Collocating convolutions

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 $\mathcal {O}({[\log (\varepsilon )]^{2\nu + 2}})$.