Symmetry Analysis of Differential Equation with Lie Algebra

Tang, Hor Sin (2019) Symmetry Analysis of Differential Equation with Lie Algebra. Final Year Project (Bachelor), Tunku Abdul Rahman University College.

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Abstract

Lie symmetry is a powerful tool to solve different type of differential equations. Lie symmetry is used to help us to find the invariant solution of some complicated form of differential equation by reducing the order of the particular equation. In Chapter1, we will give a brief of introduction of differential equations and the meaning of the symmetry followed by the problem statement and objective of this project. In the next chapter, we will provide the contents of the literature review of this project. In Chapter3, we will show the way of proving a group to be Lie group by satisfying the four axioms and some examples of invariant of differential equation. In Chapter4, we will further explain about the infinitesimal transformation and the Lie’s invariance condition for Ordinary Differential Equation (ODE). In Chapter4 section 4.4, there are also some methods to solve different type of ODEs methods including Linear, Bernoulli, Homogeneous, Exact and Riccati equation. This chapter will also review some solutions of ODEs equation by applying the invariance condition to find the infinitesimal generator and introduce the canonical coordinates for the equation. Besides that, in Chapter5, we will focus in Partial Differential Equation (PDE) and also the extended infinitesimal transformation. This chapter will show the method of solving first-order and second-order PDE. Lastly, the conclusion and the suggestion of future work for this project is included in Chapter6.