Exactly solvable variable parametric Burgers type models

TL;DR

This paper introduces exactly solvable variable parametric Burgers equations in one dimension and provides two approaches for solving their initial value problems, linking them to classical Sturm-Liouville problems.

Contribution

It presents novel methods for solving variable parametric Burgers models, including a relationship-based approach and a generalized Cole-Hopf transform, expanding solvability to a broad class.

Findings

Two approaches for solving variable parametric Burgers equations are developed.

Solutions reduce to solving second order linear ODEs with time-dependent coefficients.

Results connect Burgers models with classical Sturm-Liouville problems.

Abstract

Exactly solvable variable parametric Burgers type equations in one-dimension are introduced, and two different approaches for solving the corresponding initial value problems are given. The first one is using the relationship between the variable parametric models and their standard counterparts. The second approach is a direct linearization of the variable parametric Burgers model to a variable parametric parabolic model via a generalized Cole-Hopf transform. Eventually, the problem of finding analytic and exact solutions of the variable parametric models reduces to that of solving a corresponding second order linear ODE with time dependent coefficients. This makes our results applicable to a wide class of exactly solvable Burgers type equations related with the classical Sturm-Liouville problems for the orthogonal polynomials.