Analytic solutions of the heat equation

An Approximation of Solutions of the First Fourier Problem for the Heat Equation With an Application of the Orthogonalization Procedure

“…By the formula where Hj(z) is the Hermite polynomial (see [1] or [3]), and by the recurrence formula for the Hermite polynomials…”

Existence of Gauss Interpolation Formulas for the One-Dimensional Heat Equation

“…Widder [15] showed that the set {uk}k=0 is complete, in the maximum norm, in the space of solutions to (1) which are analytic in a neighborhood of the origin; i.e., if uix, t) is a solution to Eq. (1) which is analytic for \x\ < c, \t\ < c, then u can be approximated arbitrarily closely by a finite linear combination of the functions {«fc}¡°=0.…”

Boundary behavior of random valued heat polynomial expansions

“…Differentiating ΣΣU^M* term by term n times yields Y£= n k\l{k -n)\a k t k - Because of the bounds obtained in the preceding paragraph it can be shown that the series of (4.1) can be differentiated term by term and that u(x, t) is a C°°-solution to the heat equation in the closed strip 111 <£ 1. Since both u(0 9 1) and du/dx(0, t), as functions of ί on ( -1,1), are given by their Maclaurin expansions, u has a heat polynomial expansion in \t\ < 1 (see [5]). Thus…”

“…Since u(0, t) and du/dx(0, t) are both given by their Maclaurin expansions in 111 < 1, u possesses a heat polynomial expansion in the strip 111 < 1 (see [5]). Thus for 11 \ < 1, u(x, t) = Σ~=o α n v Λ (&, *); a^ = (e~^n\)l(2n)\, a 2n+1 = 0.…”