On the numerical solution of the Lane-Emden, Bratu and Troesch equations.

Masters Degree. University of KwaZulu-Natal, Pietermaritzburg.Many engineering and physics problems are modelled using differential equations, which may be highly nonlinear and difficult to solve analytically. Numerical techniques are often used to obtain approximate solutions. In this study, we consider the solution of three nonlinear ordinary differential equations; namely, the initial value Lane-Emden equation, the boundary value Bratu equation, and the boundary value Troesch problem. For the Lane- Emden equation, a comparison is made between the accuracy of solutions using the finite difference method and the multi-domain spectral quasilinearization method along with the exact solution. We found that the multi-domain spectral quasilinearization method gave a better solution. For the Bratu problem, a comparison is made between the spectral quasilinearization method and the higher-order spectral quasilinearization method. The higher-order spectral quasilinearization method gave more accurate results. The Troesch problem is solved using the higher-order spectral quasilinearization method and the finite difference method. The solutions obtained are compared in terms of accuracy. Overall, the higher-order spectral quasilinearization method and multi-domain spectral quasilinearization method gave the accurate solutions, making these two methods to be the most reliable for these three problems

On decoupled quasi-linearization methods for solving systems of nonlinear boundary value problems.

M. Sc. University of KwaZulu-Natal, Pietermaritzburg 2014.In this dissertation, a comparative study is carried out on three spectral based numerical methods which are the spectral quasilinearization method (SQLM), the spectral relaxation method (SRM) and the spectral local linearization method (SLLM). The study is carried out by applying the numerical methods on systems of differential equations modeling uid ow problems. Residual error analysis is used in determining the speed of convergence, convergence rate and accuracy of the methods. In Chapter 1, all the terminologies and methods that are applied throughout the course of the study are introduced. In Chapter 2, the SRM, SLLM and SQLM are applied on an unsteady free convective heat and mass transfer on a stretching surface in a porous medium with suction/injection. In Chapter 3, the SRM, SLLM and SQLM are applied on an unsteady boundary layer ow due to a stretching surface in a rotating uid. In Chapter 4, the SRM, SLLM and SQLM are used to solve an unsteady three-dimensional MHD boundary layer ow and heat transfer over an impulsively stretching plate. The purpose of this study is to assess the performance of the spectral based numerical methods when solving systems of differential equations. The performance of the methods are measured in terms of computational efficiency (in terms of time taken to generate solutions), accuracy and rate of convergence. The ease of development and implementation of the associated numerical algorithms are also considered

Chebyshev spectral pertutrbation based method for solving nonlinear fluid flow problems.

M. Sc. University of KwaZulu-Natal, Pietermaritzburg 2014.In this dissertation, a modi cation of the classical perturbation techniques for solving nonlinear ordinary di erential equation (ODEs) and nonlinear partial di erential equations (PDEs) is presented. The method, called the Spectral perturbation method (SPM) is a series expansion based technique which extends the use of the standard perturbation scheme when combined with the Chebyshev spectral method. The SPM solves a sequence of equations generated by the perturbation series approximation using the Chebyshev spectral methods. This dissertation aims to demonstrate that, in contrast to the conclusions earlier drawn by researchers about perturbation techniques, a perturbation approach can be e ectively used to generate accurate solutions which are de ned under the Williams and Rhyne (1980) transformation. A quasi-linearisation technique, called the spectral quasilinearisation method (SQLM) is used for validation purpose. The SQLM employs the quasilinearisation approach to linearise nonlinear di erential equations and the resulting equations are solved using the spectral methods. Furthermore, a spectral relaxation method (SRM) which is a Chebyshev spectral collocation based method that decouples and rearrange a system of equations in a Gauss - Seidel manner is also presented. In the SRM, the di erential equations are decoupled, rearranged and the resulting sequence of equations are numerically integrated using the Chebyshev spectral collocation method. The techniques were used to solve mathematical models in uid dynamics. This study consists of an introductory chapter which gives the description of the methods and a brief overview of the techniques used in developing the SPM, SQLM and the SRM. In Chapter 2, the SPM is used to solve the equations that model magnetohydrodynamics (MHD) stagnation point ow and heat transfer problem from a stretching sheet in the presence of heat source/sink and suction/injection in porous media. Using similarity transformations, the governing partial differential equations are transformed into ordinary di erential equations. Series solutions for small velocity ratio and asymptotic solutions for large velocity ratio were generated and the results were also validated against those obtained using the SQLM. In Chapter 3, the SPM was used to solve the momentum, heat and mass transfer equations describing the unsteady MHD mixed convection ow over an impulsively stretched vertical surface in the presence of chemical reaction e ect. The governing partial di erential equations are reduced into a set of coupled non similar equations and then solved numerically using the SPM. In order to demonstrate the accuracy and e ciency of the SPM, the SPM numerical results are compared with numerical results generated using the SRM and a good agreement between the two methods was observed up to eight decimal digits which is a reasonable level of accuracy. Several simulation are conducted to ascertain the accuracy of the SPM and the SRM. The computational speed of the SPM is demonstrated by comparing the SPM computational time with the SRM computational time. A residual error analysis is also conducted for the SPM and the SRM, in order to further assess the accuracy of the SPM. In Chapter 4, the SPM was used to solve the equations modelling the unsteady three-dimensional MHD ow and mass transfer in a porous space previously reported in literature. E ciency and accuracy of the SPM is shown by validating the SPM results against the results obtained using the SRM and the results were found to be in good agreement. The computational speed of the SPM is demonstrated by comparing the SPM and the SRM computational time. In order to further assess the accuracy of the SPM, a residual error analysis is conducted for the SPM and the SRM. In Chapter 2, we show that the SPM can be used as an alternative to the standard perturbation methods to get numerical solutions for strongly nonlinear boundary value problems. Also, it is demonstrated in Chapter 2 that the SPM is e cient even in the case where the perturbation parameter is large, as the convergence rate is seen to improve with increase in the large parameter value. In Chapters 3 and 4, the study shows that SPM is more e cient in terms of computational speed when compared with the SRM. The study also highlighted that the SPM can be used as an e cient and reliable tool for solving strongly nonlinear partial di erential equations de ned under the Williams and Rhyne (1980) transformation. In addition, the study shows that accurate results can be obtained using the perturbation method and thus, the conclusions earlier drawn by researchers regarding the accuracy of the perturbation method is corrected

Computational and numerical analysis of differential equations using spectral based collocation method.

Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.In this thesis, we develop accurate and computationally efficient spectral collocation-based methods, both modified and new, and apply them to solve differential equations. Spectral collocation-based methods are the most commonly used methods for approximating smooth solutions of differential equations defined over simple geometries. Procedurally, these methods entail transforming the gov erning differential equation(s) into a system of linear algebraic equations that can be solved directly. Owing to the complexity of expanding the numerical algorithms to higher dimensions, as reported in the literature, researchers often transform their models to reduce the number of variables or narrow them down to problems with fewer dimensions. Such a process is accomplished by making a series of assumptions that limit the scope of the study. To address this deficiency, the present study explores the development of numerical algorithms for solving ordinary and partial differential equations defined over simple geometries. The solutions of the differential equations considered are approximated using interpolating polynomials that satisfy the given differential equation at se lected distinct collocation points preferably the Chebyshev-Gauss-Lobatto points. The size of the computational domain is particularly emphasized as it plays a key role in determining the number of grid points that are used; a feature that dictates the accuracy and the computational expense of the spectral method. To solve differential equations defined on large computational domains much effort is devoted to the development and application of new multidomain approaches, based on decomposing large spatial domain(s) into a sequence of overlapping subintervals and a large time interval into equal non-overlapping subintervals. The rigorous analysis of the numerical results con firms the superiority of these multiple domain techniques in terms of accuracy and computational efficiency over the single domain approach when applied to problems defined over large domains. The structure of the thesis indicates a smooth sequence of constructing spectral collocation method algorithms for problems across different dimensions. The process of switching between dimensions is explained by presenting the work in chronological order from a simple one-dimensional problem to more complex higher-dimensional problems. The preliminary chapter explores solutions of or dinary differential equations. Subsequent chapters then build on solutions to partial differential i equations in order of increasing computational complexity. The transition between intermediate dimensions is demonstrated and reinforced while highlighting the computational complexities in volved. Discussions of the numerical methods terminate with development and application of a new method namely; the trivariate spectral collocation method for solving two-dimensional initial boundary value problems. Finally, the new error bound theorems on polynomial interpolation are presented with rigorous proofs in each chapter to benchmark the adoption of the different numerical algorithms. The numerical results of the study confirm that incorporating domain decomposition techniques in spectral collocation methods work effectively for all dimensions, as we report highly accurate results obtained in a computationally efficient manner for problems defined on large do mains. The findings of this study thus lay a solid foundation to overcome major challenges that numerical analysts might encounter

Numerical study of convective fluid flow in porous and non-porous media.

Ph. D. University of KwaZulu-Natal, Pietermaritzburg 2015.Abstract available in PDF file

Convective heat and mass transfer in boundary layer flow through porous media saturated with nanofluids.

Doctor of Philosophy in Mathematics. University of KwaZulu-Natal, Pietermaritzburg 2016.The thesis is devoted to the study of flow, heat and mass transfer processes, and crossdiffusion effects in convective boundary layer flows through porous media saturated with nanofluids. Of particular interest is how nanofluids perform as heat transfer fluids compared to traditional fluids such as oil and water. Flow in different geometries and subject to various source terms is investigated. An important aspect of the study and understanding of transport processes is the solution of the highly non-linear coupled differential equations that model both the flow and the heat transportation. In the literature, various analytical and numerical methods are available for finding solutions to fluid flow equations. However, not all these methods give accurate solutions, are stable, or are computationally efficient. For these reasons, it is important to constantly devise numerical schemes that work more efficiently, including improving the performance of existing schemes, to achieve accuracy with less computational effort. In this thesis the systems of differential equations that describe the fluid flow and other transport processes were solved numerically using both established and recent numerical schemes such as the spectral relaxation method and the spectral quasilinearization method. These spectral methods have been used only in a limited number of studies. There is therefore the need to test and prove the accuracy and general application of the methods in a wider class of boundary value problems. The accuracy, convergence, and validity of the solutions obtained using spectral methods, have been established by careful comparison with solutions for limiting cases in the published literature, or by use of a different solution method. In terms of understanding the physically important variables that impact the flow, we have inter alia, investigated the significance of different fluid and physical parameters, and how changes in these parameters affect the skin friction coefficient, the heat and mass transfer rates and the fluid properties. Some system parameters of interest in this study include the nanoparticle volume fraction, the Hartmann number, thermal radiation, Brownian motion, the heat generation, the Soret and Dufour effects, and the Prandtl and Schmidt number. The dependency of the heat, mass transfer and skin friction coefficients on these parameters has been quantified and discussed. In this thesis, we show that nanofluids have a significant impact on heat and mass transfer processes compared with traditional heat transfer fluids

Overlapping grid spectral collocation methods for nonlinear differential equations modelling fluid flow problems.

Doctoral Degree. University of KwaZulu-Natal, Durban.The focus of this thesis is on computational grid-manipulation to enhance the accuracy, convergence and computational efficiency of spectral collocation methods for the solution of differential equations in fluid mechanics. The need to develop highly accurate, convergent and computationally efficient numerical techniques for solving nonlinear problems is an ever-recurring theme in numerical mathematics. Spectral methods have been shown in the literature to be more accurate and efficient than some common numerical methods, such as finite difference methods. However, their accuracy deteriorates as the computational domain increases and when the number of grid points reaches a certain critical value. The spectral collocation algorithm produces dense matrix equations, for which there is no known efficient solution method. These deficiencies necessitate the development of spectral techniques that produce less dense matrix equations using fewer grid points. This thesis presents a new improvement in spectral collocation methods with particular application to nonlinear differential equations that model problems arising in fluid mechanics. The improvement described in this thesis requires the use of overlapping grids when descritizing the solution domain for Chebyshev spectral collocation method. The thesis is presented in two related subdivisions. In Part A, the overlapping grid approach is used only in space variable when solving nonlinear ordinary and partial differential equations. Subsequently, the overlapping grid approach is used in both the space and time variables in the solution of partial differential equations. This thesis is also devoted to analysing solutions of fluid flow models through various practical geometries with particular interest in non-Newtonian fluid flows. The physics of these fluid flows is studied through parametric studies on the effects of diverse thermophysical parameters on the fluid properties, changes in shear stresses, and heat and mass transport. Key findings, are inter alia, that the overlapping multi-domain spectral techniques are computationally efficient, produce stable and accurate results using a small number of grid points in each subinterval and in the entire computational domain. Using the overlapping grids yields less dense coefficient matrices that invert easily. Changes in thermophysical parameters has significant consequences for the fluid properties, and heat and mass transfer processes

Double-diffusive convection flow in a porous medium saturated with a nanofluid.

In this work, we studied heat and mass transfer in a nanofluid flow over a stretching sheet. Fluid flow in different flow geometries was studied and a co-ordinate transformation was used to transform the governing equations into non-dimensional non-similar boundary layer equations. These equations were then solved numerically using both established and recent techniques such as the spectral relaxation and spectral quasi-linearization methods. Numerical solutions for the heat transfer, mass transfer and skin friction coefficients have been presented for different system parameters, such as heat generation, Soret and Dufour effects, chemical reaction, thermal radiation influence, the local Grashof number, Prandtl number, Eckert number, Hartmann number and the Schmidt number. The dependency of the skin friction, heat and mass transfer coefficients on these parameters has been quantified and discussed. The accuracy, and validity of the spectral relaxation and spectral quasi-linearization methods has been established

Numerical Simulation

Nowadays mathematical modeling and numerical simulations play an important role in life and natural science. Numerous researchers are working in developing different methods and techniques to help understand the behavior of very complex systems, from the brain activity with real importance in medicine to the turbulent flows with important applications in physics and engineering. This book presents an overview of some models, methods, and numerical computations that are useful for the applied research scientists and mathematicians, fluid tech engineers, and postgraduate students

On Extending the Quasilinearization Method to Higher Order Convergent Hybrid Schemes Using the Spectral Homotopy Analysis Method

We propose a sequence of highly accurate higher order convergent iterative schemes by embedding the quasilinearization algorithm within a spectral collocation method. The iterative schemes are simple to use and significantly reduce the time and number of iterations required to find solutions of highly nonlinear boundary value problems to any arbitrary level of accuracy. The accuracy and convergence properties of the proposed algorithms are tested numerically by solving three Falkner-Skan type boundary layer flow problems and comparing the results to the most accurate results currently available in the literature. We show, for instance, that precision of up to 29 significant figures can be attained with no more than 5 iterations of each algorithm