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Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations. (English) Zbl 1093.65096
The authors propose a numerical method for a parabolic equation with delay. The spatial derivative is handled with spectral collocation, and the resulting system of ordinary differential equations is solved by waveform relaxation, which is a form of Picard iteration. It is shown that the method converges.

65M70Spectral, collocation and related methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
35R10Partial functional-differential equations
35K55Nonlinear parabolic equations
Full Text: DOI
[1] Burrage, K.: Parallel and sequential methods for ordinary differential equations. (1995) · Zbl 0838.65073
[2] Burrage, K.; Jackiewicz, Z.; Nørsett, S. P.; Renaut, R.: Preconditioning waveform relaxation iterations for differential systems. Bit 36, 54-76 (1996) · Zbl 0864.65049
[3] Burrage, K.; Jackiewicz, Z.; Renaut, R.: The performance of preconditioned waveform relaxation techniques for pseudospectral methods. Numer. methods partial differential equations 12, 245-263 (1996) · Zbl 0863.65063
[4] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A.: Spectral methods in fluid mechanics. (1988) · Zbl 0658.76001
[5] Cohen, D. S.; Hagan, P. S.; Simpson, H. C.: Spatial structures in predator-prey communities with hereditary effects and diffusion. Math. biosci. 44, 167-177 (1979) · Zbl 0399.92016
[6] Fornberg, B.: A practical guide to pseudospectral methods. (1996) · Zbl 0844.65084
[7] G. Friesecke, A global bifurcation result for a delay-diffusion equation, Tech. Report No. 92, University of Bonn · Zbl 0763.35008
[8] Friesecke, G.: Exponentially growing solutions for a delay-diffusion equation with negative feedback. J. differential equations 98, 1-18 (1992) · Zbl 0763.35008
[9] Green, D.; Stech, H. W.: Diffusion and hereditary effects in a class of population models. Differential equations and applications in ecology, epidemics, and population problems, 19-28 (1981)
[10] Higham, D. J.; Sardar, T. K.: Existence and stability of fixed points for a discretized nonlinear reaction -- diffusion equation with delay. Appl. numer. Math. 18, 155-173 (1995) · Zbl 0834.65079
[11] Jackiewicz, Z.; Kwapisz, M.: Convergence of waveform relaxation methods for differential-algebraic systems. SIAM J. Numer. anal. 33, 2303-2317 (1996) · Zbl 0889.34064
[12] Jackiewicz, Z.; Kwapisz, M.; Lo, E.: Waveform relaxation methods for functional-differential systems of neutral type. J. math. Anal. appl. 207, 255-285 (1997) · Zbl 0874.65056
[13] Kuang, Y.: Delay differential equations with applications in population dynamics. (1993) · Zbl 0777.34002
[14] E. Lelarasmee, The Waveform Relaxation Method for the Time Domain Analysis of Large Scale Nonlinear Dynamical Systems, PhD thesis, University of California, Berkeley, CA, 1982
[15] Lelarasmee, E.; Ruehli, A.; Sangiovanni-Vincentelli, A.: The waveform relaxation methods for time domain analysis of large scale integrated circuits. IEEE trans. CAD IC syst. 1, 131-145 (1982)
[16] Luckhaus, S.: Global boundedness for a delay-differential equation. Trans. amer. Math. soc. 294, 767-774 (1986) · Zbl 0604.35040
[17] Ludwig, D.; Jones, D. D.; Holling, C. S.: Qualitative analysis of insect outbreak systems: the spruce budworm and forest. J. anim. Ecol. 47, 315-332 (1978)
[18] Murray, J. D.: Spatial structures in predator-prey communities --- a nonlinear time delay diffusional model. Math. biosci. 31, 73-85 (1976) · Zbl 0335.92002
[19] Murray, J. D.: Mathematical biology. (1993) · Zbl 0779.92001
[20] Murray, J. D.: Mathematical biology. I. an introduction. (2002) · Zbl 1006.92001
[21] T.K. Sardar, Dynamics of Constant and Variable Stepsize Methods for a Nonlinear Reaction -- Diffusion Model with Delay, PhD thesis, University of Dundee, Dundee, 1996
[22] Trefethen, L. N.: Spectral methods in Matlab. (2000) · Zbl 0953.68643
[23] Verhulst, P. F.: Notice sur la loi que la population suit dans son accroissement. Corr. math. Phys. 10, 113-121 (1838)
[24] Verhulst, P. F.: Recherche mathémathiques sur le loi d’accroissement de la population. Nouveau mém. Acad. roy. Sci. belles lett. Bruxelles 18, 3-38 (1845)
[25] Welfert, B. D.: On the eigenvalues of second-order pseudospectral differentiation operators. Comput. methods appl. Mech. engrg. 116, 281-292 (1994) · Zbl 0824.65083
[26] Welfert, B. D.: Generation of pseudospectral differentiation matrices. I. SIAM J. Numer. anal. 34, 1640-1657 (1997) · Zbl 0889.65013
[27] Wright, E. M.: A non-linear difference-differential equation. J. reine angew. Math. 494, 66-87 (1955) · Zbl 0064.34203
[28] Zubik-Kowal, B.; Vandewalle, S.: Waveform relaxation for functional-differential equations. SIAM J. Sci. comput. 21, 207-226 (1999) · Zbl 0945.65107
[29] Zubik-Kowal, B.: Chebyshev pseudospectral method and waveform relaxation for differential and differential-functional parabolic equations. Appl. numer. Math. 34, 309-328 (2000) · Zbl 0948.65102