Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

A sinc-collocation method for
initial value problems


Authors: Timothy S. Carlson, Jack Dockery and John Lund
Journal: Math. Comp. 66 (1997), 215-235
MSC (1991): Primary 65L05, 65L60
DOI: https://doi.org/10.1090/S0025-5718-97-00789-8
MathSciNet review: 1372000
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A collocation procedure is developed for the initial value problem $u'(t) = f(t,u(t))$, $u(0) = 0$, using the globally defined sinc basis functions. It is shown that this sinc procedure converges to the solution at an exponential rate, i.e., $ \mathcal { O} (M^{2} \exp (-\kappa \sqrt {M}) )$ where $\kappa > 0$ and $2M$ basis functions are used in the expansion. Problems on the domains $\mathbb {R} = (-\infty ,\infty )$ and $\mathbb {R} ^{+} = (0,\infty )$ are used to illustrate the implementation and accuracy of the procedure.


References [Enhancements On Off] (What's this?)

  • 1. B. Bialecki, Sinc-collocation methods for two-point boundary value problems, IMA J. Numer. Anal. 11 (1991), 357-375. MR 92f:65086
  • 2. T. S. Carlson, Sinc methods for Burgers' equation, Ph.D. thesis, Montana State University, 1995.
  • 3. N. Eggert, M. Jarratt, and J. Lund, Sinc function computation of the eigenvalues of Sturm-Liouville problems, J. Comput. Phys. 69 (1987), no. 1, 209-229. MR 89c:65090
  • 4. U. Grenander and G. Szegö, Toeplitz forms and their applications, 2nd ed., Chelsea Publishing Co., New York, 1984. MR 88b:42031
  • 5. J. Lund and K. L. Bowers, Sinc methods for quadrature and differential equations, SIAM, Philadelphia, 1992. MR 93i:65004
  • 6. J. Lund and B. V. Riley, A sinc-collocation method for the computation of the eigenvalues of the radial Schrödinger equation, IMA J. Numer. Anal. 4 (1984), 83-98. MR 86f:65134
  • 7. L. Lundin and F. Stenger, Cardinal type approximations of a function and its derivatives, SIAM J. Math. Anal. 10 (1979), 139-160. MR 81c:41043
  • 8. K. M. McArthur, A collocative variation of the Sinc-Galerkin method for second order boundary value problems, Computation and Control (K. Bowers and J. Lund, eds.), Birkhäuser, Boston, 1989, pp. 253-261. CMP 90:10
  • 9. A. C. Morlet, Convergence of the sinc method for a fourth-order ordinary differential equation with an application, SIAM J. Numer. Anal. 32 (1995), 1475-1503. MR 96f:65097
  • 10. F. Stenger, Numerical methods based on sinc and analytic functions, Springer-Verlag, New York, 1993. MR 94k:65003
  • 11. F. Stenger, B. Barkey, and R. Vakili, Sinc convolution approximate solution of Burgers' equation, Computation and Control III (K. Bowers and J. Lund, eds.), Birkhäuser, Boston, 1993, pp. 341-354. CMP 94:04

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 65L05, 65L60

Retrieve articles in all journals with MSC (1991): 65L05, 65L60


Additional Information

Timothy S. Carlson
Affiliation: Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501
Email: tim@santafe.edu

Jack Dockery
Affiliation: Department of Mathematics, Montana State University, Bozeman, Montana 59717
Email: umsfjdoc@math.montana.edu

John Lund
Affiliation: Department of Mathematics, Montana State University, Bozeman, Montana 59717
Email: umsfjlun@math.montana.edu

DOI: https://doi.org/10.1090/S0025-5718-97-00789-8
Received by editor(s): February 27, 1995
Received by editor(s) in revised form: November 2, 1995, and January 26, 1996
Additional Notes: The first author was supported in part by the Office of Naval Research under contract ONR-00014-89-J-1114.
The second author was supported in part by the National Science Foundation grants OSR-93-50-546 and DMS-94-04-160.
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society