On polynomial solutions to Fokker-Planck and sinked density evolution equations

TL;DR

This paper provides analytical solutions for density evolution models with specific coefficients, expressing eigenfunctions via hypergeometric functions, and extends classical orthogonal polynomials to include sink terms, with applications to Langevin equations.

Contribution

It introduces new analytical solutions for density evolution equations with sink terms, generalizing classical orthogonal polynomials and applying MacRobert's proof for normalization.

Findings

Eigenfunctions expressed in hypergeometric functions

Discrete spectra generalize orthogonal polynomials

Analytical solutions for Bertalanffy-Richards Langevin equation

Abstract

We analytically solve for the time dependent solutions of various density evolution models. With specific forms of the diffusion, drift and sink coefficients, the eigenfunctions can be expressed in terms of hypergeometric functions. We obtain the relevant discrete and continuous spectra for the eigenfunctions. With non-zero sink terms the discrete spectra eigenfunctions are generalisations of well known orthogonal polynomials: the so-called associated-Laguerre, Bessel, Fisher-Snedecor and Romanovski functions. We use a MacRobert's proof to obtain closed form expressions for the continuous normalisation of the Romanovski density function. Finally, we apply our results to obtain the analytical solutions associated with the Bertalanffy-Richards Langevin equation.