Delay-independent stability of Euler method for nonlinear one-dimensional diffusion equation with constant delay^?

Abstract

This paper is concerned with delay-independent asymptotic stability of a numerical process that arises after discretization of a nonlinear one-dimensional diffusion equation with a constant delay by the Euler method. Explicit sufficient and necessary conditions for the Euler method to be asymptotically stable for all delays are derived. An additional restriction on spatial stepsize is required to preserve the asymptotic stability due to the presence of the delay. A numerical experiment is implemented to confirm the results.

Keywords

Partial functional differential equation / asymptotic stability / Euler method

References

[1] Bellen A, Zennaro M. Numerical Methods for Delay Differential Equations. Oxford: Clarendon Press, 2003

[2] Chan W C, Green D. Stablility and bifurcation in delay diffusion models. Diff Equa Dynam Syst, 1993, 1(1): 87-100

[3] Green D, Stech H W. Diffusion and Hereditary Effects in a Class of Population Models in Differential Equations and Applications in Ecology, Epidemics, and Population Problems. New York: Academic Press, 1981

[4] Haberman R. Applied Partial Differential Equations: with Fourier Series and Boundary Value Problems. 4th ed. New Jersey: Pearson Prentice Hall, 2004

[5] Higham D J, Sardar T. Existence and stability of fixed points for a discretized nonlinear reaction-diffusion equation with delay. Appl Numer Math, 1995, 18(1-3): 155-173

[6] Huang C, Vandewalle S. An analysis of delay-dependent stability for ordinary and partial differential equations with fixed and distributed delay. SIAM J Sci Comp, 2004, 25(5): 1608-1632

[7] Liu M, Spijker M N. The stability of the θ-methods in the numerical solution of delay differential equations. IMA J Numer Anal, 1990, 10(1): 31-48

[8] Turyn L. A partial functional differential equations. J Math Anal Appl, 2001, 263(1): 1-13

[9] Wu J. Theory and Applications of Partial Functional Differential Equations. New York: Springer-Verlag, 1996

[10] Zubik-Kowal B. Stability in the numerical solution of linear parabolic equations with delay term. BIT, 2001, 41(1): 191-206

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