Exact and numerical stability analysis of reaction-diffusion equations with distributed delays

Abstract

This paper is concerned with the stability analysis of the exact and numerical solutions of the reaction-diffusion equations with distributed delays. This kind of partial integro-differential equations contains time memory term and delay parameter in the reaction term. Asymptotic stability and dissipativity of the equations with respect to perturbations of the initial condition are obtained. Moreover, the fully discrete approximation of the equations is given. We prove that the one-leg θ-method preserves stability and dissipativity of the underlying equations. Numerical example verifies the efficiency of the obtained method and the validity of the theoretical results.

Keywords

Reaction-diffusion equations / distributed delay / dissipativity / asymptotic stability

References

[1] Baker C T H. A perspective on the numerical treatment of Volterra equations. J Comput Appl Math, 2000, 125: 217–249

[2] Berezansky L, Braverman E, Idels L.Nicholson’s blowflies differential equations revisited: Main results and open problems. Appl Math Model, 2000, 34: 1405–1417

[3] Branco J R, Ferreira J A, Silva P. Non-Fickian delay reaction-diffusion equations: Theoretical and numerical study. Appl Numer Math, 2010, 60: 531–549

[4] Brunner H. Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge: Cambridge University Press, 2004

[5] Chen H, Zhang C J. Boundary value methods for Volterra integral and integrodifferential equations. Appl Math Comput, 2011, 218(6): 2619–2630

[6] Chen H, Zhang C J. Convergence and stability of extended block boundary value methods for Volterra delay integro-differential equations. Appl Numer Math, 2012, 62(2): 141–154

[7] Cushing J M. Integro-Differential Equations and Delay Models in Population Dynamics. Lecture Notes in Biomath, Vol 20. New York: Springer-Verlag, 1977

[8] Fakhar-Izadi F, Dehghan M. The spectral methods for parabolic Volterra integrodifferential equations. J Comput Appl Math, 2011, 235: 4032–4046

[9] Gan S Q. Dissipativity of θ-methods for nonlinear Volterra delay-integro-differential equations. J Comput Appl Math, 2007, 206: 898–907

[10] 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-London: Academic Press, 1981, 19–28

[11] Greenwell-Yanik C E, Fairweather G. Analyses of spline collocation methods for parabolic and hyperbolic problems in the two space variables. SIAM J Numer Anal, 1986, 23: 282–296

[12] Gurney W S, Blythe S P, Nisbet R M. Nicholson’s blowflies. Nature, 1980, 287: 17–21

[13] Higham D J, Sardar T. Existence and stability of fixed points for a discretised nonlinear reaction-diffusion equation with delay. Appl Numer Math, 1995, 18: 155–173

[14] Houwen P Jvan der, Sommeijer B P, Baker C T H. On the stability of predictorcorrector methods for parabolic equations with delay. IMA J Numer Anal, 1986, 6:1–23

[15] Huang C M. Delay-dependent stability of high order Runge-Kutta methods. Numer Math, 2009, 111: 377–387

[16] Huang C M, Chang Q S. Linear stability of general linear methods for systems of neutral delay differential equations. Appl Math Lett, 2001, 14: 1017–1021

[17] Huang C M, Chang Q S. Stability analysis of numerical methods for systems of functional-differential and functional equations. Comput Math Appl, 2002, 44(5-6): 717–729

[18] Jackiewicz Z, Zubik-Kowal B. Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations. Appl Numer Math, 2006, 56: 433–443

[19] Kauthen J P. The method of lines for parabolic partial integro-differential equations. J Integral Equations Appl, 1992, 4: 69–81

[20] Larsson S, Thomee V, Wahlbin L B. Numerical solution of parabolic integrodifferential equations by the discontinuous Galerkin method. Math Comp, 1998, 67: 45–71

[21] Li D F, Zhang C J, Wang W S. Long time behavior of non-Fickian delay reactiondiffusion equations. Nonlinear Anal Real World Appl, 2012, 13: 1401–1415

[22] Li S F. A review of theoretical and numerical analysis for nonlinear stiff Volterra functional differential equations. Front Math China, 2009, 4(1): 23–48

[23] Li S F. Numerical Analysis for Stiff Ordinary Differential Equations and Functional Differential Equations. Xiangtan: Xiangtan University Press, 2010 (in Chinese)

[24] Ma J T. Finite element methods for partial Volterra integro-differential equations on two-dimensional unbounded spatial domains. Appl Math Comput, 2007, 186: 598–609

[25] Sachs E W, Strauss A K. Efficient solution of a partial integro-differential equation in finance. Appl Numer Math, 2008, 58: 687–703

[26] Tian H J. Asymptotic stability of numerical methods for linear delay parabolic differential equations. Comput Math Appl, 2008, 56: 1758–1765

[27] Tian H J, Guo Ni, Shen A L. Dissipativity of delay functional differential equations with bounded lag. J Math Anal Appl, 2009, 355: 778–782

[28] Wang W S, Li S F. Dissipativity of Runge-Kutta methods for neutral delay differential equations with piecewise constant delay. Appl Math Lett, 2008, 21: 983–991

[29] Wang W S,Zhang C J. Preserving stability implicit Euler method for nonlinear Volterra and neutral functional differential equations in Banach space. Numer Math, 2010,115(3): 451–474

[30] Wang X, Li Z. Dynamics for a type of general reaction-diffusion model. Nonlinear Anal, 2007, 67: 2699–2711

[31] Wen L P, Yu Y X, Li S F. Dissipativity of Runge-Kutta methods for Volterra functional differential equations. Appl Numer Math, 2011, 61: 368–381

[32] Wen L P, Yu Y X, Wang W S. Generalized Halanay inequalities for dissipativity of Volterra functional differential equations. J Math Anal Appl, 2008, 347: 169–178

[33] Wu J H. Theory and Application of Partial Functional Differential Equation. Applied Mathematical Sciences, Vol 119. Berlin: Springer, 1996

[34] Wu S F, Gan S Q. Analytical and numerical stability of neutral delay integro-differential equations and neutral delay partial differential equations. Comput Math Appl, 2008, 55: 2426–2443

[35] Yan Y, Fairweather G. Orthogonal spline collocation methods for some partial integrodifferential equations. SIAM J Numer Anal, 1992, 29: 755–768

[36] Yanik E G, Fairweather G. Finite element methods for parabolic and hyperbolic partial integro-differential equations. Nonlinear Anal, 1988, 12: 785–809

[37] Zacher R. Boundedness of weak solutions to evolutionary partial integrodifferential equations with discontinuous coefficients. J Math Anal Appl, 2008, 348: 137–149

[38] Zhang C J, Vandewalle S. Stability analysis of Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations. IMA J Numer Anal, 2004, 24: 193–214

[39] Zhang C J, Vandewalle S. Stability criteria for exact and discrete solutions of neutral multidelay-integro-differential equations. Adv Comput Math, 2008, 28(4): 383–399

[40] Zhao J J, Xu Y, Liu M Z. Stability analysis of numerical methods for linear neutral Volterra delay-integro-differential system. Appl Math Comput, 2005, 167: 1062–1079

[41] Zubik-Kowal B, Vandewalle S. Waveform relaxation for functional-differential equations. SIAM J Sci Comput, 1999, 21: 207–226

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg