Nonlinear Delay Partial Differential Equations
Exact methods:
A. D. Polyanin, A. I. Zhurov. Functional constraints method for constructing exact solutions to delay reaction-diffusion equations and more complex nonlinear equations.Communications in Nonlinear Science and Numerical Simulation,Vol. 19, No. 3, pp. 417–430, 2014. (See also: A new method for constructing exact solutions to nonlinear delaypartial differential equations. arXiv:1304.5473v1 [nlin.SI] 19 Apr 2013.)A. D. Polyanin, A. I. Zhurov. Non-linear instability and exact solutions to some delay reaction-diffusion systems.International Journal of Non-Linear Mechanics, Vol. 62, pp. 33–40, 2014.A. D. Polyanin, A. I. Zhurov. Nonlinear delay reaction-diffusion equations with varying transfer coefficients: Exact methods and new solutions,Applied Mathematics Letters ,Vol. 37, pp. 43–48, 2014.A. D. Polyanin, A. I. Zhurov. The functional constraints method: Application to non-linear delay reaction-diffusion equations with varying transfer coefficients,International Journal of Non-Linear Mechanics ,Vol. 67, pp. 267–277, 2014.A. D. Polyanin, A. I. Zhurov. The generating equations method: Constructing exact solutions to delay reaction-diffusion systems and other non-linear coupled delay PDEs,International Journal of Non-Linear Mechanics,Vol. 71, pp. 104–115, 2015.S. V. Meleshko, S. Moyo.On the complete group classification of the reaction--diffusionequation with a delay. Journal of Mathematical Analysis and Applications, Vol. 338, pp. 448–466, 2008.
Numerical methods:
Q. He, L. Kang, D. J. Evans.Convergence and stability of the finite difference scheme for nonlinear parabolic systems with time delay.Numerical Algorithms, Vol. 16, No. 2, pp. 129–153, 1997.C. V. Pao.Numerical methods for systems of nonlinear parabolic equations with time delays.Journal of Mathematical Analysis and Applications,Vol. 240, No. 1, pp. 249–279, 1999.Z. Jackiewicza, B. Zubik-Kowal.Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations.Applied Numerical Mathematics,Vol. 56, No. 3-4, pp. 433–443, 2006.Q. Zhang, C. Zhang.A compact difference scheme combined with extrapolation techniques for solving a class of neutral delayparabolic differential equations.Applied Mathematics Letters,Vol. 26, No. 2, pp. 306–312, 2013.Q. Zhang, C. Zhang.A new linearized compact multisplitting scheme for the nonlinear convection-reaction-diffusion equations with delay.Communications in Nonlinear Science and Numerical Simulation,Vol. 18, No. 12, pp. 3278–3288, 2013.
See also:
B. Zubik-Kowal. Delay partial differential equations.Scholarpedia, 3(4):2851, 2008.P. M. Jordan, W. Dai, R.E. Mickens. A note on the delayed heat equation: Instability with respect to initial data. Mechanics Research Communications, 2008, Vol. 35, pp. 414-420.R. S. Ismagilov, N. A. Rautian, V. V. Vlasov.Examples of very unstable linear partial functional differential equations.arXiv: arXiv:1402.4107 [math.AP] 17 Feb 2014.Delay Reaction-Diffusion Equations - EqWorld
