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 對於此次的研究中，針對解決離散空間上的反應擴散方程行進波問題。首先使用基於隱型Runge-Kutta演算程序(Implicit Runge-Kutta)、配置法則(Collocation Method) 等泛函微分方程(Functional Differential Equations, FDE)技巧，以上述數值計算方法處理典型 bistable型離散空間上的反應擴散方程。其中包含以延續法(Continuation Method)之數值技巧作為解決行進波問題的對策。並在文章最後列舉兩個實際實驗結果的呈現。
 In this work, traveling wave solutions for reaction-diffusion equations on a discrete spatial domain are considered. We use the collocation method based on k-stage implicit Runge-Kutta scheme to compute numerically the functional differential equation which is the profile equation of some typical bistable spatial discrete reaction diffusion equation. Numerical techniques for solving the traveling wave equations include the continuation method. Finally, some numerical results are presented.
 1 Introduction 12 Preliminaries 22.1 Implicit Runge-Kutta Scheme and Collocation Method . . . . 22.2 The linear case . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Qusilinearization for Nonlinear Case . . . . . . . . . . . . . . . 63 Applications : Traveling Wave Solution Problems 73.1 Boundary Functions and Boundary Conditions . . . . . . . . . 83.2 DDE Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Numerical Results 125 Conclusions 13
 [1] Kate A Abell, Christopher E Elmer, A. R Humphries, and Erik S VanVleck, Computation of mixed type functional di?erential boundary valueproblems,SIAMJ.Appl.Dyn.Syst.4(2005),no.3,755–781(electronic).[2] Paolo Arena, Maide Bucolo, Stefano Fazzino, Luigi Fortuna, and MattiaFrasca, The cnn paradigm: shapes and complexity, Internat. J. Bifur.Chaos Appl. Sci. Engrg. 15 (2005), no. 7, 2063–2090.[3] PeterWBatesandAdamChmaj, A discrete convolution model for phasetransitions, Arch. Ration. Mech. Anal. 150 (1999), no. 4, 281–305.[4] Jonathan Bell, Some threshold results for models of myelinated nerves,Math. Biosci. 54 (1981), no. 3-4, 181–190.[5] Henjin Chi, Jonathan Bell, and Brian Hassard, Numerical solution ofa nonlinear advance-delay-differential equation from nerve conductiontheory, J. Math. Biol. 24 (1986), no. 5, 583–601.[6] Leon O Chua, Cnn: a vision of complexity, Internat. J. Bifur. ChaosAppl. Sci. Engrg. 7 (1997), no. 10, 2219–2425.[7] LeonOChua,MartinHasler, GeorgeSMoschytz, andJacquesNeirynck,Autonomous cellular neural networks: a unified paradigm for patternformation and active wave propagation, IEEE Trans. Circuits Systems IFund. Theory Appl. 42 (1995), no. 10, 559–577.[8] Christopher E Elmer and Erik S Van Vleck, Computation of travelingwaves for spatially discrete bistable reaction-diffusion equations, Appl.Numer. Math. 20 (1996), no. 1-2, 157–169.[9] Christopher E Elmer and Erik S Van Vleck,Analysis and computation of travelling wave solutions of bistabledifferential-di?erence equations, Nonlinearity 12 (1999), no. 4, 771–798.[10] Christopher E Elmer and Erik S Van Vleck, A variant of newton’s method for the computation of travelingwaves of bistable di?erential-di?erence equations, Journal of Dynamicsand Differential Equations 14 (2002), no. 3, 493–517.[11] Thomas Erneux and Gr’egoire Nicolis, Propagating waves in discretebistable reaction-di?usion systems, Phys. D 67 (1993), no. 1-3, 237–244.[12] James Keener and James Sneyd, Mathematical physiology. vol. i: Cel-lular physiology, 8/ (2009), xxvi+470+A2+R45+I29.[13] , Mathematical physiology. vol. ii: Systems physiology, 8/(2009),i–xxvi, 471–974, A1–A2, R1–R45 and I1–I29.
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 1 反應擴散方程的非平面傳動波 2 網格動態系統的行進波

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