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Initial conditions for integrable-square solutions to singular differential equations

Published online by Cambridge University Press:  14 November 2011

Philip W. Walker
Philip W. Walker
Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 77204-3476, U.S.A.
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Synopsis

This paper deals with determining in a constructive manner those members of a linear space of functions which are of integrable-square. The space considered is the set of solutions to an ordinary differential equation, and the solutions of integrable-square are delineated by way of initial conditions. Numerical procedures for implementing the construction are discussed, and application is made to the deficiency index problem. Results from some specific computations are given.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 120 , Issue 1-2 , 1992 , pp. 117 - 127
Copyright
Copyright © Royal Society of Edinburgh 1992

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References

1Atkinson, F. V.. Dicrete and Continuous Boundary Problems (New York: Academic Press, 1964).Google Scholar
2Bulirsch, R. and Stoer, J.. Numerical treatment of ordinary differential equations by extrapolation methods. Numer. Math. 8 (1966), 1–13.CrossRefGoogle Scholar
3Coddington, E. A. and Levinson, N.. Theory of Ordinary Differential Equations (New York: McGraw-Hill, 1955).Google Scholar
4Dunford, N. and Schwartz, J. T.. Linear Operators, part II (New York: John Wiley-Interscience, 1963).Google Scholar
5Everitt, W. N. and Race, D.. Some remarks on linear ordinary quasi-differential expressions. Proc. London Math. Soc. (3) 54 (1987), 300–320.Google Scholar
6Everitt, W. N. and Zettl, A.. Generalized symmetric ordinary differential expressions I: the general theory. Nieuw Arch. Wisk. (3) 27 (1979), 363–397.Google Scholar
7Gear, C. W.. Numerical Initial Value Problems in Ordinary Differential Equations (Englewood Cliffs, New Jersey: Prentice-Hall, 1971).Google Scholar
8Hinton, D. B., Krall, A. M. and Shaw, J. K.. Boundary conditions for differential operators with intermediate deficiency index. Appl. Anal. 25 (1987), 43–53.CrossRefGoogle Scholar
9Hinton, D. B. and Shaw, J. K.. Hamiltonian systems of limit point or limit circle type with both endpoints singular. J. Differential Equations 50 (1983), 444–464.CrossRefGoogle Scholar
10Krall, A. M.. On nth order Stieltjes differential boundary operators and Stieltjes differential boundary systems. J. Differential Equations 24 (1977), 253–267.Google Scholar
11Lambert, J. D. and Shaw, B.. A method for the numerical solution of y' = f (x, y) based on self-adjusting non-polynomial interpolant. Math. Comp. 20 (1966), 11–20.Google Scholar
12Martin, E. M.. On the Theoretical and Numerical Determination of Deficiency Indices of Ordinary Differential Equations (M.Sc. thesis, University of Dundee, 1969).Google Scholar
13Naimark, M. A.. Linear Differential Operators, Part II, English edn (New York: Frederick Ungar, 1968).Google Scholar
14Neuberger, J. W.. A constructive lemma for the deficiency index problem. Proc. 1986–87Focused Research Program on Spectral Theory and Boundary Value Problems, Vol. 2, eds Kaper, H. G., Kwong, M. K. and Zettl, A., ANL87-26, pp. 149–152 (Argonne, Illinois: Argonne National Laboratory, 1987).Google Scholar
15Press, W. H., Flannery, B. P., Teukolsky, S. A. and Vettering, W. K.. Numerical Recipes (Cambridge: Cambridge University Press, 1986).Google Scholar
16Shaw, B.. Modified multistep methods based on a non-polynomial interpolant, J. Assoc. Comput. Mach. 14 (1967), 143–154.CrossRefGoogle Scholar
17Shaw, B.. Some multistep formulae for special high order differential equations. Numer. Math. 9 (1967), 367–378.CrossRefGoogle Scholar
18Riesz, F. and Sz.-Nagy, B.. Functional Analysis (New York: Frederick Ungar, 1955).Google Scholar
19Titchmarsh, E. C.. Eigenfunction Expansions Associated with Second-order Differential Equations, Part I, 2nd edn (Oxford: Oxford University Press, 1962).CrossRefGoogle Scholar
20Walker, P. W.. A vector-matrix formulation for formally symmetric ordinary differential equations with application to solutions of integrable-square. J. London Math. Soc. (2) 9 (1974), 151–159.CrossRefGoogle Scholar
21Walker, P. W.. The square integrable span of locally square integrable functions, Spectral Theory of Differential Operators. North-Holland Math. Stud. 55 (1981), 375–378.CrossRefGoogle Scholar
22Weidmann, J.. Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathematics 1258 (Berlin: Springer, 1987).CrossRefGoogle Scholar