Finite difference methods for Burgers-Huxley equation and biofilm formation
- Authors: Tijani, Yusuf Olatunji
- Date: 2023-12
- Subjects: Burgers equation , Terrestrial heat flow , Applied mathematics
- Language: English
- Type: Doctorial theses , text
- Identifier: http://hdl.handle.net/10948/62732 , 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. , Thesis (DPhil) -- Faculty of Science, School of Computer Science, Mathematics, Physics and Statistics , 2023
- Full Text:
- Date Issued: 2023-12
- Authors: Tijani, Yusuf Olatunji
- Date: 2023-12
- Subjects: Burgers equation , Terrestrial heat flow , Applied mathematics
- Language: English
- Type: Doctorial theses , text
- Identifier: http://hdl.handle.net/10948/62732 , 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. , Thesis (DPhil) -- Faculty of Science, School of Computer Science, Mathematics, Physics and Statistics , 2023
- Full Text:
- Date Issued: 2023-12
Finite element simulations of shear aggregation as a mechanism to form platinum group elements (PGEs) in dyke-like ore bodies
- Authors: Mbandezi, Mxolisi Louis
- Date: 2002
- Subjects: Platinum group , Magmas , Shear flow , Geophysics , Terrestrial heat flow
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5561 , http://hdl.handle.net/10962/d1018249
- Description: This research describes a two-dimensional modelling effort of heat and mass transport in simplified intrusive models of sills and their feeder dykes. These simplified models resembled a complex intrusive system such as the Great Dyke of Zimbabwe. This study investigated the impact of variable geometry to transport processes in two ways. First the time evolution of heat and mass transport during cooling was investigated. Then emphasis was placed on the application of convective scavenging as a mechanism that leads to the formation of minerals of economic interest, in particular the Platinum Group Elements (PGEs). The Navier-Stokes equations employed generated regions of high shear within the magma where we expected enhanced collisions between the immiscible sulphide liquid particles and PGEs. These collisions scavenge PGEs from the primary melt, aggregate and concentrate it to form PGEs enrichment in zero shear zones. The PGEs scavenge; concentrate and 'glue' in zero shear zones in the early history of convection because of viscosity and dispersive pressure (Bagnold effect). The effect of increasing the geometry size enhances scavenging, creates bigger zero shear zones with dilute concentrate of PGEs but you get high shear near the roots of the dyke/sill where the concentration will not be dilute. The time evolution calculations show that increasing the size of the magma chamber results in stronger initial convection currents for large magma models than for small ones. However, convection takes, approximately the same time to cease for both models. The research concludes that the time evolution for convective heat transfer is dependent on the viscosity rather than on geometry size. However, conductive heat transfer to the e-folding temperature was almost six times as long for the large model (M4) than the small one (M2). Variable viscosity as a physical property was applied to models 2 and 4 only. Video animations that simulate the cooling process for these models are enclosed in a CD at the back of this thesis. These simulations provide information with regard to the emplacement history and distribution of PGEs ore bodies. This will assist the reserve estimation and the location of economic minerals.
- Full Text:
- Date Issued: 2002
- Authors: Mbandezi, Mxolisi Louis
- Date: 2002
- Subjects: Platinum group , Magmas , Shear flow , Geophysics , Terrestrial heat flow
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5561 , http://hdl.handle.net/10962/d1018249
- Description: This research describes a two-dimensional modelling effort of heat and mass transport in simplified intrusive models of sills and their feeder dykes. These simplified models resembled a complex intrusive system such as the Great Dyke of Zimbabwe. This study investigated the impact of variable geometry to transport processes in two ways. First the time evolution of heat and mass transport during cooling was investigated. Then emphasis was placed on the application of convective scavenging as a mechanism that leads to the formation of minerals of economic interest, in particular the Platinum Group Elements (PGEs). The Navier-Stokes equations employed generated regions of high shear within the magma where we expected enhanced collisions between the immiscible sulphide liquid particles and PGEs. These collisions scavenge PGEs from the primary melt, aggregate and concentrate it to form PGEs enrichment in zero shear zones. The PGEs scavenge; concentrate and 'glue' in zero shear zones in the early history of convection because of viscosity and dispersive pressure (Bagnold effect). The effect of increasing the geometry size enhances scavenging, creates bigger zero shear zones with dilute concentrate of PGEs but you get high shear near the roots of the dyke/sill where the concentration will not be dilute. The time evolution calculations show that increasing the size of the magma chamber results in stronger initial convection currents for large magma models than for small ones. However, convection takes, approximately the same time to cease for both models. The research concludes that the time evolution for convective heat transfer is dependent on the viscosity rather than on geometry size. However, conductive heat transfer to the e-folding temperature was almost six times as long for the large model (M4) than the small one (M2). Variable viscosity as a physical property was applied to models 2 and 4 only. Video animations that simulate the cooling process for these models are enclosed in a CD at the back of this thesis. These simulations provide information with regard to the emplacement history and distribution of PGEs ore bodies. This will assist the reserve estimation and the location of economic minerals.
- Full Text:
- Date Issued: 2002
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