Investigations into the ranks of regular graphs

**Authors:**Garner, Charles R.**Date:**2012-08-17**Subjects:**Graph theory , Graphic methods , Spectral theory (Mathematics) , Eigenvalues**Type:**Thesis**Identifier:**uj:2622 , http://hdl.handle.net/10210/6069**Description:**Ph.D. , In this thesis, the ranks of many types of regular and strongly regular graphs are determined. Also determined are ranks of regular graphs under unary operations: the line graph, the complement, the subdivision graph, the connected cycle, the complete subdivision graph, and the total graph. The binary operations considered are the Cartesian product and the complete product. The ranks of the Cartesian product of regular graphs have been investigated previously in [BBD1]; here, we summarise and extend those results to include more regular graphs. We also examine a special nonregular graph, the path. Ranks of paths and products of graphs involving paths are presented as well**Full Text:**

**Authors:**Garner, Charles R.**Date:**2012-08-17**Subjects:**Graph theory , Graphic methods , Spectral theory (Mathematics) , Eigenvalues**Type:**Thesis**Identifier:**uj:2622 , http://hdl.handle.net/10210/6069**Description:**Ph.D. , In this thesis, the ranks of many types of regular and strongly regular graphs are determined. Also determined are ranks of regular graphs under unary operations: the line graph, the complement, the subdivision graph, the connected cycle, the complete subdivision graph, and the total graph. The binary operations considered are the Cartesian product and the complete product. The ranks of the Cartesian product of regular graphs have been investigated previously in [BBD1]; here, we summarise and extend those results to include more regular graphs. We also examine a special nonregular graph, the path. Ranks of paths and products of graphs involving paths are presented as well**Full Text:**

An analytic approach to the spectral characterization of the radical

**Authors:**Bower, Hendri**Date:**2016**Subjects:**Density functionals , Banach algebras , Spectral theory (Mathematics) , Proof theory , Zemanek, Jaroslav**Language:**English**Type:**Masters (Thesis)**Identifier:**http://hdl.handle.net/10210/124927 , uj:20975**Description:**Abstract: Please refer to full text to view abstract , M.Sc. (Mathematics)**Full Text:**

**Authors:**Bower, Hendri**Date:**2016**Subjects:**Density functionals , Banach algebras , Spectral theory (Mathematics) , Proof theory , Zemanek, Jaroslav**Language:**English**Type:**Masters (Thesis)**Identifier:**http://hdl.handle.net/10210/124927 , uj:20975**Description:**Abstract: Please refer to full text to view abstract , M.Sc. (Mathematics)**Full Text:**

The new spectral Adomian decomposition method and its higher order based iterative schemes for solving highly nonlinear two-point boundary value problems

**Authors:**Mdziniso, Madoda Majahonkhe**Date:**2014-07-01**Subjects:**Boundary value problems , Spectral theory (Mathematics)**Type:**Thesis**Identifier:**uj:11638 , http://hdl.handle.net/10210/11351**Description:**M.Sc. (Applied Mathematics) , A comparison between the recently developed spectral relaxation method (SRM) and the spectral local linearisation method (SLLM) is done for the first time in this work. Both spectral hybrid methods are employed in finding the solution to the non isothermal mass and heat balance model of a catalytic pellet boundary value problem (BVP) with finite mass and heat transfer resistance, which is a coupled system of singular nonlinear ordinary differential equations (ODEs). The SRM and the SLLM are applied, for the first time, to solve a problem with singularities. The solution by the SRM and the SLLM are validated against the results by bvp4c, a well known matlab built-in procedure for solving BVPs. Tables and graphs are used to show the comparison. The SRM and the SLLM are exceptionally accurate with the SLLM being the fastest to converge to the correct solution. We then construct a new spectral hybrid method which we named the spectral Adomian decomposition method (SADM). The SADM is used concurrently with the standard Adomian decomposition method (ADM) to solve well known models arising in fluid mechanics. These problems are the magneto hydrodynamic (MHD) Jeffery-Hamel flow model and the Darcy-Brinkman- Forchheimer momentum equations. The validity of the results by the SADM and ADM are verified by the exact solution and bvp4c solution where applicable. A simple alteration of the SADM is made to improve the performance.**Full Text:**

**Authors:**Mdziniso, Madoda Majahonkhe**Date:**2014-07-01**Subjects:**Boundary value problems , Spectral theory (Mathematics)**Type:**Thesis**Identifier:**uj:11638 , http://hdl.handle.net/10210/11351**Description:**M.Sc. (Applied Mathematics) , A comparison between the recently developed spectral relaxation method (SRM) and the spectral local linearisation method (SLLM) is done for the first time in this work. Both spectral hybrid methods are employed in finding the solution to the non isothermal mass and heat balance model of a catalytic pellet boundary value problem (BVP) with finite mass and heat transfer resistance, which is a coupled system of singular nonlinear ordinary differential equations (ODEs). The SRM and the SLLM are applied, for the first time, to solve a problem with singularities. The solution by the SRM and the SLLM are validated against the results by bvp4c, a well known matlab built-in procedure for solving BVPs. Tables and graphs are used to show the comparison. The SRM and the SLLM are exceptionally accurate with the SLLM being the fastest to converge to the correct solution. We then construct a new spectral hybrid method which we named the spectral Adomian decomposition method (SADM). The SADM is used concurrently with the standard Adomian decomposition method (ADM) to solve well known models arising in fluid mechanics. These problems are the magneto hydrodynamic (MHD) Jeffery-Hamel flow model and the Darcy-Brinkman- Forchheimer momentum equations. The validity of the results by the SADM and ADM are verified by the exact solution and bvp4c solution where applicable. A simple alteration of the SADM is made to improve the performance.**Full Text:**

Riesz- en Fredholmteorie in Banach-algebras

**Authors:**Vermaak, Jacobus Andries**Date:**2014-09-11**Subjects:**Banach algebras , Banach spaces , Spectral theory (Mathematics) , Freeholm operators**Type:**Thesis**Identifier:**uj:12268 , http://hdl.handle.net/10210/12031**Description:**M.Sc. (Mathematics) , Please refer to full text to view abstract**Full Text:**

**Authors:**Vermaak, Jacobus Andries**Date:**2014-09-11**Subjects:**Banach algebras , Banach spaces , Spectral theory (Mathematics) , Freeholm operators**Type:**Thesis**Identifier:**uj:12268 , http://hdl.handle.net/10210/12031**Description:**M.Sc. (Mathematics) , Please refer to full text to view abstract**Full Text:**

On the role of subharmonic functions in the spectral theory of general Banach algebras

**Authors:**Moolman, Ruan**Date:**2010-02-23T08:13:24Z**Subjects:**Banach algebras , Spectral theory (Mathematics) , Algebraic functions**Type:**Thesis**Identifier:**uj:6625 , http://hdl.handle.net/10210/3026**Description:**M.Sc.**Full Text:**

**Authors:**Moolman, Ruan**Date:**2010-02-23T08:13:24Z**Subjects:**Banach algebras , Spectral theory (Mathematics) , Algebraic functions**Type:**Thesis**Identifier:**uj:6625 , http://hdl.handle.net/10210/3026**Description:**M.Sc.**Full Text:**

On approximate identities and topological divisors of zero in Banach algebras

**Authors:**Hasse, Melanie Alexandra**Date:**2016**Subjects:**Banach algebras , Spectral theory (Mathematics) , Tensor products , Topological algebras**Language:**English**Type:**Masters (Thesis)**Identifier:**http://hdl.handle.net/10210/244008 , uj:25229**Description:**M.Sc. , Abstract: Please refer to full text to view abstract**Full Text:**

**Authors:**Hasse, Melanie Alexandra**Date:**2016**Subjects:**Banach algebras , Spectral theory (Mathematics) , Tensor products , Topological algebras**Language:**English**Type:**Masters (Thesis)**Identifier:**http://hdl.handle.net/10210/244008 , uj:25229**Description:**M.Sc. , Abstract: Please refer to full text to view abstract**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:**

A multi-domain implementation of the pseudo-spectral method and compact finite difference schemes for solving time-dependent differential equations

**Authors:**Mathale, Dyke**Date:**2019**Subjects:**Differential equations - Numerical solutions , Spectral theory (Mathematics) , Finite differences**Language:**English**Type:**Masters (Thesis)**Identifier:**http://hdl.handle.net/10210/401292 , uj:33529**Description:**Abstract : In this dissertation, we introduce new numerical methods for solving time-dependant differential equations. These methods involve dividing the domain of the problem into multiple sub domains. The nonlinearity of the differential equations is dealt with by using a Gauss-Seidel like relaxation or quasilinearisation technique. To solve the linearized iteration schemes obtained we use either higher order compact finite difference schemes or spectral collocation methods and we call the resulting methods the multi-domain compact finite difference relaxation method (MD-CFDRM), multi-domain compact finite difference quasilinearisation method (MD-CFDQLM) and multi-domain bivariate spectral quasilinearisation method (MD-BSQLM) respectively. We test the applicability of these methods in a wide variety of differential equations. The accuracy is compared against other methods as well as other results from literature. The MD-CFDRM is used to solve famous chaotic systems and hyperchaotic systems. Chaotic and hyperchaotic systems are characterized by high sensitivity to small perturbation on initial data and rapidly changing solutions. Such rapid variations in the solution pose tremendous problems to a number of numerical approximations. We modify the CFDs to be able to deal with such systems of equations. We also used the MD-CFDQLM to solve the nonlinear evolution partial differential equations, namely, the Fisher’s equation, Burgers- Fisher equation, Burgers-Huxley equation and the coupled Burgers’ equations over a large time domain. The main advantage of this approach is that it offers better accuracy on coarser grids which significantly improves the computational speed of the method for large time domain. We also studied the generalized Kuramoto-Sivashinsky (GKS) equations. The KS equations exhibit chaotic behaviour under certain conditions. We used the multi-domain bivariate spectral quasilinearisation method (MD-BSQLM) to approximate the numerical solutions for the generalized KS equations. , M.Sc. (Pure and Applied Mathematics)**Full Text:**

**Authors:**Mathale, Dyke**Date:**2019**Subjects:**Differential equations - Numerical solutions , Spectral theory (Mathematics) , Finite differences**Language:**English**Type:**Masters (Thesis)**Identifier:**http://hdl.handle.net/10210/401292 , uj:33529**Description:**Abstract : In this dissertation, we introduce new numerical methods for solving time-dependant differential equations. These methods involve dividing the domain of the problem into multiple sub domains. The nonlinearity of the differential equations is dealt with by using a Gauss-Seidel like relaxation or quasilinearisation technique. To solve the linearized iteration schemes obtained we use either higher order compact finite difference schemes or spectral collocation methods and we call the resulting methods the multi-domain compact finite difference relaxation method (MD-CFDRM), multi-domain compact finite difference quasilinearisation method (MD-CFDQLM) and multi-domain bivariate spectral quasilinearisation method (MD-BSQLM) respectively. We test the applicability of these methods in a wide variety of differential equations. The accuracy is compared against other methods as well as other results from literature. The MD-CFDRM is used to solve famous chaotic systems and hyperchaotic systems. Chaotic and hyperchaotic systems are characterized by high sensitivity to small perturbation on initial data and rapidly changing solutions. Such rapid variations in the solution pose tremendous problems to a number of numerical approximations. We modify the CFDs to be able to deal with such systems of equations. We also used the MD-CFDQLM to solve the nonlinear evolution partial differential equations, namely, the Fisher’s equation, Burgers- Fisher equation, Burgers-Huxley equation and the coupled Burgers’ equations over a large time domain. The main advantage of this approach is that it offers better accuracy on coarser grids which significantly improves the computational speed of the method for large time domain. We also studied the generalized Kuramoto-Sivashinsky (GKS) equations. The KS equations exhibit chaotic behaviour under certain conditions. We used the multi-domain bivariate spectral quasilinearisation method (MD-BSQLM) to approximate the numerical solutions for the generalized KS equations. , M.Sc. (Pure and Applied Mathematics)**Full Text:**

- «
- ‹
- 1
- ›
- »