Solution of the Neutral Kimura equation with two integral constraints

TL;DR

This paper derives explicit solutions to the neutral Kimura equation using Gegenbauer polynomials, ensuring they satisfy multiple integral constraints, and analyzes their long-term behavior in population genetics.

Contribution

It introduces a novel method to solve the Kimura equation with integral constraints using Gegenbauer polynomials and establishes new relations for these polynomials.

Findings

Explicit solutions in terms of Gegenbauer polynomials

New relations for Gegenbauer polynomials satisfying integral constraints

Analysis of long-term asymptotic behavior

Abstract

The Kimura equation is a degenerated partial differential equation of drift-diffusion type used in population genetics. Its solution is required to satisfy not only the equation but a series of conservation laws formulated as integral constraints. In this work, we consider a population of two types evolving without mutation or selection, the so-called neutral evolution. We obtain explicit solutions in terms of Gegenbauer polynomials. To satisfy the integral constraints it is necessary to prove new relations satisfied by the Gegenbauer polynomials. The long-term in time asymptotic is also studied.