Uniform convergence of wavelet solution to the sideways heat equation

We consider the problem u _t (x, t), 0 ≤ x < 1, t ≥ 0, where the Cauchy data g(t) is given at x = 1. This is an ill-posed problem in the sense that a small disturbance on the boundary g(t) can produce a big alteration on its solution (if it exists). We shall define a wavelet solution to obtain the well-posed approximating problem in the scaling space V _j . In the previous papers, the theoretical results concerning the error estimate are L ²-norm and the solutions aren’t stable at x = 0. However, in practice, the solution is usually required to be stable at the boundary. In this paper we shall give uniform convergence on interval x ∈ 0, 1].