Initial-to-Interface Maps for the Heat Equation on Composite Domains

Abstract

A map from the initial conditions to the function and its first spatial derivative evaluated at the interface is constructed for the heat equation on finite and infinite domains with n interfaces. The existence of this map allows changing the problem at hand from an interface problem to a boundary value problem which allows for an alternative to the approach of finding a closed-form solution to the interface problem.

Global existence for Laplace reaction-diffusion equations

“…First, u(t) is a real function. Indeed, if u(t) is a solution to (8), then its complex conjugate u(t) is also a solution lying in (16) and having the same initial value. Uniqueness of solution then implies that u(t) = u(t).…”

“…Mikhailov-Özişik [14] and Salt [15] construct solutions for two and three-dimensional linear equations. Meanwhile, Sheils-Deconinck [16] constructs a mapping from the initial functions to the trace functions of the solutions on the interfaces. We hope that the techniques obtained in this paper together with those of handling (5) will open researches for the nonlinear problems, i.e., (5)- (6).…”

The time-dependent Schrödinger equation with piecewise constant potentials

“…The existence of this map allows one to change the problem at hand from an interface problem to a boundary-value problem (BVP) which allows for an alternative to the approach of finding a closed-form solution to the interface problem. This was explored previously by the authors for the heat equation in [32].…”

The unified transform for evolution equations on the half‐line with time‐periodic boundary conditions*

“…For such equations, among the most important results obtained via this method are the following: (i) Linear equations formulated on the half-line or a finite interval have been analyzed by Deconinck, Fokas, Pelloni, and collaborators. [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] (ii) Numerical techniques for linear equations are developed in Refs. 18-22. (iii) Novel results in spectral theory are derived in Refs.…”