Convection-Diffusion-Reaction equation with similarity solutions

TL;DR

This paper explores similarity solutions of generalized convection-diffusion-reaction equations with variable coefficients, reducing them to solvable ordinary differential equations and providing examples of exactly solvable systems.

Contribution

It introduces a method to find similarity solutions for complex convection-diffusion-reaction equations with variable coefficients, linking them to known solvable equations.

Findings

Derived a reduction of PDEs to ODEs using similarity variables

Identified classes of exactly solvable convection-diffusion-reaction systems

Constructed equivalent systems with identical similarity solutions

Abstract

We consider similarity solutions of the generalized convection-diffusion-reaction equation with both space- and time-dependent convection, diffusion and reaction terms. By introducing the similarity variable, the reaction-diffusion equation is reduced to an ordinary differential equation. Matching the resulting ordinary differential equation with known exactly solvable equations, one can obtain corresponding exactly solvable convection-diffusion-reaction systems. Some representative examples of exactly solvable systems are presented. We also describe how an equivalent convection-diffusion-reaction system can be constructed which admits the same similarity solution of another convection-diffusion-reaction system.