Eigenvalue problems on infinite intervals
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 by Peter A. Markowich PDF
 Math. Comp. 39 (1982), 421441 Request permission
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.References

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 10.1016/00219991(77)901024
 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 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 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 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, Regional Conference Series in Applied Mathematics, No. 24, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976. 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 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 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 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 10.1016/00219991(80)901503
 John E. Osborn, Spectral approximation for compact operators, Math. Comput. 29 (1975), 712–725. MR 0383117, DOI 10.1090/S00255718197503831173
 Friedrich Stummel, Diskrete Konvergenz linearer Operatoren. I, Math. Ann. 190 (1970/71), 45–92 (German). MR 291870, DOI 10.1007/BF01349967
 Friedrich Stummel, Diskrete Konvergenz linearer Operatoren. II, Math. Z. 120 (1971), 231–264 (German). MR 291871, DOI 10.1007/BF01117498
Additional Information
 © Copyright 1982 American Mathematical Society
 Journal: Math. Comp. 39 (1982), 421441
 MSC: Primary 34B25
 DOI: https://doi.org/10.1090/S00255718198206696376
 MathSciNet review: 669637