×

On the reduction of Holt’s problem to a finite interval. (English) Zbl 0625.65070

Holt’s problem: \(x''-p(t)x=0\), \(0<t<\infty\), \(x(0)=b\), \(x(\infty)=0\), \(p(t)=t^ 2(1+p_ 0t^ 2)\), \(p_ 0=2m+1\), m nonnegative integer, is solved using Abramov’s variant of the “chasing” (or “factorization” or “sweeping”) method, replacing the infinite interval by \(0<t<T\), and the condition \(x(\infty)=0\) by \(x'(t^*)-t^*{\tilde \alpha}(t^*)x(t^*)=0,\) \(0<t^*<T\). This condition appears instead of the usual one: \(x'(T)-T\alpha (T)x(T)=0,\) \({\tilde \alpha}\)(t) being an approximation of \(\alpha\) (t). The method seems advantageous when a small interval \([0,t^*]\) is used. The authors construct an error estimate for both reductions.
Reviewer: A.de Castro

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Abramov, A.A.: A variant of the sweeping method. USSR J. Comput. Math. Math. Phys.1, 349-351 (1961) (in Russian)
[2] Abramov, A.A., Birget, E.S., Konyukhova, N.B., Ulyanova, V.I.: On methods of numerical solution of boundary value problems for systems of linear ordinary differential equations. In: Rozsa, P. (ed.), Colloquia Mathematica Societatis János Bolyai 22. Numerical Methods, Keszthely, (Hungary). pp. 33-67. Amsterdam, Oxford, New York: North Holland 1977
[3] Abramov, A.A., Dyshko, A.L., Konyukhova, N.B., Pak, T.V., Parijskij, B.S.: Computation of prolate spheroidal functions by solving the corresponding differential equations. USSR J Comput. Math. and Math. Phys.24, 3-18 (1984) (in Russian) · Zbl 0539.65006
[4] Agarwal, R.P.: On Gel’fand’s method of chasing for solving multipoint boundary value problems. In: Equadiff 6. Proceedings of the International Conference on Differential Equations and their Applications (Vosmanský, J., Zlámal, M., (eds.). pp. 267-274. Brno, Aug. 26-30, 1985 Brno: J.E. Purkyne University
[5] Agarwal, R.P., Gupta, R.C.: On the solution of Holt’s problem. BIT24, 342-346 (1984) · Zbl 0551.65058 · doi:10.1007/BF02136032
[6] Bailey, P.B., Shampine, L.F.: Automatic solution of Sturm-Liouville eigenvalue problems. In: Childs, B., Scott, M., Daniel, J.W., Nelson, P. (eds.) Codes for boundary-value problems in ordinary differential equations. Proceedings of a Working Conference May 14-17. 1978 Lect. Notes Comput. Sci. vol. 76. Berlin, Heidelberg, New York: Springer 1979
[7] Berezin, J.S., Zhidkov, N.P.: Methods of computation, vol. 2. Oxford: Pergamon 1965. · Zbl 0122.12903
[8] Beryland, O.S., Gavrilova, R.I., Prudnikov, A.P.: Tables of integral error functions and hermite polynomials. New York: Macmillan 1962 · Zbl 0105.11304
[9] Birger, E.S.: On the error estimation for the replacement of the boundedness condition for the solution of a linear differential equation on the infinite interval. USSR. J. Comput. Math. and Math. Phys.8, 674-676 (1968) (in Russian) · Zbl 0203.48204
[10] Birger, E.S., Lyalikova, N.B.: On finding solutions with give condition at the infinity for certain systems of ordinary differential equations. Part 1.USSR J. Comput. Math. and Math. Phys.5, 979-990 (1965) (in Russian)
[11] Holt, J.F.: Numerical solution of nonlinear two-point boundary problems by finite difference methods. Commun. ACM7, 366-373 (1964) · Zbl 0123.11805 · doi:10.1145/512274.512291
[12] Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher transcendental functions. New York Toronto London: Mc Graw-Hill Book Comp. 1953 · Zbl 0052.29502
[13] Markowich, P.: A theory for the approximation of solutions of boundary value problems on infinite intervals. SIAM J Math. Anal.13, 484-513 (1982) · Zbl 0498.34007 · doi:10.1137/0513033
[14] Osborne, M.R.: On shooting methods for boundary value problems. J. Math. Anal. Appl.27, 417-433 (1969) · Zbl 0177.20402 · doi:10.1016/0022-247X(69)90059-6
[15] Roberts, S.M., Shipman, J.S.: Multipoint solution of two-point boundary value problems. J. Optimization Theory Appl.7, 301-318 (1971) · Zbl 0202.15905 · doi:10.1007/BF00928709
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.