Approximation of the probability density function of the randomized heat equation with non-homogeneous boundary conditions

TL;DR

This paper develops a method to approximate the probability density function of solutions to the randomized heat equation with non-homogeneous boundary conditions on general intervals, extending existing results for homogeneous boundary cases.

Contribution

It introduces a novel approach using variable transformations and the Random Variable Transformation technique to handle non-homogeneous boundary conditions.

Findings

Provides conditions for uniform and pointwise approximation of the density function.

Shows how to compute expectation and variance from the approximations.

Includes numerical examples with Gaussian and non-Gaussian initial conditions.

Abstract

This paper deals with the randomized heat equation defined on a general bounded interval $[L_1,L_2]$ and with non-homogeneous boundary conditions. The solution is a stochastic process that can be related, via changes of variable, with the solution stochastic process of the random heat equation defined on $[0,1]$ with homogeneous boundary conditions. Results in the extant literature establish conditions under which the probability density function of the solution process to the random heat equation on $[0,1]$ with homogeneous boundary conditions can be approximated. Via the changes of variable and the Random Variable Transformation technique, we set mild conditions under which the probability density function of the solution process to the random heat equation on a general bounded interval $[L_1,L_2]$ and with non-homogeneous boundary conditions can be approximated uniformly or pointwise. Furthermore, we provide sufficient conditions in order that the expectation and the variance of the solution stochastic process can be computed from the proposed approximations of the probability density function. Numerical examples are performed in the case that the initial condition process has a certain Karhunen-Lo\`eve expansion, being Gaussian and non-Gaussian.