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.