#### 517.9:532:536 Nonlinear delay reaction-diffusion equations with varying transfer coefficients: generalized and functional separable solutions

##### Polyanin A. D. (Bauman Moscow State Technical University/Ishlinsky Institute for Problems in Mechanics/MEPhI), Zhurov A. I. (Cardiff University/Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences)

###### DELAY REACTION-DIFFUSION EQUATIONS, VARYING TRANSFER COEFFICIENTS, EXACT SOLUTIONS, GENERALIZED SEPARABLE SOLUTIONS, FUNCTIONAL SEPARABLE SOLUTIONS, TIME-VARYING DELAY, NONLINEAR DELAY PARTIAL DIFFERENTIAL EQUATIONS

doi: 10.18698/2309-3684-2015-4-337

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 , withn (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.

[1] Wu J. Theory and applications of partial functional differential equations. New York, Springer-Verlag, 1996.
[2] 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.
[3] 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.
[4] 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.
[5] 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.
[6] 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.
[7] 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.
[8] 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.
[9] 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.
[10] 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.
[11] 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.
[12] 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.
[13] 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.
[14] Cao J. New results concerning exponential stability and periodic solutions of delayed cellular neural networks. Phys. Lett. A, 2003, vol. 307, pp. 136–147.
[15] 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.
[16] 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.
[17] 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.
[18] 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.
[19] 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.
[20] Zhao Н. Exponential stability and periodic oscillatory of bidirectional associative memory neural network involving delays. Neurocomputing, 2006, vol. 69, pp. 424–448.
[21] 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.
[22] 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.
[23] 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).
[24] 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.
[25] Kudryashov N.A. Teoreticheskaya i matematicheskaya fizika — Theor. & Math. Phys., 1993, vol. 94, no. 2, pp. 211–218.
[26] 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.
[27] 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.
[28] 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).
[29] 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.
[30] Polyanin A.D., Zaitsev V.F. Handbook of nonlinear partial differential equations. 2nd ed. Boca Raton, Chapman & Hall/CRC Press, 2012.
[31] 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.
[32] 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.
[33] 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.
[34] 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.
[35] 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.
[36] 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.
[37] 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.
[38] Bellman R., Cooke K.L. Differential-difference equations [in Russian: Bellman R., Cooke K.L. Differentsialno-raznostnye uravneniya. Moscow, Mir Publ., 1967].
[39] Hale J. Functional differential equations. New York, Springer-Verlag, 1977.
[40] Driver R.D. Ordinary and delay differential equations. New York, Springer- Verlag, 1977.
[41] Kolmanovskii V., Myshkis А. Applied theory of functional differential equations. Dordrecht, Kluwer, 1992.
[42] KuangY. Delay differential equations with applications in population dynamics. Boston, Academic Press, 1993.
[43] Smith H.L. An introduction to delay differential equations with applications to the life sciences. New York, Springer, 2010.
[44] 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.
[45] 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.
[46] 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.
[47] 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.
[48] 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.
[49] 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.
[50] 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).
[51] 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.
[52] 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.
[53] Grundland A.M., Infeld E. A family of nonlinear Klein — Gordon equations and their solutions. J. Math. Phys., 1992, vol. 33, pp. 2498–2503.
[54] Miller W., Rubel L.A. Functional separation of variables for Laplace equations in two dimensions. Physica A, 1993, vol. 26, pp. 1901–1913.
[55] Zhdanov R.Z. Separation of variables in the nonlinear wave equation. Physica A, 1994, vol. 27, pp. L291–L297.
[56] 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.
[57] Pucci Е., Saccomandi G. Evolution equations, invariant surface conditions and functional separation of variables. Physica D: Nonlinear Phenomena, 2000, vol. 139, pp. 28–47.
[58] 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