Lie Symmetry and Lie Bracket in Solving Differential Equation Models of Functional Materials: A Survey

Abstract

Functional materials are becoming an increasingly important part of our daily life, e.g. they used for sensing, actuation, computing, energy conversion. These materials often have  unique physical, chemical, and structural characteristic involving very complex phase.  Many mathematical model have been devised to study the complex behavior of functional materials. Some of the models have been proven powerful in predicting the behavior of new materials built upon the composites of existing materials. One of  mathematical methods used to model the behavior of the materials is the differential equation. Very often the resulting differential equations are very complicated so that most methods failed in obtaining the exact solutions of the problems. Fortunately, a relatively new approach via Lie symmetry gives a new hope in obtaining or at least understanding the behavior of the solutions, which is needed to understand the behavior of the materials being modeled. In this paper we present a survey on the use of Lie symmetry and related concepts (such as  Lie algebra, Lie group, etc) in modeling the behavior of functional materials and discuss some fundamental results of the Lie symmetry theory which often used in solving differential equations. The survey shows that the use of Lie symmetry and alike have been accepted in many field and gives an alternative approach in studying the complex behavior of functional materials.