Collocating convolutions

Author:
Frank Stenger

Journal:
Math. Comp. **64** (1995), 211-235

MSC:
Primary 65D30; Secondary 41A35, 65N35, 65R10

MathSciNet review:
1270624

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Abstract: An explicit method is derived for collocating either of the convolution integrals or , where , a subinterval of . The collocation formulas take the form or , where **g** is an *m*-vector of values of the function *g* evaluated at the "Sinc points", and are explicitly described square matrices of order *m*, and , for arbitrary . The components of the resulting vectors **p** (resp., **q**) approximate the values of *p* (resp., *q*) at the Sinc points, and may then be used in a Sinc interpolation formula to approximate *p* and *q* at arbitrary points on (a, b). The procedure offers a new method of approximating the solutions to (definite or indefinite) convolution-type integrals or integral equations as well as solutions of partial differential equations that are expressed in terms of convolution-type integrals or integral equations via the use of Green's functions. If *u* is the solution of a partial differential equation expressed as a *v*-dimensional convolution integral over a rectangular region **B**, and if *u* is analytic and of class on the interior of each line segment in **B**, then the complexity of computing an -approximation of *u* by the method of this paper is .

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1995-1270624-7

Keywords:
Indefinite integral convolution

Article copyright:
© Copyright 1995
American Mathematical Society