Reduction of balance laws in (3+1)--dimensions to autonomous conservation laws by means of equivalence transformations

TL;DR

This paper investigates how equivalence transformations can reduce complex (3+1)-dimensional balance laws with arbitrary functions to simpler autonomous conservation laws, facilitating analysis and solution of physical models.

Contribution

It introduces a method to transform a broad class of balance laws into autonomous conservation laws using equivalence transformations, extending previous approaches.

Findings

Identified transformations mapping balance laws to autonomous conservation laws

Demonstrated the method on a physical problem example

Provided a framework for simplifying complex PDE systems

Abstract

A class of partial differential equations (a conservation law and four balance laws), with four independent variables and involving sixteen arbitrary continuously differentiable functions, is considered in the framework of equivalence transformations. These are point transformations of differential equations involving arbitrary elements and live in an augmented space of independent, dependent and additional variables representing values taken by the arbitrary elements. Projecting the admitted symmetries into the space of independent and dependent variables, we determine some finite transformations mapping the system of balance laws to an equivalent one with the same differential structure but involving different arbitrary elements; in particular, the target system we want to recover is an autonomous system of conservation laws. An application to a physical problem is considered.