Eigenvalue problems on infinite intervals
Author:
Peter A. Markowich
Journal:
Math. Comp. 39 (1982), 421441
MSC:
Primary 34B25
DOI:
https://doi.org/10.1090/S00255718198206696376
MathSciNet review:
669637
Fulltext PDF Free Access
Abstract  References  Similar Articles  Additional Information
Abstract: This paper is concerned with eigenvalue problems for boundary value problems of ordinary differential equations posed on an infinite interval. Problems of that kind occur for example in fluid mechanics when the stability of laminar flows is investigated. Characterizations of eigenvalues and spectral subspaces are given, and the convergence of approximating problems, which are derived by reducing the infinite interval to a finite but large one and by imposing additional boundary conditions at the far end, is proved. Exponential convergence is shown for a large class of problems.

C. de Boor & B. Swartz, Collocation Approximation to Eigenvalues of an Ordinary Differential Equation. The Principle of the Thing, MRC Tech. Report #1937, Madison, Wisc., 1980.
 Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, Interscience Publishers John Wiley & Sons New YorkLondon, 1963. With the assistance of William G. Bade and Robert G. Bartle. MR 0188745 R. D. Grigorieff, "Diskrete Approximation von Eigenwertproblemen. I, II," Numer. Math., v. 24, 1975, pp. 355374, 415433.
 Chester E. Grosch and Steven A. Orszag, Numerical solution of problems in unbounded regions: coordinate transforms, J. Comput. Phys. 25 (1977), no. 3, 273–295. MR 488870, DOI https://doi.org/10.1016/00219991%2877%29901024
 Frank R. de Hoog and Richard Weiss, Difference methods for boundary value problems with a singularity of the first kind, SIAM J. Numer. Anal. 13 (1976), no. 5, 775–813. MR 440931, DOI https://doi.org/10.1137/0713063
 Frank R. de Hoog and Richard Weiss, On the boundary value problem for systems of ordinary differential equations with a singularity of the second kind, SIAM J. Math. Anal. 11 (1980), no. 1, 41–60. MR 556495, DOI https://doi.org/10.1137/0511003
 F. R. de Hoog and R. Weiss, An approximation theory for boundary value problems on infinite intervals, Computing 24 (1980), no. 23, 227–239 (English, with German summary). MR 620090, DOI https://doi.org/10.1007/BF02281727
 Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, SpringerVerlag New York, Inc., New York, 1966. MR 0203473
 Herbert B. Keller, Numerical solution of two point boundary value problems, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976. Regional Conference Series in Applied Mathematics, No. 24. MR 0433897
 Bertil Gustafsson, HeinzOtto Kreiss, and Arne Sundström, Stability theory of difference approximations for mixed initial boundary value problems. II, Math. Comp. 26 (1972), 649–686. MR 341888, DOI https://doi.org/10.1090/S00255718197203418883
 Marianela Lentini and Herbert B. Keller, Boundary value problems on semiinfinite intervals and their numerical solution, SIAM J. Numer. Anal. 17 (1980), no. 4, 577–604. MR 584732, DOI https://doi.org/10.1137/0717049 P. A. Markowich, Randwertprobleme auf Unendllichen Intervallen, Dissertation, TU Wien, 1980. A. Markowich, "Analysis of boundary value problems on infinite intervals," SIAM J. Appl. Math., v. 42, 1982, pp. 549557.
 Peter A. Markowich, A theory for the approximation of solutions of boundary value problems on infinite intervals, SIAM J. Math. Anal. 13 (1982), no. 3, 484–513. MR 653468, DOI https://doi.org/10.1137/0513033
 B. S. Ng and W. H. Reid, On the numerical solution of the OrrSommerfeld problem: asymptotic initial conditions for shooting methods, J. Comput. Phys. 38 (1980), no. 3, 275–293. MR 609434, DOI https://doi.org/10.1016/00219991%2880%29901503
 John E. Osborn, Spectral approximation for compact operators, Math. Comput. 29 (1975), 712–725. MR 0383117, DOI https://doi.org/10.1090/S00255718197503831173
 Friedrich Stummel, Diskrete Konvergenz linearer Operatoren. I, Math. Ann. 190 (1970/71), 45–92 (German). MR 291870, DOI https://doi.org/10.1007/BF01349967
 Friedrich Stummel, Diskrete Konvergenz linearer Operatoren. II, Math. Z. 120 (1971), 231–264 (German). MR 291871, DOI https://doi.org/10.1007/BF01117498
Retrieve articles in Mathematics of Computation with MSC: 34B25
Retrieve articles in all journals with MSC: 34B25
Additional Information
Keywords:
Boundary value problems of linear equations,
spectral theory of boundary value problems,
boundedness of solutions,
asymptotic expansion,
theoretical approximation of solutions
Article copyright:
© Copyright 1982
American Mathematical Society