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
收藏
页码：523 / 572
页数：49
相关论文
共 38 条
  • [1] Abell K.A., Elmer C.E., Humphries A.R., Vleck E.S.V., Computation of mixed type functional differential boundary value problems, SIAM J. Appl. Dyn. Sys., (2005)
  • [2] Bader G., Solving boundary value problems for functional differential equations by collocation, Proceedings in Scientific Computing, Numerical Boundary Value ODEs, 5, pp. 227-243, (1985)
  • [3] Bates P.W., Chen X., Chmaj A., Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal, 35, pp. 520-546, (2003)
  • [4] Bates P.W., Chmaj A., A discrete convolution model for phase transitions, Arch. Rational Mech. Anal., 150, pp. 281-305, (1999)
  • [5] Bell J., Some threshold results for models of Myelinated Nerves, Math. Biosci., 54, pp. 181-190, (1981)
  • [6] Cahn J.W., Theory of crystal growth and interface motion in crystalline materials, Acta Met., 8, pp. 554-562, (1960)
  • [7] Cahn J.W., Mallet-Paret J., Van Vleck E.S., Traveling wave solutions for systems of ODE's on a two-dimensional spatial lattice, SIAM J. Appl. Math., 59, pp. 455-493, (1999)
  • [8] Cash J.R., Moore G., Wright R.W., An automatic continuation strategy for the solution of singularly perturbed linear two-point boundary value problems, J. Comp. Phys., 122, pp. 266-279, (1995)
  • [9] Chi H., Bell J., Hassard B., Numerical solution of a nonlinear advance-delay differential equation from nerve conduction theory, J. Math. Bio., 24, pp. 583-601, (1986)
  • [10] Chow S.N., Mallet-Paret J., Shen W., Traveling waves in lattice dynamical systems, J. Diff. Eq., 149, pp. 248-291, (1998)
1234