A heat approximation. (Q1582522)

The author studies the following boundary value problem: Let \(M\) be a set of points \((x,t)\in \mathbb R^ 2\) such that \(t\in (a,b)\) and \(x>\varphi(t)\), where \(\varphi \) is a continuous function of bounded variation on a compact interval \([a,b]\). Let \(g\) be a function continuous on \([a,b]\) such that \(g(a)=0\). The problem is to find a function \(h\) continuous and bounded on \(\overline {M}\) that fullfils on M the heat equation and is such that \(h(x,a)=0\) for \(x\geq \varphi (a)\) and \(h(\varphi (t),t)=g(t)\) for \(t\in[a,b]\). This Fourier problem is solved by means of a boundary integral equation. A simple numerical method for this integral equation is described and its convergence proved. Then, the author shows how to use it to approximate the solution of the Fourier problem and estimate the error. A numerical example is given and discussed in detail.