The Riccati Differential Equation and a Diffusion-Type Equation

TL;DR

This paper derives explicit solutions for certain diffusion-type equations with variable coefficients, providing Green functions in elementary terms and addressing both homogeneous and non-homogeneous cases.

Contribution

It presents a novel explicit construction of solutions and Green functions for diffusion equations with variable coefficients on the real line.

Findings

Explicit Green functions expressed via elementary functions

Solutions applicable to non-homogeneous equations

Analysis of special and limiting cases

Abstract

We construct an explicit solution of the Cauchy initial value problem for certain diffusion-type equations with variable coefficients on the entire real line. The corresponding Green function (heat kernel) is given in terms of elementary functions and certain integrals involving a characteristic function, which should be found as an analytic or numerical solution of the second order linear differential equation with time-dependent coefficients. Some special and limiting cases are outlined. Solution of the corresponding non-homogeneous equation is also found.