199, no. 3–4, pp. 425–436.
 Lu H.T., Chung F.L., He Z.Y. Some sufficient conditions for global exponential stability of delayed Hopfield neural networks. Neural Networks, 2004, vol. 17, pp. 537–544.
 Cao J.D., Ho D.W.C. A general framework for global asymptotic stability analysis of delayed neural networks based on LMI approach. Chaos, Solitons & Fractals, 2005, vol. 24, pp. 1317–1329.
 Liao X.X., Wang J., Zeng Z. Global asymptotic stability and global exponential stability of delayed cellular neural networks. IEEE Trans. Circ. Syst II, 2005, vol. 52, no. 7, pp. 403–409.
 Song O.K., Cao J.D. Global exponential stability and existence of periodic solutions in BAM networks with delays and reaction diffusion terms. Chaos, Solitons & Fractals, 2005, vol. 23, no. 2, pp. 421–430.
 Zhao Н. Exponential stability and periodic oscillatory of bidirectional associative memory neural network involving delays. Neurocomputing, 2006, vol. 69, pp. 424–448.
 Wang L., Gao Y. Global exponential robust stability of reaction-diffusion interval neural networks with time-varying delays. Physics Letters A, 2006, vol. 350, pp. 342–348.
 Lu J.G. Global exponential stability and periodicity of reaction-diffusion delayed recurrent neural networks with Dirichlet boundary conditions. Chaos, Solitons and Fractals, 2008, vol. 35, pp. 116–125.
 Dorodnitsyn V.A. Zhurnal vychislitelnoi matematiki i matematicheskoy fiziki – J. of Computational Mathematics and Mathematical Physics, 1982, vol. 22, no. 6, pp. 1393–1400. (in Russian).
 Nucci М.С., Clarkson Р.А. The nonclassical method is more general than the direct method for symmetry reductions. An example of the Fitzhugh — Nagumo equation. Phys. Lett. A, 1992, vol. 164, pp. 49–56.
 Kudryashov N.A. Teoreticheskaya i matematicheskaya fizika — Theor. & Math. Phys., 1993, vol. 94, no. 2, pp. 211–218.
 Galaktionov V.A. Quasilinear heat equations with first-order sign-invariants and new explicit solutions. Nonlinear Analys., Theory, Meth. and Applications, 1994, vol. 23, pp. 1595–1621.
 Ibragimov N.H., ed. CRC handbook of Lie group analysis of differential equations. Vol. 1. Symmetries, exact solutions and conservation laws. Boca Raton, CRC Press, 1994.
 Polyanin A.D., Zaitsev V.F., Zhurov A.I. Metody resheniya nelineinykh uravneniy matematicheskoy fiziki [Solution methods for nonlinear equations of mathematical physics and mechanics]. Moscow, Fizmatlit Publ., 2005 (in Russian).
 Galaktionov V.A., Svirshchevskii S.R. Exact solutions and invariant subspaces of nonlinearpartial differential equations in mechanics and physics. Boca Raton, Chapman & Hall/CRC Press, 2006.
 Polyanin A.D., Zaitsev V.F. Handbook of nonlinear partial differential equations. 2nd ed. Boca Raton, Chapman & Hall/CRC Press, 2012.
 Polyanin A.D., Zhurov A.I. Exact separable solutions of delay reactiondiffusion equations and other nonlinear partial functional-differential equations. Communications in Nonlinear Science and Numerical Simulation, 2014, vol. 19, pp. 409–416.
 Polyanin A.D., Zhurov A.I. Functional constraints method for constructing exact solutions to delay reaction-diffusionequations and more complex nonlinear equations. Communications in Nonlinear Science and Numerical Simulation, 2014, vol. 19, pp. 417–430.
 Polyanin A.D., Zhurov A.I. New generalized and functional separable solutions to non-linear delay reaction-diffusion equations. International Journal of Non-Linear Mechanics, 2014, vol. 59, pp. 16–22.
 Polyanin A.D., Zhurov A.I. Non-linear instability and exact solutions to some delay reaction-diffusion systems. International Journal of Non-Linear Mechanics, 2014, vol. 62, pp. 33–40.
 Polyanin A.D., Sorokin V.G., Vyazmin A.V. Matematicheskoe modelirovanie I chislennye metody — Mathematical Modelling and Numerical Methods, 2014, no. 4, p. 53–73.
 Polyanin A.D., Sorokin V.G. Nonlinear delay reaction-diffusion equations: Traveling-wave solutions in elementary functions. Applied Mathematics Letters, 2015, vol. 46, p. 38–43.
 Polyanin A.D., Zhurov A.I. Nonlinear delay reaction-diffusion equations with varying transfer coefficients: Exact methods and new solutions. Applied Mathematics Letters, 2014, vol. 37, pp. 43–48.
 Bellman R., Cooke K.L. Differential-difference equations [in Russian: Bellman R., Cooke K.L. Differentsialno-raznostnye uravneniya. Moscow, Mir Publ., 1967].
 Hale J. Functional differential equations. New York, Springer-Verlag, 1977.
 Driver R.D. Ordinary and delay differential equations. New York, Springer- Verlag, 1977.
 Kolmanovskii V., Myshkis А. Applied theory of functional differential equations. Dordrecht, Kluwer, 1992.
 KuangY. Delay differential equations with applications in population dynamics. Boston, Academic Press, 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. Communications in Nonlinear Science and Numerical Simulation, 2012, vol. 17, pp. 4978–4987.
 Polyanin A.D., Zhurov A.I. Exact solutions of non-linear differentialdifference equations of a viscous fluid with finite relaxation time. International Journal of Non-Linear Mechanics, 2013, vol. 57, pp. 116–122.
 Polyanin A.D., Zhurov A.I. Generalized and functional separable solutions to nonlinear delay Klein — Gordon equations. Communications in Nonlinear Science and Numerical Simulation, 2014, vol. 19, no. 8, pp. 2676–2689.
 He Q., Kang L., Evans D.J. Convergence and stability of the finite difference scheme for nonlinear parabolic systems with time delay. Numerical Algorithms, 1997, vol. 16, no. 2, pp. 129–153.
 Pao C.V. Numerical methods for systems of nonlinear parabolic equations with time delays. Journal of Mathematical Analysis and Applications, 1999, vol. 240, no. 1, pp. 249–279.
 Jackiewicza Z., Zubik-Kowal B. Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations. Applied Numerical Mathematics, 2006, vol. 56, no. 3–4, pp. 433–443.
 Bratsun D. A., Zakharov A. P. Vestnik Permskogo universiteta. ser. matematika, mekhanika, informatika — Bulletin of Perm University. Series Mathematics, Mechanics, Computer Science, 2012, vol. 4, no. 12, pp. 32–41 (in Russian).
 Zhang Q., Zhang C. A compact difference scheme combined with extrapolation techniques for solving a class of neutral delay parabolic differential equations. Applied Mathematics Letters, 2013, vol. 26, no. 2, pp. 306–312.
 Zhang Q., Zhang C. A new linearized compact multisplitting scheme for the nonlinear convection-reaction-diffusion equations with delay. Communications in Nonlinear Science and Numerical Simulation, 2013, vol. 18, no. 12, pp. 3278–3288.
 Grundland A.M., Infeld E. A family of nonlinear Klein — Gordon equations and their solutions. J. Math. Phys., 1992, vol. 33, pp. 2498–2503.
 Miller W., Rubel L.A. Functional separation of variables for Laplace equations in two dimensions. Physica A, 1993, vol. 26, pp. 1901–1913.
 Zhdanov R.Z. Separation of variables in the nonlinear wave equation. Physica A, 1994, vol. 27, pp. L291–L297.
 Doyle Ph.W., Vassiliou P.J. Separation of variables for the 1-dimensional nonlinear diffusion equation. International Journal of Non-Linear Mechanics, 1998, vol. 33, pp. 315–326.
 Pucci Е., Saccomandi G. Evolution equations, invariant surface conditions and functional separation of variables. Physica D: Nonlinear Phenomena, 2000, vol. 139, pp. 28–47.
 Andreev V.K., Kaptsov O.V., Pukhnachov V.V., Rodionov А.А. Applications of Group-Theoretical Methods in Hydrodynamics. Dordrecht, Kluwer, 1998.
Polyanin A., Zhurov A. Nonlinear delay reaction-diffusion equations with varying transfer coefficients: generalized and functional separable solutions. Маthematical Modeling and Coтputational Methods, 2015, №4 (8), pp. 3-37
Количество скачиваний: 344