  In this paper we study the randomized heat equation with homogeneous boundary
conditions. The diffusion coeffcient is assumed to be a random variable and the
initial condition is treated as a stochastic process. The solution of this
randomized partial differential equation problem is a stochastic process, which
is given by a random series obtained via the classical method of separation of
variables. Any stochastic process is determined by its finite-dimensional joint
distributions. In this paper, the goal is to obtain approximations to the
probability density function of the solution (the first finite-dimensional
distributions) under mild conditions. Since the solution is expressed as a
random series, we perform approximations of its probability density function.
We use two approaches: broadly speaking, first, dealing with the random Fourier
coefficients of the random series, and second, taking advantage of the
Karhunen-Loeve expansion of the initial condition stochastic process. Finally,
several numerical examples illustrating the potentiality of our findings with
regard to both approaches are presented.
