Continuation Methods for Nonlinear Eigenvalue Problems via a Sinc-Galerkin Scheme

  • Jack D. Dockery
  • Nancy J. Lybeck
Part of the Progress in Systems and Control Theory book series (PSCT, volume 15)


In this paper we will be looking at semilinear boundary value problems of the form
$$ \begin{gathered} \mathcal{L}u(x)\bar = u^{''} (x) + cu(x) = f(x,u(x),\lambda ),a < x < b \hfill \\ u(a) = u(b) = 0, \hfill \\ \end{gathered} $$
where c is a fixed constant and A is a parameter. Here, a and b need not be finite. This type of problem often arises as the equilibrium problem for a scalar evolution equation. The purpose of this paper is to illustrate the use of the Sinc-Galerkin method for one-parameter problems such as (1.1).


Discrete System Continuation Method Solution Branch Half Line Sinc Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    E. L. ALLGOWER and K. GEORG, Numerical Continuation Methods: An Introduction ,Springer-Verlag, New York, 1990.CrossRefGoogle Scholar
  2. [2]
    J. CHAUVETT and F. STENGER, “The Approximate Solution of the Nonlinear Equation Au = u - u3,” J. Math. Anal. Appl ,v. 51, 1975, pp. 229–242.CrossRefGoogle Scholar
  3. [3]
    E. J. DOEDEL, “ AUTO: A Program for the Automatic Bifurcation Analysis of Autonomous Systems,” Cong. Num. ,v. 30, 1981, pp. 265–284. (Proc. 10th Manitoba Conf. on Num. Math. and Comp., Univ. of Manitoba, Winnipeg, Canada, 1980).Google Scholar
  4. [4]
    R. GLOWINSKI, H. B. KELLER and L. RHEINHART, “Continuation Conjuate Gradient Methods for the Least-Squares Solution of Nonlinear Boundary Value Problems,” SIAM J. Sci. Stat. Comput. ,v. 6, 1985, pp. 793–832.CrossRefGoogle Scholar
  5. [5]
    T. M. HAGSTROM and H. B. KELLER, “Asymptotic Boundary Conditions and Numerical Methods for Nonlinear Elliptic Problems on Unbounded Domains,”Math. Comp. ,v. 48, 1987, pp.449–470.CrossRefGoogle Scholar
  6. [6]
    C. K. R. T. JONES and T. KUEPPER, “On the Infinitely Many Solutions of a Semilinear Elliptic Equation,” SIAM J. Math. Anal , v. 17, 1986, pp.803–835.CrossRefGoogle Scholar
  7. [7]
    H. B. KELLER, “Numerical Solution of Bifurcation and Nonlinear Eigenvalue Problems,” In: Applications of Bifurcation Theory ,ed. P.H. Rabinowitz, Academic Press, New York, 1977, pp. 359–384.Google Scholar
  8. [8]
    J. LUND and K. L. BOWERS, Sinc Methods for Quadrature and Dif ferential Equations. ,Siam, Philadelphia, PA, 1992.CrossRefGoogle Scholar
  9. [9]
    G.H. RYDER, Boundary value problems for a class of nonlinear differential equation; Pacific J. Math ,v. 22, 1967, pp. 477–503.CrossRefGoogle Scholar
  10. [10]
    F. STENGER, “Numerical Methods Based on Whittaker Cardinal, or Sinc Functions,” SIAM Review, v. 23 No.2, April, 1981.Google Scholar
  11. [11]
    F. STENGER, “A Sinc-Galerkin Method of Solution of Boundary Value Problems,” Math. Comp. ,v. 33, 1979, pp. 85–109.Google Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Jack D. Dockery
    • 1
  • Nancy J. Lybeck
    • 1
  1. 1.Department of Mathematical SciencesMontana State UniversityBozemanUSA

Personalised recommendations