## A sinc-collocation method for initial value problems

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- by Timothy S. Carlson, Jack Dockery and John Lund PDF
- Math. Comp.
**66**(1997), 215-235 Request permission

## 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

- Baruch Cahlon and Louis J. Nachman,
*Numerical methods for discontinuous linear boundary value problems with deviation arguments*, J. Math. Anal. Appl.**154**(1991), no. 2, 529–542. MR**1088649**, DOI 10.1016/0022-247X(91)90056-6 - T. S. Carlson,
*Sinc methods for Burgers’ equation*, Ph.D. thesis, Montana State University, 1995. - Norman Eggert, Mary Jarratt, and John Lund,
*Sinc function computation of the eigenvalues of Sturm-Liouville problems*, J. Comput. Phys.**69**(1987), no. 1, 209–229. MR**892259**, DOI 10.1016/0021-9991(87)90163-X - Ulf Grenander and Gábor Szegő,
*Toeplitz forms and their applications*, 2nd ed., Chelsea Publishing Co., New York, 1984. MR**890515** - John Lund and Kenneth L. Bowers,
*Sinc methods for quadrature and differential equations*, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR**1171217**, DOI 10.1137/1.9781611971637 - John R. Lund and Bruce V. Riley,
*A sinc-collocation method for the computation of the eigenvalues of the radial Schrödinger equation*, IMA J. Numer. Anal.**4**(1984), no. 1, 83–98. MR**740786**, DOI 10.1093/imanum/4.1.83 - L. Lundin and F. Stenger,
*Cardinal-type approximations of a function and its derivatives*, SIAM J. Math. Anal.**10**(1979), no. 1, 139–160. MR**516759**, DOI 10.1137/0510016 - 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. - Anne C. Morlet,
*Convergence of the sinc method for a fourth-order ordinary differential equation with an application*, SIAM J. Numer. Anal.**32**(1995), no. 5, 1475–1503. MR**1352199**, DOI 10.1137/0732067 - Frank Stenger,
*Numerical methods based on sinc and analytic functions*, Springer Series in Computational Mathematics, vol. 20, Springer-Verlag, New York, 1993. MR**1226236**, DOI 10.1007/978-1-4612-2706-9 - 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.

## 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
- 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. - © Copyright 1997 American Mathematical Society
- 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