## A novel numerical solver for nonlinear boundary value problems, with applications to the forced Gardner equation

thesis

posted on 03.04.2019 by ANDREW CRAIG CULLEN#### thesis

In order to distinguish essays and pre-prints from academic theses, we have a separate category. These are often much longer text based documents than a paper.

Nonlinear differential equations appear within a wide range of fields, however, solving these problems is particularly difficult. This work charts the development of a suite of new, numerical tools for solving nonlinear differential equations. These tools significantly outperform currently existing methods, in both the amount of time required to solve the problems, and the range of problems that can be approached.
After analysing the properties of these tools, they are applied to the forced Gardner equation. This problem arises in geophysical fluid dynamics, and has particular implications to climate change and extreme weather events.

### Categories

- Geophysical Fluid Dynamics
- Numerical Analysis
- Ordinary Differential Equations, Difference Equations and Dynamical Systems
- Computational Physics
- Atmospheric Sciences
- Applied Computer Science
- Computation Theory and Mathematics
- Fluidisation and Fluid Mechanics
- Simulation and Modelling
- Applied Mathematics not elsewhere classified
- Numerical Solution of Differential and Integral Equations
- Numerical and Computational Mathematics not elsewhere classified
- Analysis of Algorithms and Complexity
- Numerical Computation
- Fluid Physics

### Keyword(s)

HomotopyAnalysisMethodHomotopy Analysis MethodGegenbauerUltrasphericalGegenbauer Homotopy Analysis Methodnumericalnonlinearcomputational physicsSpectralHAMGHAMSHAMalgorithmsKorteweg-de VriesKdVfKdVmKdVGardnercomputationoperationscomputational complexityfluidswavesrossbyatmosphericcomplexity analysis

### History

#### Campus location

Australia#### Principal supervisor

Simon Rex Clarke#### Year of Award

2018#### Department, School or Centre

Mathematical Sciences#### Course

Doctor of Philosophy#### Degree Type

DOCTORATE#### Faculty

Faculty of Science### Licence

### Exports

### Categories

- Geophysical Fluid Dynamics
- Numerical Analysis
- Ordinary Differential Equations, Difference Equations and Dynamical Systems
- Computational Physics
- Atmospheric Sciences
- Applied Computer Science
- Computation Theory and Mathematics
- Fluidisation and Fluid Mechanics
- Simulation and Modelling
- Applied Mathematics not elsewhere classified
- Numerical Solution of Differential and Integral Equations
- Numerical and Computational Mathematics not elsewhere classified
- Analysis of Algorithms and Complexity
- Numerical Computation
- Fluid Physics

### Keyword(s)

HomotopyAnalysisMethodHomotopy Analysis MethodGegenbauerUltrasphericalGegenbauer Homotopy Analysis Methodnumericalnonlinearcomputational physicsSpectralHAMGHAMSHAMalgorithmsKorteweg-de VriesKdVfKdVmKdVGardnercomputationoperationscomputational complexityfluidswavesrossbyatmosphericcomplexity analysis