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。

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.

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

Figure 1.................................................12
Figure 2.................................................13
Figure 3.................................................13
Figure 4.................................................14
Figure 5.................................................15
Figure 6.................................................16
Figure 7.................................................16
Figure 8.................................................17
Figure 9.................................................18
Figure 10................................................19
Figure 11................................................21
Figure 12................................................23
Figure 13................................................24
Figure 14................................................24
Figure 15................................................25
Figure 16................................................26
Figure 17................................................27
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

[1] Bates, P. W., and Chmaj, A. (1999). A discrete convolution model for phase transitions. Arch. Rational Mech. Anal. 150, 281-305.
[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.