- Title
- Finite difference methods for Burgers-Huxley equation and biofilm formation
- Creator
- Tijani, Yusuf Olatunji
- Subject
- Burgers equation
- Subject
- Terrestrial heat flow
- Subject
- Applied mathematics
- Date Issued
- 2023-12
- Date
- 2023-12
- Type
- Doctorial theses
- Type
- text
- Identifier
- http://hdl.handle.net/10948/62732
- Identifier
- vital:72934
- Description
- In this thesis, we constructed some versions of finite difference scheme for the Burgers-Huxley equation and for a set of partial differential equations (PDEs) arising in biofilm formation. The Burgers-Huxley equation serves as a fundamental model that describes the interaction between reaction mechanisms, convection effects, and diffusion transport. It has applications in the study of wave mechanics, population dynamics, physiology, fluid mechanics to list but a few. The study of biofilm formation is becoming increasingly important due to micro-organisms (i.e. bacteria) forming a protected mode from the host defense mechanism which may result in alteration in the host gene transcription and growth rate. Applications can be found useful in the treatment of bacterial infections, contamination of foods and water quality. We designed two nonstandard finite difference and two exponential finite difference schemes for the Burgers-Huxley equation. Numerical experiments with six cases and in three different regimes were studied. We show that the nonstandard scheme preserves the properties of the continuous equation which include positivity and boundedness. The stability region of the explicit exponential scheme was obtained and we outlined the algorithm for the implicit exponential scheme. The performance of the four schemes are compared in regard to absolute error, relative error, L1 and L∞ norms. For a singularly perturbed Burgers-Huxley equation, a novel nonstandard finite difference technique is constructed. It is demonstrated numerically that the NSFD scheme outperforms the classical scheme by comparing maximum pointwise errors and rate of convergence. We then solved the 2D Burgers-Huxley equation using four novel nonstandard finite difference schemes (NSFD1, NSFD2, NSFD3 and NSFD4). The numerical profiles from NSFD1 and NSFD2 exhibit some deviation from the exact profile. Our quest for a better performing scheme led to the modification of NSFD1 using the remainder effect technique. NSFD4 was designed by employing the time splitting approach. All the schemes preserve the properties of the continuous model (positivity and boundedness). The performance of all the schemes are analysed. We construct three nonstandard finite difference schemes for the equations modelling biomass equation and coupled substrate-biomass system of equations respectively. We checked the accuracy of our scheme by the conservation of physical properties (positivity, boundedness, biofilm formation and annihilation) since an analytical solution is not available. We show the instability, lack of conservation of physical properties by the classical scheme. Our proposed scheme shows good performance when compared with other results in the literature. The results here give more insight into the benefits of the nonstandard finite difference approximations.
- Description
- Thesis (DPhil) -- Faculty of Science, School of Computer Science, Mathematics, Physics and Statistics , 2023
- Format
- computer
- Format
- online resource
- Format
- application/pdf
- Format
- 1 online resource (vi, 220 pages)
- Format
- Publisher
- Nelson Mandela University
- Publisher
- Faculty of Science
- Language
- English
- Rights
- Nelson Mandela University
- Rights
- All Rights Reserved
- Rights
- Open Access
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Collections
NMU School of Computer Science, Mathematics, Physics and Statistics
| Thumbnail | File | Description | Size | Format | |||
|---|---|---|---|---|---|---|---|
| View Details Download | SOURCE1 | 221356053_TIJANI YO._.pdf | 9 MB | Adobe Acrobat PDF | View Details Download |