EqWorld

The World of Mathematical Equations

Exact Solutions Methods Software For Authors Math Forums

Methods > Delay Partial Differential Equations

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 delay partial 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--diffusion equation 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 delay parabolic 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