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Mathematics of Computation

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



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 $ p(x) = \smallint _a^xf(x - t)g(t)dt$ or $ q(x) = \smallint _x^bf(t - x)g(t)dt$, where $ x \in (a,b)$, a subinterval of $ \mathbb{R}$. The collocation formulas take the form $ {\mathbf{p}} = F({A_m}){\mathbf{g}}$ or $ {\mathbf{q}} = F({B_m}){\mathbf{g}}$, where g is an m-vector of values of the function g evaluated at the "Sinc points", $ {A_m}$ and $ {B_m}$ are explicitly described square matrices of order m, and $ F(s) = \smallint _0^c\exp [ - t/s]f(t)dt$, for arbitrary $ c \in [(b - a),\infty ]$. 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 $ {\mathbf{Li}}{{\mathbf{p}}_\alpha }$ on the interior of each line segment in B, then the complexity of computing an $ \varepsilon $-approximation of u by the method of this paper is $ \mathcal{O}({[\log (\varepsilon )]^{2\nu + 2}})$.

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Keywords: Indefinite integral convolution
Article copyright: © Copyright 1995 American Mathematical Society

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