For usual uncertain heat equations, it is challenging to acquire their analytic solutions. A forward difference Euler method has been used to compute the uncertain heat equations\' numerical solutions. Nevertheless, the Euler scheme is instability in some cases. This paper proposes an implicit task to overcome this disadvantage, namely the Crank-Nicolson method, which is unconditional stability. An example shows that the Crank-Nicolson scheme is more stable than the previous scheme (Euler scheme). Moreover, the Crank-Nicolson method is also applied to compute two characteristics of uncertain heat equation\'s solution-expected value and extreme value. Some examples of uncertain heat equations are designed to show the availability of the Crank-Nicolson method.

In this paper, we derive several a posteriori error estimators for generalized diffusion equation with delay in a convex polygonal domain. The Crank-Nicolson method for time discretization is used and a continuous, piecewise linear finite element space is employed for the space discretization. The a posteriori error estimators corresponding to space discretization are derived by using the interpolation estimates. Two different continuous, piecewise quadratic reconstructions are used to obtain the error due to the time discretization. To estimate the error in the approximation of the delay term, linear approximations of the delay term are used in a crucial way. As a consequence, a posteriori upper and lower error bounds for fully discrete approximation are derived for the first time. In particular, long-time a posteriori error estimates are obtained for stable systems. Numerical experiments are presented which confirm our theoretical results.