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系統識別號 U0002-1608200822564800
中文論文名稱 某些非線性混合型網格微分方程的行進波之數值解
英文論文名稱 Numerical Traveling Wave Solutions of Some Nonlinear Mixed-type Lattice Differential Equations
校院名稱 淡江大學
系所名稱(中) 數學學系碩士班
系所名稱(英) Department of Mathematics
學年度 96
學期 2
出版年 97
研究生中文姓名 黃清郎
研究生英文姓名 Ching-Lang Huang
電子信箱 a694150250@gmail.com
學號 694150250
學位類別 碩士
語文別 英文
口試日期 2008-07-16
論文頁數 38頁
口試委員 指導教授-楊定揮
委員-許正雄
委員-楊智烜
中文關鍵字 耦合輪廓方程式  有限差分法  牛頓迭代  連續法 
英文關鍵字 Coupling profile equation  Finite difference method  Continuation method  Newton's iteration 
學科別分類 學科別自然科學數學
中文摘要 我們使用有限差分法來計算二維的網格型微分方程之行進波波前解。特別地,在方程式中非線性的反應函數為雙穩定的類型,且其擴散項具有函數耦合之特性。在特徵方程式的某些適當條件下,我們證明了正向波速的存在性,它能夠幫助我們近似在輪廓方程式邊界上的漸近行為。最後我們將以牛頓法解,由有限差分法所導出之非線性代數方程。在牛頓迭代中,為了克服尋找好的初始解困難,我們採用了參數連續法。
英文摘要 We present a finite difference method for computing traveling wave front solutions of a two-dimensional lattice differential equations. In particular, the nonlinear reaction function is bi-stable type and the diffusion term is with function-couple. Under some suitable conditions on the characteristic equation, we prove the existence of the positive wave speed. It can help us to approximate the asymptotically behavior on the boundaries of profile equation. Newton's method is used to find the solution of nonlinear algebraic equations inducing by the finite difference method. To overcome the difficulty of finding a good initial solution of Newton's iteration, the continuation method is implemented.
論文目次 Contents

1 Introduction..................................................................................................................1
2 Scheme of Solution Approximation.................................................................................2
2.1 Existence of Characteristic Roots................................................................................3
2.2 Approximate Solutions outside the Finite Interval..........................................................4
2.3 Finite Difference Method............................................................................................8
2.4 Continuation Method.................................................................................................11
3 Numerical Results.......................................................................................................12
3.1 The Coupling Function g is Identity............................................................................12
3.2 The Coupling Function g is a Hyper-Tangent Map........................................................23
Appendixes...................................................................................................................29
A The Algorithms...........................................................................................................29
B Data of Polar Figures..................................................................................................32
B-1 Identity Map to Coupling g........................................................................................32
B-2 Hyper-Tangent Map to Coupling g..............................................................................35
References....................................................................................................................37


Graph of contents

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Figure 18................................................27
Figure 19................................................28


Table of contents

Table 1..................................................14
Table 2..................................................17
Table 3..................................................20
Table 4..................................................22
Table 5..................................................25
Table 6..................................................28
Table A..................................................32
Table B..................................................33
Table C..................................................34
Table D..................................................35
Table E..................................................36

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[2] Bell, J. (1981). Some threshold results for models of Myelinated Nerves. Math. Biosci. 54, 181-190.
[3] Cahn, J. W. (1960), Theory of crystal growth and interface motion in crystalline materials. Acta Met. 8, 554-562.
[4] C. E. Elmer and E. S. Van Vleck, Computation of traveling waves for spatially discrete bistable reaction-diffusion equations, Applied Numerical Mathematics, 20 (1996), pp. 157-169.[5] C.-H. Hsu and S.-S. Lin, Existence and multiplicity of traveling waves in a lattice dynamical system, J. Diff. Eqns., 164 (2000), pp. 431-450.
[6] Chua, L. O. and Roska, T. (1993). The CNN paradigm. IEEE Trans. Circuits Syst. 40, 147-156.
[7] E. Wasserstorm, Numerical solution by the continuation method, IAM Review, 15 (1973), pp. 89-119.
[8] Erneux, T., and Nieolis, G. (1993). Propagating waves in discrete bistable reaction-diffusion systems. Physica D 67, 237-244.
[9] H. Chi, J. Bell, and B. Hassard, Numerical solution of a nonlinear advance delay differential equation from nerve conduction theory, Journal of Mathematical Biology, 24 (1986), pp. 583-601.
[10] H. J. Hupkes and S. M. Verduyn Lunel, Analysis of Newton's method to compute travelling waves in discrete media, Journal of Dynamics and Differential Equations, Vol. 17, No. 3, July 2005.
[11] J. Wu and X. Zou, Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations, Journal of Differential Equations, 135 (1997), pp. 315-357.
[12] Keener, J., and Sneed, J. (1998). Mathematical Physiology. Springer-Verlag, New York.
[13] Laplante, J, P., and Erneux, T. (1992). Propagation failure in arrays of coupled bistable chemical reactors. J. Phys. Chem. 96, 4 931-934.
[14] L. N. Trefethen, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, Cornell University, 1996.
[15] L. O. Chua and L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits Syst., 35 (1988), pp. 1257-1272.
[16] L. O. Chua and L. Yang, Cellular neural networks: Applications, IEEE Trans. Circuits Syst., 35 (1988), pp. 1273-1290.
[17] P. W. Bates, X. Chen, and A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), pp. 520-546.
[18] R. L. Burden and J. D. Faires, Numerical Analysis, 8th edition, Brooks-Cole, 2005.
[19] S. Ma, X. Liao, and J. Wu, Traveling wave solutions for planar lattice differential systems with applications to neural networks, Journal of Differential Equations, 182 (2002), pp. 269-297.
[20] Y.-C. Lin, Numerical computation for traveling wave solutions of lattice differential equations, National Central University, 2005.
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