Analysis of Newton's method to compute travelling waves in discrete media

被引：31

作者：

Hupkes H.J. ^{[1
]}

Lunel S.M.V. ^{[1
]}

机构：

[1] Mathematisch Instituut, Universiteit Leiden, Leiden, RA 2300

来源：

Journal of Dynamics and Differential Equations | 2005年 / 17卷 / 3期

关键词：

Bistable lattice differential equations; Computation of travelling waves; Discrete media; Functional differential equations; Ising model; Myelinated nerve fibers; Newton's method; Numerical computation;

D O I：

10.1007/s10884-005-5809-z

中图分类号：

学科分类号：

摘要：

We present a variant of Newton's method for computing travelling wave solutions to scalar bistable lattice differential equations. We prove that the method converges to a solution, obtain existence and uniqueness of solutions to such equations with a small second order term and study the limiting behaviour of such solutions as this second order term tends to zero. The robustness of the algorithm will be discussed using numerical examples. These results will also be used to illustrate phenomena like propagation failure, which are encountered when studying lattice differential equations. We finish by discussing the broad application range of the method and illustrate that higher dimensional systems exhibit richer behaviour than their scalar counterparts. © 2005 Springer Science+Business Media, Inc.

引用

页码：523 / 572

页数：49

共 38 条

[31]

Rudin W., Real and Complex Analysis, (1974)

[32]

Rudin W., Functional Analysis., (1991)

[33]

Strunz T., Elmer F.J., Driven Frenkel-Kontorova model I. Uniform sliding states and dynamical domains of different particle densities, Phys. Rev. e, 58, pp. 1601-1611, (1998)

[34]

Strunz T., Elmer F.J., Driven Frenkel-Kontorova model II. Chaotic sliding and nonequilibrium melting and freezing, Phys. Rev. e, 58, pp. 16120-21620, (1998)

[35]

Wu J., Zou X., Asymptotic and periodic boundary value problems of mixed FDE's and wave solutions of lattice differential equations, J. Diff. Eg., 135, pp. 315-357, (1997)

[36]

Zinner B., Existence of traveling wavefront solutions for the discrete Nagumo Equation, J. Diff Eg., 96, pp. 1-27, (1992)

[37]

Zinner B., Stability of traveling wavefronts for the discrete Nagumo Equation, J. Diff. Eg., 96, pp. 1-27, (1992)

[38]

Zinner B., Harris G., Hudson W., Traveling wavefronts for the discrete Fisher's equations, J. Diff. Eg., 105, pp. 46-62, (1993)

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