Nonclassical equivalence transformations associated with a parameter identification problem

TL;DR

This paper explores nonclassical equivalence transformations for parameter identification in PDEs, enabling dimension reduction, special solutions, and new domain analysis, exemplified by nonlinear heat conduction.

Contribution

It introduces an approach to use nonclassical symmetry reductions in parameter identification problems, extending existing methods and updating computational tools for this purpose.

Findings

Enables reduction of PDEs in parameter identification

Allows analysis on new domain types for heat conduction

Provides a computational routine for determining equations

Abstract

A special class of symmetry reductions called nonclassical equivalence transformations is discussed in connection to a class of parameter identification problems represented by partial differential equations. These symmetry reductions relate the forward and inverse problems, reduce the dimension of the equation, yield special types of solutions, and may be incorporated into the boundary conditions as well. As an example, we discuss the nonlinear stationary heat conduction equation and show that this approach permits the study of the model on new types of domains. Our MAPLE routine GENDEFNC which uses the package DESOLV (authors Carminati and Vu) has been updated for this propose and its output is the nonlinear partial differential equation system of the determining equations of the nonclassical equivalence transformations.