A note on the multi-stage spectral relaxation method for chaos control and synchronization
- Dlamini, Phumlani Goodwill, Khumalo, Melusi, Motsa, Sandile Sydney
- Authors: Dlamini, Phumlani Goodwill , Khumalo, Melusi , Motsa, Sandile Sydney
- Date: 2014
- Subjects: Multistage spectral relaxation method , Chaos control and synchronization , Numerical methods
- Type: Article
- Identifier: uj:5430 , http://hdl.handle.net/10210/12049
- Description: In this study, we present and apply a new, accurate and easy to implement numerical method to realize and verify the synchronization between two identical chaotic Lorenz, Genesio-Tesi, Rössler, Chen and Rikitake systems. The proposed method is called the multi-stage spectral relaxation method (MSRM). We utilize the active control technique for the synchronization of these systems. To illustrate the effectiveness of the method, simulation results are presented and compared with results obtained using the Runge-Kutta (4, 5) based MATLAB solver, ode45.
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- Authors: Dlamini, Phumlani Goodwill , Khumalo, Melusi , Motsa, Sandile Sydney
- Date: 2014
- Subjects: Multistage spectral relaxation method , Chaos control and synchronization , Numerical methods
- Type: Article
- Identifier: uj:5430 , http://hdl.handle.net/10210/12049
- Description: In this study, we present and apply a new, accurate and easy to implement numerical method to realize and verify the synchronization between two identical chaotic Lorenz, Genesio-Tesi, Rössler, Chen and Rikitake systems. The proposed method is called the multi-stage spectral relaxation method (MSRM). We utilize the active control technique for the synchronization of these systems. To illustrate the effectiveness of the method, simulation results are presented and compared with results obtained using the Runge-Kutta (4, 5) based MATLAB solver, ode45.
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Numerical simulation of finite-time blow-up in nonlinear ODEs, reaction-diffusion equations and VIDEs
- Authors: Dlamini, Phumlani Goodwill
- Date: 2012-11-02
- Subjects: Midpoint-implicit Euler method , Differential equations, nonlinear
- Type: Thesis
- Identifier: uj:7316 , http://hdl.handle.net/10210/8054
- Description: M.Sc. , There have been an extensive study on solutions of differential equations modeling physical phenomena that blows up in finite time. The blow-up time often represents an important change in the properties of such models and hence it is very important to compute it as accurate as possible. In this work, an adaptive in time numerical method for computing blow-up solutions for nonlinear ODEs is introduced. The method is named implicit midpoint-implicit Euler method (IMIE) and is based on the implicit Euler and the implicit midpoint method. The method is used to compute blow-up time for different examples of ODEs, PDEs and VIDEs. The PDEs studied are reaction-diffusion equations whereby the method of lines is first used to discretize the equation in space to obtain a system of ODEs. Quadrature rules are used to approximate the integral in the VIDE to get a system of ODEs. The IMIE method is then used then to solve the system of ODEs. The results are compared to results obtained by the PECEIE method and Matlab solvers ode45 and ode15s. The results show that the IMIE method gives better results than the PECE-IE and ode15s and compares quite remarkably with the 4th order ode45 yet it is of order 1 with order 2 superconvergence at the mesh points.
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- Authors: Dlamini, Phumlani Goodwill
- Date: 2012-11-02
- Subjects: Midpoint-implicit Euler method , Differential equations, nonlinear
- Type: Thesis
- Identifier: uj:7316 , http://hdl.handle.net/10210/8054
- Description: M.Sc. , There have been an extensive study on solutions of differential equations modeling physical phenomena that blows up in finite time. The blow-up time often represents an important change in the properties of such models and hence it is very important to compute it as accurate as possible. In this work, an adaptive in time numerical method for computing blow-up solutions for nonlinear ODEs is introduced. The method is named implicit midpoint-implicit Euler method (IMIE) and is based on the implicit Euler and the implicit midpoint method. The method is used to compute blow-up time for different examples of ODEs, PDEs and VIDEs. The PDEs studied are reaction-diffusion equations whereby the method of lines is first used to discretize the equation in space to obtain a system of ODEs. Quadrature rules are used to approximate the integral in the VIDE to get a system of ODEs. The IMIE method is then used then to solve the system of ODEs. The results are compared to results obtained by the PECEIE method and Matlab solvers ode45 and ode15s. The results show that the IMIE method gives better results than the PECE-IE and ode15s and compares quite remarkably with the 4th order ode45 yet it is of order 1 with order 2 superconvergence at the mesh points.
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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.
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- 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.
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Spectral relaxation method and spectral quasilinearization method for solving unsteady boundary layer flow problems
- Motsa, Sandile Sydney, Dlamini, Phumlani Goodwill, Khumalo, Melusi
- Authors: Motsa, Sandile Sydney , Dlamini, Phumlani Goodwill , Khumalo, Melusi
- Date: 2014
- Subjects: Nonlinear partial differential equations , Spectral quasilinearization method , Partial differential equations , Spectral relaxation method
- Type: Article
- Identifier: uj:5431 , http://hdl.handle.net/10210/12050
- Description: Nonlinear partial differential equations (PDEs) modelling unsteady boundary-layer flows are solved by the spectral relaxation method (SRM) and the spectral quasilinearization method (SQLM). The SRM and SQLM are Chebyshev pseudospectral based methods that have been successfully used to solve nonlinear boundary layer flow problems described by systems of ordinary differential equations. In this paper application of these methods is extended, for the first time, to systems of nonlinear PDEs that model unsteady boundary layer flow. The new extension is tested on two problems: boundary layer flow caused by an impulsively stretching plate and a coupled four-equation system that models the problem of unsteady MHD flow and mass transfer in a porous space. Numerous simulation experiments are conducted to determine the accuracy and compare the computational performance of the proposed methods against the popular Keller-box finite difference scheme which is widely accepted as being one of the ideal tools for solving nonlinear PDEs that model boundary layer flow problems. The results indicate that the methods are more efficient in terms of computational accuracy and speed compared with the Keller-box.
- Full Text:
- Authors: Motsa, Sandile Sydney , Dlamini, Phumlani Goodwill , Khumalo, Melusi
- Date: 2014
- Subjects: Nonlinear partial differential equations , Spectral quasilinearization method , Partial differential equations , Spectral relaxation method
- Type: Article
- Identifier: uj:5431 , http://hdl.handle.net/10210/12050
- Description: Nonlinear partial differential equations (PDEs) modelling unsteady boundary-layer flows are solved by the spectral relaxation method (SRM) and the spectral quasilinearization method (SQLM). The SRM and SQLM are Chebyshev pseudospectral based methods that have been successfully used to solve nonlinear boundary layer flow problems described by systems of ordinary differential equations. In this paper application of these methods is extended, for the first time, to systems of nonlinear PDEs that model unsteady boundary layer flow. The new extension is tested on two problems: boundary layer flow caused by an impulsively stretching plate and a coupled four-equation system that models the problem of unsteady MHD flow and mass transfer in a porous space. Numerous simulation experiments are conducted to determine the accuracy and compare the computational performance of the proposed methods against the popular Keller-box finite difference scheme which is widely accepted as being one of the ideal tools for solving nonlinear PDEs that model boundary layer flow problems. The results indicate that the methods are more efficient in terms of computational accuracy and speed compared with the Keller-box.
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