Exact solution of mixed problems for variable coefficient one-dimensional diffusion equation

This paper deals with diffusion problems modeled by the equation a(t)uxx = ut, x > 0, t > 0, u(x, 0) = c(x) together with the boundary condition u(0, t) = b(t) or ux(0, t) = b(t). By using Fourier transforms, existence conditions and exact solutions of the above mixed problems are given.