- Title
- Solution of some cross-diffusion equations in biosciences using finite difference methods and artificial neural networks
- Creator
- de Waal, Gysbert Nicolaas
- Subject
- Mathematics
- Subject
- Difference equations
- Subject
- Functional equations
- Date Issued
- 2024-12
- Date
- 2024-12
- Type
- Master's theses
- Type
- text
- Identifier
- http://hdl.handle.net/10948/68829
- Identifier
- vital:77121
- Description
- In this dissertation, three cross-diffusion models which require positivity-preserving solutions are solved using standard and nonstandard finite difference methods and physics-informed neural networks. The three models are a basic reaction-diffusion-chemotaxis model, a convective predator-prey pursuit and evasion model, and a two-dimensional Keller-Segel chemotaxis model. All three models involve systems of nonlinear coupled partial differential equations subject to some specified initial and boundary conditions for which no exact solution is known. It is not possible to obtain the stability region of the standard finite difference methods for the three models theoretically and therefore a range of values of temporal step size at a given spatial step size for reasonable solutions is obtained by running some numerical experiments. It is observed that the standard finite difference schemes are not always positivity-preserving, and this is why nonstandard finite difference schemes are necessary. Chapter 1 provides some background detail on partial differential equations, cross-diffusion equations, finite difference methods, and artificial neural networks. In Chapter 2, the cross-diffusion models considered in this dissertation are provided, namely a basic reaction-diffusion–chemotaxis model for two cases, a convective predator-prey pursuit and evasion model, and a two-dimensional Keller-Segel chemotaxis model for two cases. In Chapter 3, the basic reaction-diffusion–chemotaxis model is solved for the two cases using some standard and nonstandard finite difference schemes. It is determined that the standard methods give reasonable positivity-preserving numerical solutions if the temporal step size, 𝑘, is chosen such that 𝑘 ≤ 0.25 with the spatial step size, ℎ, fixed at ℎ = 1.0. Two nonstandard finite difference methods abbreviated as NSFD1 and NSFD2 are considered from Chapwanya et al. (2014). It is shown that NSFD1 preserves the positivity of the continuous model if the following criteria are satisfied: 𝜙(𝑘)[𝜓(ℎ)]2=12𝛾≤12𝜎+𝛽 and 𝛽≤𝜎. NSFD1 is modified to obtain NSFD2, which is positivity-preserving if 𝑅=𝜙(𝑘)[𝜓(ℎ)]2=12𝛾 and 2𝜎𝑅≤1, that is, 𝜎≤𝛾. In this work, it is shown that NSFD2 does not always achieve consistency, and it is proven that consistency can be achieved if 𝛽→0 and 𝑘ℎ2→0. Lastly, it is demonstrated that the numerical rate of convergence in time of the finite difference methods is approximately one for case 2 of the chemotaxis model. In Chapter 4, one standard and two nonstandard finite difference methods are constructed to solve the convective predator-prey model. Through some numerical experiments, it is observed that reasonable
- Description
- Thesis (MSc) -- Faculty of Science, School of Computer Science, Mathematics, Physics and Statistics, 2024
- Format
- computer
- Format
- online resource
- Format
- application/pdf
- Format
- 1 online resource (156 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 | de Waal, GN.pdf | 11 MB | Adobe Acrobat PDF | View Details Download |