Electronic Thesis and Dissertation Repository

Thesis Format

Integrated Article

Degree

Doctor of Philosophy

Program

Applied Mathematics

Supervisor

Reid, Gregory

Abstract

PDE models arising in applications can be over-determined, and usually have non-trivial integrability conditions. Differential elimination methods can be applied to these systems, which involve a finite number of differentiations and eliminations to reveal hidden constraints, and form the determining equations of symmetries. Normally, such PDE systems have approximate parameters, and the direct application of well-defined symbolic differential elimination algorithms on such systems is prone to instability since these algorithms strongly depend on the ordering of the variables involved (similar to the instability of naïve Gaussian elimination).

To avoid such instability, numerical differential elimination methods are necessary. In this thesis, the Geometric Involutive Form is mainly used to perform numerical differential elimination of linear homogeneous DE systems. However, the application of GIF to the whole system leads to singular value decomposition on large matrices. To reduce the size of matrices and increase efficiency, this thesis presents an important idea of partitioning the whole differential equation system into exact and approximate subsystems. After that, applying symbolic methods to the exact subsystem and numerical techniques to the approximate subsystem ensures stability. Finally, checking that the jointly prolonged system is geometrically involutive is done by computing dimension criteria of the approximate subsystem and the differential Hilbert function of the exact subsystem. This approach enhances the efficiency in simplifying PDE systems involving both exact and approximate parameters.

Another contribution of this thesis is the establishment of a foundation for constructing and analyzing local approximate Lie symmetry algebras of differential equations. This thesis includes the definition of local approximate Lie symmetry of DEs, the computation of structure constants of the approximate Lie symmetry algebra of DEs, and the evaluation of the reliability of computed results.

Summary for Lay Audience

This research explores innovative ways to simplify complicated partial differential equation systems. PDEs are used to describe various physical phenomena, such as heat flow, fluid dynamics, and wave propagation, making them essential in fields like engineering and physics. However, usually over-determined PDE systems arising from modeling have nontrivial integrability conditions and involve approximate parameters, and the direct application of well-defined symbolic differential elimination algorithms on such systems lacks numerical stability. Therefore, a numerical differential elimination method is needed.

In this thesis, a new approach is developed to improve the efficiency of the numerical differential elimination method by partitioning a PDE system into two subsystems: one that can be solved exactly, and the other which requires numerical methods. This partition allows us to apply symbolic algorithms to the exact subsystem while applying advanced numerical methods to the approximate subsystem. This approach not only makes the process more efficient but also increases the accuracy of the solutions.

Moreover, this thesis defines local approximate Lie symmetry of differential equations and introduces a method to compute structure constants of such Lie symmetry algebra. These symmetries can help simplify the equations and can reveal deeper insights into the system’s behavior. By evaluating the computed result, this thesis provides a new idea to assess the reliability of approximate results, ensuring they are both meaningful and useful.

The methods developed in this thesis can be applied to a wide range of problems, offering a powerful tool for scientists and engineers who need to solve complicated DE systems. This research not only advances the field of mathematics but also has practical implications for improving the design and optimization of various technological systems.

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