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

Abstract:

Let $C = \{ (x(s),t(s)):a \leqslant s \leqslant b\}$ be a Jordan arc in the x-t plane satisfying $(x(a),t(a)) = (a,{t_ \ast }),(x(b),t(b)) = (b,{t_\ast })$, and $t(s) < {t_\ast }$ when $a < s < b$. Let $a < {x_\ast } < b$. We prove the existence of Gauss interpolation formulas for C and the point $({x_\ast },{t_\ast })$, for solutions u of the one-dimensional heat equation, ${u_t} = {u_{xx}}$. Such formulas approximate $u({x_\ast },{t_\ast })$ 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.