Chapter 16 - The Solution in a General Region with Temperature-Boundary Specification: The Method of Perron-Poincaré

Summary

INTRODUCTION

We consider the problem

where f, g, h, s₁, and s₂ are continuous functions of their respective arguments, f(a) = g(0), and f(b) = h(0). (See Fig. 16.1.1.) Let

and let B_T denote its boundary. For what follows it is convenient to define

Denoting by F the triple (g, f, h) on B_T, Problem (16.1.1) condenses to the Statement

From the definition of D_T, we see that for, any point (x, t) ∈ D_T, there exists a closed polygonal region P with one side a line segment contained in {T} ∩ D_T, another a line segment contained in {τ} ∩ D_T, 0 < τ < t, and the remaining boundary of P consisting of two nonintersecting polygonal curves whose line segments are not characteristics and which connect, respectively, the left endpoint of the line segment on t = τ to the left endpoint of the line segment on t = T and the right endpoint to the right endpoint, such that (x, t) ∈ P°, the interior of P.

Remark. We shall call all such polygonal regions P admissible. Moreover, we shall utilize only admissible P in the discussion below.

For all such admissible P, we define the following operator M_P on continuous functions in D_T ∪ B_T.