The multi-resolution method applied to the sideways heat equation

We consider the sideways heat equation uxx(x,t)=ut(x,t), 0⩽x<1, t⩾0. The solution u(x,t) on the boundary x=1 is a known function g(t). This is an ill-posed problem, since the solution-if it exists-does not depend continuously on the boundary, i.e., small changes on the boundary may result in big changes in the solution. In this paper, we shall use the multi-resolution method based on the Shannon MRA to obtain a well-posed approximating problem and obtain an estimate for the difference between the exact solution and the solution of the approximating problem defined in Vj.