The Fundamental Solution for the Heat Equation on the half-line with Drift and Dirichlet Boundary Condition

TL;DR

This paper derives an explicit fundamental solution for the heat equation on a half-line with drift and boundary conditions using probabilistic methods, advancing understanding of such PDEs.

Contribution

It introduces a novel probabilistic approach to explicitly solve the heat equation with drift and boundary conditions on the half-line.

Findings

Explicit fundamental solution derived

Method applicable to similar PDEs with boundary conditions

Enhances analytical tools for heat equations with drift

Abstract

By a probabilistic method we provide an explicit fundamental solution of the Cauchy problem associated to the heat equation on the half-line with constant drift and Dirichlet boundary condition at zero.