Electronic Thesis and Dissertation Repository

Thesis Format

Integrated Article

Degree

Doctor of Philosophy

Program

Applied Mathematics

Supervisor

Gregory Reid

Abstract

Differential Equations are used to mathematically express the laws of physics and models in biology, finance, and many other fields. Examining the solutions of related differential equation systems helps to gain insights into the phenomena described by the differential equations. However, finding exact solutions of differential equations can be extremely difficult and is often impossible. A common approach to addressing this problem is to analyze solutions of differential equations by using their symmetries. In this thesis, we develop algorithms based on analyzing infinitesimal symmetry features of differential equations to determine the existence of invertible mappings of less tractable systems of differential equations (e.g., nonlinear) into more tractable systems of differential equations (e.g., linear). We also characterize features of the map if it exists. An algorithm is provided to determine if there exists a mapping of a non-constant coefficient linear differential equation to one with constant coefficients. These algorithms are implemented in the computer algebra language Maple, in the form of the MapDETools package. Our methods work directly at the level of systems of equations for infinitesimal symmetries. The key idea is to apply a finite number of differentiations and eliminations to the infinitesimal symmetry systems to yield them in the involutive form, where the properties of Lie symmetry algebra can be explored readily without solving the systems. We also generalize such differential-elimination algorithms to a more frequently applicable case involving approximate real coefficients. This contribution builds on a proposal by Reid et al. of applying Numerical Algebraic Geometry tools to find a general method for characterizing solution components of a system of differential equations containing approximate coefficients in the framework of the Jet geometry. Our numeric-symbolic algorithm exploits the fundamental features of the Jet geometry of differential equations such as differential Hilbert functions. Our novel approach establishes that the components of a differential equation can be represented by certain points called critical points.

Summary for Lay Audience

Differential Equations are used to mathematically express the governing laws of physics and models in biology, finance, and other fields. However, such equations can be difficult to analyze or solve analytically or numerically. For example, they may be nonlinear, or even if they are linear, may have non-constant coefficients. In this thesis, we develop algorithms to determine whether an invertible mapping of a nonlinear system of differential equations to a linear system exists. Once existence is established, it can determine features of the map and if possible explicitly determine the mapping by integration. We also provide an algorithm to determine if a mapping of a non-constant coefficient linear differential equation onto a simple constant-coefficient differential equation exists. These algorithms are implemented in the symbolic computation language Maple, as a part of the MapDETools package. So, our methods are available to a wide audience through user-friendly interfaces. The above methods depend on analyzing the symmetry properties of the input (e.g., nonlinear) systems for features that characterize the (e.g. linear) target. The methods also employ an exact differential-elimination algorithm that applies a finite number of differentiations and eliminations to the system of differential equations for the symmetries and reduces them to the involutive form, where their properties are readily determined. We also generalize such differential-elimination algorithms to the more realistic case of input systems with approximate real coefficients. This algorithm exploits fundamental features of the Jet geometry of differential equations such as differential Hilbert functions.

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