Towards understanding crime dynamics in a heterogeneous environment : a mathematical modelling approach

**Authors:**Chikore, Tichaona.**Date:**2019**Subjects:**Crime - Research - Mathematical models , Differential equations**Language:**English**Type:**Masters (Thesis)**Identifier:**http://hdl.handle.net/10210/416835 , uj:35276**Description:**Abstract: , M.Sc. (Applied Mathematics)**Full Text:**

**Authors:**Chikore, Tichaona.**Date:**2019**Subjects:**Crime - Research - Mathematical models , Differential equations**Language:**English**Type:**Masters (Thesis)**Identifier:**http://hdl.handle.net/10210/416835 , uj:35276**Description:**Abstract: , M.Sc. (Applied Mathematics)**Full Text:**

On spectral relaxation and compact finite difference schemes for ordinary and partial differential equations

**Authors:**Dlamini, Phumlani Goodwill**Date:**2015-07-03**Subjects:**Differential equations , Finite differences , Spectral theory (Mathematics)**Type:**Thesis**Identifier:**uj:13687 , http://hdl.handle.net/10210/13883**Description:**Ph.D. (Applied Mathematics) , In this thesis we introduce new numerical methods for solving nonlinear ordinary and partial differential equations. These methods solve differential equations in a manner similar to the Gauss Seidel approach of solving linear systems of algebraic equations. First the nonlinear differential equations are linearized by simply evaluating nonlinear terms at previous iterations. To solve the linearized iteration schemes obtained we use either the spectral method or higher order compact finite difference schemes and we call the resulting methods the spectral relaxation method (SRM) and the compact finite difference relaxation method (CFD-RM) respectively. We test the applicability of these methods in a wide variety of ODEs and PDEs. The accuracy and computational efficiency in terms of CPU time is compared against other methods as well as other results from literature. We solve a range of chaotic and hyperchaotic systems of equations. Chaotic and hyperchaotic are complex dynamical systems which are characterised by rapidly changing solutions and high sensitivity to small perturbations of the initial data. As a result finding their solutions is a challenging task. We modify the proposed SRM to be able to deal with such systems of equations. We also consider chaos control and synchronization between too identical chaotic systems. We also make a comparison between the SRM and CFD-RM and between the spectral quasilinearization method (SQLM) and the compact finite difference quasilinearization method (CFD-QLM). The aim is to compare the performance between the spectral and the compact finite difference approaches in solving similarity boundary layer problems. We consider two examples. First, we consider the flow of a viscous incompressible electrically conducting fluid over a continuously shrinking sheet. We also consider a three-equation system that models the problem of unsteady free convective heat and mass transfer on a stretching surface in a porous medium in the presence of a chemical reaction. We extend the application of the SRMand SQLMto PDEs. In particular we consider two unsteady boundary layer flow problems modelled by a PDE or a system of PDEs. We solve a one dimensional unsteady boundary layer flow due to an impulsively stretching surface and the problem of unsteady three-dimensional MHD flow and mass transfer in a porous space. Results are compared with results obtained using the Keller-box method which is popular in solving unsteady boundary layer problems. We also extend the application of the CFD-RM to PDEs modelling unsteady boundary layer flows and again compare results to Keller-box results. We consider two examples, the unsteady one dimensional MHD laminar boundary layer flow due to an impulsively stretching surface, and the unsteady three-dimensional MHD flow and heat transfer over an impulsively stretching plate.**Full Text:**

**Authors:**Dlamini, Phumlani Goodwill**Date:**2015-07-03**Subjects:**Differential equations , Finite differences , Spectral theory (Mathematics)**Type:**Thesis**Identifier:**uj:13687 , http://hdl.handle.net/10210/13883**Description:**Ph.D. (Applied Mathematics) , In this thesis we introduce new numerical methods for solving nonlinear ordinary and partial differential equations. These methods solve differential equations in a manner similar to the Gauss Seidel approach of solving linear systems of algebraic equations. First the nonlinear differential equations are linearized by simply evaluating nonlinear terms at previous iterations. To solve the linearized iteration schemes obtained we use either the spectral method or higher order compact finite difference schemes and we call the resulting methods the spectral relaxation method (SRM) and the compact finite difference relaxation method (CFD-RM) respectively. We test the applicability of these methods in a wide variety of ODEs and PDEs. The accuracy and computational efficiency in terms of CPU time is compared against other methods as well as other results from literature. We solve a range of chaotic and hyperchaotic systems of equations. Chaotic and hyperchaotic are complex dynamical systems which are characterised by rapidly changing solutions and high sensitivity to small perturbations of the initial data. As a result finding their solutions is a challenging task. We modify the proposed SRM to be able to deal with such systems of equations. We also consider chaos control and synchronization between too identical chaotic systems. We also make a comparison between the SRM and CFD-RM and between the spectral quasilinearization method (SQLM) and the compact finite difference quasilinearization method (CFD-QLM). The aim is to compare the performance between the spectral and the compact finite difference approaches in solving similarity boundary layer problems. We consider two examples. First, we consider the flow of a viscous incompressible electrically conducting fluid over a continuously shrinking sheet. We also consider a three-equation system that models the problem of unsteady free convective heat and mass transfer on a stretching surface in a porous medium in the presence of a chemical reaction. We extend the application of the SRMand SQLMto PDEs. In particular we consider two unsteady boundary layer flow problems modelled by a PDE or a system of PDEs. We solve a one dimensional unsteady boundary layer flow due to an impulsively stretching surface and the problem of unsteady three-dimensional MHD flow and mass transfer in a porous space. Results are compared with results obtained using the Keller-box method which is popular in solving unsteady boundary layer problems. We also extend the application of the CFD-RM to PDEs modelling unsteady boundary layer flows and again compare results to Keller-box results. We consider two examples, the unsteady one dimensional MHD laminar boundary layer flow due to an impulsively stretching surface, and the unsteady three-dimensional MHD flow and heat transfer over an impulsively stretching plate.**Full Text:**

Engineering students’ actions in a mathematical modelling task: Mediating mathematical understanding in a computer algebra system

- Kotze, H., Spangenberg, E. D.

**Authors:**Kotze, H. , Spangenberg, E. D.**Date:**2019**Subjects:**Computer algebra system , Differential equations , Engineering students**Language:**English**Type:**Article**Identifier:**http://hdl.handle.net/10210/403549 , uj:33822 , Citation: Kotze, H. & Spangenberg, E.D. 2019. Engineering students’ actions in a mathematical modelling task: Mediating mathematical understanding in a computer algebra system. DOI: http://dx.doi.org/10.18820/2519593X/pie.v37i2.5**Description:**Abstract: Many engineering subjects rely on the interpretation of symbolic, numeric and graphic representations. Engineering students have challenges pertaining to their mathematical understanding of their actions with a computer algebra system (CAS). We investigated how a mathematical modelling task could mediate varied levels of mathematical understanding. When engineering students are exposed to a CAS environment, they habitually engage in programming activities without considering the computerised outputs. The purpose of this paper was to ascertain South African engineering students’ actions that can mediate broader levels of mathematical understanding in a CAS by utilising the Pirie- Kieren model of growth in mathematical understanding. Thirteen participants agreed to engage collaboratively in a mathematical modelling task. The task was analysed by means of content analysis following a deductive research approach. The findings disclosed that engineering students interdepend on paper-and-pen, computerised and reflective actions in their growth of mathematical understanding. Engineering students can be assisted in mediated and folding-back actions in order to fluctuate back and forth on their way to a more sound mathematical understanding. Explicit planning and sequence of subtasks can support engineering students to merge new levels of mathematics understanding with past comprehensions. Thoroughly planned modelling tasks can mediate novel levels of mathematical understanding when engineering students learn with a CAS.**Full Text:**

**Authors:**Kotze, H. , Spangenberg, E. D.**Date:**2019**Subjects:**Computer algebra system , Differential equations , Engineering students**Language:**English**Type:**Article**Identifier:**http://hdl.handle.net/10210/403549 , uj:33822 , Citation: Kotze, H. & Spangenberg, E.D. 2019. Engineering students’ actions in a mathematical modelling task: Mediating mathematical understanding in a computer algebra system. DOI: http://dx.doi.org/10.18820/2519593X/pie.v37i2.5**Description:**Abstract: Many engineering subjects rely on the interpretation of symbolic, numeric and graphic representations. Engineering students have challenges pertaining to their mathematical understanding of their actions with a computer algebra system (CAS). We investigated how a mathematical modelling task could mediate varied levels of mathematical understanding. When engineering students are exposed to a CAS environment, they habitually engage in programming activities without considering the computerised outputs. The purpose of this paper was to ascertain South African engineering students’ actions that can mediate broader levels of mathematical understanding in a CAS by utilising the Pirie- Kieren model of growth in mathematical understanding. Thirteen participants agreed to engage collaboratively in a mathematical modelling task. The task was analysed by means of content analysis following a deductive research approach. The findings disclosed that engineering students interdepend on paper-and-pen, computerised and reflective actions in their growth of mathematical understanding. Engineering students can be assisted in mediated and folding-back actions in order to fluctuate back and forth on their way to a more sound mathematical understanding. Explicit planning and sequence of subtasks can support engineering students to merge new levels of mathematics understanding with past comprehensions. Thoroughly planned modelling tasks can mediate novel levels of mathematical understanding when engineering students learn with a CAS.**Full Text:**

Symmetry methods and conservation laws applied to the Black-Scholes partial differential equation

**Authors:**McDonald, Ruth Leigh**Date:**2012-07-03**Subjects:**Applied mathematics , Symmetry (Mathematics) , Conservation laws (Mathematics) , Differential equations , Differential equations, Partial**Type:**Thesis**Identifier:**uj:8783 , http://hdl.handle.net/10210/5141**Description:**M.Sc. , The innovative work of Black and Scholes [1, 2] extended the mathematical understanding of the options pricing model, beginning the deliberate study of the theory of option pricing. Its impact on the nancial markets was immediate and unprecedented and is arguably one of the most important discoveries within nance theory to date. By just inserting a few variables, which include the stock price, risk-free rate of return, option's strike price, expiration date, and an estimate of the volatility of the stock's price, the option-pricing formula is easily used by nancial investors. It allows them to price various derivatives ( nancial instrument whose price and value are derived from the value of assets underlying them), including options on commodities, nancial assets and even pricing of employee stock options. Hence, European1 and American2 call or put options on a non-dividend-paying stock can be valued using the Black-Scholes model. All further advances in option pricing since the Black-Scholes analysis have been re nements, generalisations and expansions of the original idea presented by them.**Full Text:**

**Authors:**McDonald, Ruth Leigh**Date:**2012-07-03**Subjects:**Applied mathematics , Symmetry (Mathematics) , Conservation laws (Mathematics) , Differential equations , Differential equations, Partial**Type:**Thesis**Identifier:**uj:8783 , http://hdl.handle.net/10210/5141**Description:**M.Sc. , The innovative work of Black and Scholes [1, 2] extended the mathematical understanding of the options pricing model, beginning the deliberate study of the theory of option pricing. Its impact on the nancial markets was immediate and unprecedented and is arguably one of the most important discoveries within nance theory to date. By just inserting a few variables, which include the stock price, risk-free rate of return, option's strike price, expiration date, and an estimate of the volatility of the stock's price, the option-pricing formula is easily used by nancial investors. It allows them to price various derivatives ( nancial instrument whose price and value are derived from the value of assets underlying them), including options on commodities, nancial assets and even pricing of employee stock options. Hence, European1 and American2 call or put options on a non-dividend-paying stock can be valued using the Black-Scholes model. All further advances in option pricing since the Black-Scholes analysis have been re nements, generalisations and expansions of the original idea presented by them.**Full Text:**

Vector product and an integrable dynamical system

- Steeb, Willi-Hans, Tanski, Igor, Hardy, Yorick

**Authors:**Steeb, Willi-Hans , Tanski, Igor , Hardy, Yorick**Date:**2011**Subjects:**Vector product , Nambu mechanics , Differential equations , Integrals**Type:**Article**Identifier:**uj:5808 , ISSN 0253-6102 , http://hdl.handle.net/10210/7816**Description:**We study an autonomous system of first order ordinary differential equations based on the vector product. We show that the system is completely integrable by constructing the first integrals. The connection with Nambu mechanics is established. The extension to higher dimensions is also discussed.**Full Text:**

**Authors:**Steeb, Willi-Hans , Tanski, Igor , Hardy, Yorick**Date:**2011**Subjects:**Vector product , Nambu mechanics , Differential equations , Integrals**Type:**Article**Identifier:**uj:5808 , ISSN 0253-6102 , http://hdl.handle.net/10210/7816**Description:**We study an autonomous system of first order ordinary differential equations based on the vector product. We show that the system is completely integrable by constructing the first integrals. The connection with Nambu mechanics is established. The extension to higher dimensions is also discussed.**Full Text:**

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