Two new regularization methods for solving sideways heat equation (Q2261998)

The paper is concerned with the ill--posed problem for the heat equation: \(u_{t}(x,t)=u_{xx}(x,t)\), \(0<x<L\), \(0\leq t \leq 2\pi\); \(u(L,t)=g(t)\), \(0\leq t \leq 2\pi\); \(u(x,0)=0\), \(0<x<L\). Here, instead of \(g\), an approximation \(g^{\varepsilon} \in L^2(0,2\pi)\) is given, and \(\| g^{\varepsilon} -g\|_{L^2(0,2\pi)} \leq \varepsilon\). The authors describe and study two regularization methods, namely, the quasi-boundary value method and iteration methods to solve the problem and prove that the methods are stable under both a priori and a posteriori parameter choice rules.