Some Interpolation Formulas for Approximating the Solution of the Heat Equation.

Abstract

Interpolation formulas of the form

u(x_*1 , ... ,x_(* n),t_*) ~= Sum_{i=1}^{N} A_i u(v_i1,...,v_in, t_i)

are presented. These formulas are based on the heat polynomials of Appell. The point (x_{* 1}, ... x_{* n}t_*) is in the interior of an (n+1) dimensional region, R_n+1, and the points (v_i1,...,v_in, t_i) are on the boundary of R_n+1. These formulas can be used to approximate the solution of the heat conduction problem in R_n+1. The relationship between formulas of the above type of deqree 2 in R_n+1 and the second degree harmonic interpolation formulas of Stroud, Chen, Wang, and Mao in R_n is persented. Some higher degree formulas for special regions in R_2 are also developed.