A computational approach to approximate the probability density function of random differential equations is based on transformation of random variables and finite difference schemes. The theoretical analysis of this computational method has not been performed in the extant literature. In this paper, we deal with a particular random differential equation: a random diffusion-reaction Poisson-type problem of the form $-u^{″}\left(x\right)+\alpha u\left(x\right)=\phi \left(x\right)$, $x\in [0,1]$, with boundary conditions $u\left(0\right)=A$, $u\left(1\right)=B$. Here, α, A and B are random variables and $\phi \left(x\right)$ is a stochastic process. The term $u\left(x\right)$ is a stochastic process that solves the random problem in the sample path sense. Via a finite difference scheme, we approximate $u\left(x\right)$ with a sequence of stochastic processes in both the almost sure and ${\mathrm{L}}^p$ senses. This allows us to find mild conditions under which the probability density function of $u\left(x\right)$ can be approximated. Illustrative examples are included.

近似计算随机微分方程的概率密度函数的方法是基于随机变量的转换和有限差分方案。现有文献中尚未对该计算方法进行理论分析。在本文中，我们处理一个特殊的随机微分方程：形式为随机扩散反应的泊松型问题$-{ü}^{\text{'}\text{'}}（X）+\alpha ü（X）=\phi （X）$， $X\in [0，\mathrm{1个}]$，具有边界条件 $ü（0）=\mathrm{一种}$， $ü（\mathrm{1个}）=乙$。在这里，α，A和B是随机变量，$\phi （X）$是一个随机过程。术语$ü（X）$是一个随机过程，可以解决样本路径意义上的随机问题。通过有限差分方案，我们近似$ü（X）$ 在几乎肯定和不合理的情况下都有一系列随机过程 ${\mathrm{大号}}^p$感官。这使我们能够找到温和条件，在该条件下$ü（X）$可以近似。包括说明性示例。