Does the analytic solution of explicit scheme increase with time? I would solve $U_t=U_{xx}$. To do that I do the approximation $U(x,t)\approx u_{p,q}$ and use the explicit scheme $$\frac{u_{p,q+1}-u_{p,q}}{\delta t}=\frac{u_{p-1,q}-2u_{p,q}+u_{p+1,q}}{(\delta x)^2}.$$ My question is does $u_{p,q}$ increase in time? So the question boils down to $u_{p,q+1}-u_{p,q}>0$? Which is true if $u_{p-1,q}-2u_{p,q}+u_{p+1,q}>0$. Which I think cannot be determined. 
 A: I think the answer to this questions might be related to a linear stability analysis of the explicit Euler method you have used. Indeed, (if I remember well) explicit Euler method is linearly unstable (the numerical solution blows up as $q$ tends to infinity) for $r > 1/2$, where $r$ is the numerical diffusivity:
$$r = \alpha \, \delta t/ \delta x^2,$$ 
where $\alpha = k/\rho c_p$ is thermal diffusivity. In your problem, $\alpha = 1$.
I hope this is useful to you. Cheers.
Edit: let me be a bit more specific. I didn't notice that you were asking for the analytical solution. Indeed, the analytical solution can be found in terms of separation of variables as follows:
$$U(x,t) = \sum^\infty_{n=0} \psi_n(t) \phi_n(x),$$ where the eigenfunctions $\phi_n(x)$ depend on the boundary conditions whilst the Fourier coefficients $\psi_n(t)$ depend on the initial condition, so your problem strongly depends on both the boundary and initial conditions. This problem could (or could not) be coupled with the numerical stability problem.
