Summary: The canonical diffusion problem for \(x\geq 0\), \(t\geq 0\), \[ u_ t(x,t)=\kappa^ 2(q)u_{xx}(x,t)+F(q,x,t,u), \]
\[ u(x,0)=f(x,q),\quad u(0,t)=0,\quad u_ x(\cdot,t)\text{ in } L_ 2[0,\infty), \] is studied in this paper. Here q is a fixed vector and may represent unknown parameters. The basic problems are to compute an approximate solution for fixed q and perhaps to identify q by using measurements at discrete spatial and time points. Unknown parameters can occur in the diffusion coefficient, in the initial function, or in the input function F. Attention will be given to a particular structural property for the functions f and F which will induce the same property on the solution u and which is useful in numerical computations.