Existence of Gauss interpolation formulas for the one-dimensional heat equation

Abstract

Let C = { ( x ( s ) , t ( s ) ) : a ⩽ s ⩽ b } C = \{ (x(s),t(s)):a \leqslant s \leqslant b\} be a Jordan arc in the x-t plane satisfying ( x ( a ) , t ( a ) ) = ( a , t ∗ ) , ( x ( b ) , t ( b ) ) = ( b , t ∗ ) (x(a),t(a)) = (a,{t_ \ast }),(x(b),t(b)) = (b,{t_\ast }) , and t ( s ) > t ∗ t(s) > {t_\ast } when a > s > b a > s > b . Let a > x ∗ > b a > {x_\ast } > b . We prove the existence of Gauss interpolation formulas for C and the point ( x ∗ , t ∗ ) ({x_\ast },{t_\ast }) , for solutions u of the one-dimensional heat equation, u t = u x x {u_t} = {u_{xx}} . Such formulas approximate u ( x ∗ , t ∗ ) u({x_\ast },{t_\ast }) in terms of a linear combination of its values on C. The formulas are characterized by the requirement that they are exact for as many basis functions (the heat polynomials) as possible.

Gauss Interpolation Formulas and Totally Positive Kernels

“…As in [1], the proof uses the concept of topological degree to establish the existence of a solution to a system of A equations in A unknowns.…”

“…The formula (1), where the points (xk, tk) lie on C and the weights Ak aie positive, is characterized by the requirement that it be exact for as many "basis functions" as possible. In [1] we proved the existence of w-point formulas which are exact for all heat polynomials of degree n = 2m -I, and that this is best possible, in the sense that no m-point formula is exact for all heat polynomials of degree n x > 2m -1. Such formulas were called Gauss interpolation formulas, because of their similarity to Gaussian quadrature formulas.…”

“…Karlin and Studden [3]). These facts can be used to simplify the proof of the result from [1] where a = <x(x, t) is a prescribed continuous function on the curve C. Such a formula could be used, for example, in the case where the data au + bux = / is known on C, for a and b fixed continuous functions with a > 0. One first obtains the formula (4) corresponding to a = b/a, and then applies it to the data f/a.…”

An algorithm for Gauss harmonic formulas

“…Barrow [4,5] discussed similar formulas for the heat equation. Finally Johnson and Riess [6] gave a closed form for formulas on a circular region.…”

Gauss interpolation formulas and totally positive kernels

“…Introduction. In a recent paper [1] we discussed formulas of the form m (1) u(x*, t*) -Z Ak»iXk-tk) k=l for approximating solutions to the heat equation…”