The two solutions solve different problems for the same equation.
The Fourier series solution solves the heat equation u_t=u_xx on a bounded interval [a,b] with an initial condition at t=0 of the form u(x,0)=f(x), a <=x<=b, and boundary conditions at both ends of the interval. These conditions can be of different types, leading to different series expansions. The most general (homogeneous) conditions are of the form
\alpha u(a)+\beta u_x(a)=\alpha u(b)+\beta u_x(b)=0, \alpha^2+\beta^2!=0.
If \beta=0 they are called Dirichlet conditions; if \alpha=0 Neumann conditions, if both \alpha and \beta are non zero, Robin conditions. Conditions can also be mixed: of one type on one end, of another type on the other end.
The convolution solutions solves the pure initial value problem, or Cauchy problem, on the whole real line, with initial value u(x,0)=f(x), x\in R.
I don't have a complete answer, but just some preliminary thoughts: the idea behind using Fourier analysis to solve constant coefficient linear PDEs is to transform a partial differential equation into an ordinary differential equation. In symbols, suppose $\psi:I\times\mathbb{R}^n\to \mathbb{C}$ solves the PDE
$$ \sum_{0\leq i \leq N} P_i(\nabla) \partial_t^i \psi = 0 $$
where $P_i$ are constant coefficient polynomials, the Fourier transform "gives an equation"
$$ \sum_{0 \leq i \leq N} P_i(\xi) \partial_t^i \hat{\psi} = 0~. $$
The problem is: how do you interpret this equation? To treat it as an ODE, you need to treat $\hat{\psi}$ as a map $I \to X$ where $X$ is, say, the Hilbert space of $L^2$ functions over $\mathbb{R}^n$ or some such.
Now, in your case of prescribing boundary conditions for the interval $[0,1]$, if your boundary conditions were time independent, and if the boundary conditions plays well with the Fourier transform, then you can again recover the ODE formulation. (The solvability of the ODE, as you noted, depends on which Hilbert space you use and the properties of the polynomials $P_i$ on the Hilbert space.)
But if your boundary conditions are time dependent, then an immediate problem is that the Hilbert space in which $\hat{\psi}$ lives will be time-dependent. So the naive application of "Fourier" methods won't make sense. Geometrically the case where $X$ is a fixed Hilbert space is the analogue of solving an ODE on the trivial vector bundle $V$ over $I$ with trivial connection. The case where $X$ also varies with time can be thought of as having some sort of an attempt at writing down an ODE on an arbitrary vector bundle $V$ over $I$. Without specifying the connection, even the notion of an ODE is not well-defined.
To put it differently, since a connection over a curve is just an identification of the fibres (roughly speaking), what you need to use an analogue of the Fourier method is a collection of 1-parameter families of functions $\phi_i(t;x)$, such that
- For each fixed $t$, the functions $\phi_i(t;x)$ forms an ON basis of some appropriate Hilbert space
- Each $\phi_i(t;x)$ solves the PDE you are looking at
Just directly assuming the trace of $\phi_i$ on constant $t$ slices are the trigonometric functions is probably not the right way to go in general.
Not having thought about this problem in detail before, I don't have much more to say. But I suspect that the suitability of individual boundary conditions need to be examined on a case by case basis.
Best Answer
The concept of a non-characteristic surface for a PDE or a system of PDE's is useful primarily for only establishing the existence and uniqueness of real analytic or formal power series solutions to the initial value problem using the Cauchy-Kovalevsky theorem.
The generalization of this to the smooth category is the class of hyperbolic PDE's, where you need the initial hypersurface to be more than characteristic. It has to be space-like.
Parabolic equations are a set of PDE's for which the initial value problem in time is well-posed only in the smooth category and not in the real analytic category. Indeed, if you try to apply the Cauchy-Kovalevsky theorem, the $t = c$ hypersurface is characteristic. From the point of view of this theorem, the initial value problem for the standard heat operator is well-posed only for hypersurfaces that are noncharacteristic with respect to the space-like Laplacian. The time derivative is lower order and does not even appear in the symbol.
However, in any useful application of parabolic equations, you want smooth solutions to the initial value problem in time. The explicit formula for the heat kernel shows that the solution is not necessarily real analytic in time. For parabolic equations, the study of solutions that are real analytic or have a power series expansion in the time variable is of little interest. You want to use a weaker category of solutions.