Most real world phenomena is modeled by ordinary and/or partial differential equations. Most of these equations are highly nonlinear and exact solutions are not always possible. Exact solutions always give a good account of the physical nature of the phenomena modeled. However, existing analytical methods can only handle a limited range of these equations. Semi-numerical and numerical methods give approximate solutions where exact solutions are impossible to find. However, some common numerical methods give low accuracy and may lack stability. In general, the character and qualitative behaviour of the solutions may not always be fully revealed by numerical approximations, hence the need for improved semi-numerical methods that are accurate, computational efficient and robust. In this study we introduce innovative techniques for finding solutions of highly nonlinear coupled boundary value problems. These techniques aim to combine the strengths of both analytical and numerical methods to produce efficient hybrid algorithms. In this work, the homotopy analysis method is blended with spectral methods to improve its accuracy. Spectral methods are well known for their high levels of accuracy. The new spectral homotopy analysis method is further improved by using a more accurate initial approximation to accelerate convergence. Furthermore, a quasi-linearisation technique is introduced in which spectral methods are used to solve the linearised equations. The new techniques were used to solve mathematical models in fluid dynamics. The thesis comprises of an introductory Chapter that gives an overview of common numerical methods currently in use. In Chapter 2 we give an overview of the methods used in this work. The methods are used in Chapter 3 to solve the nonlinear equation governing two-dimensional squeezing flow of a viscous fluid between two approaching parallel plates and the steady laminar flow of a third grade fluid with heat transfer through a flat channel. In Chapter 4 the methods were used to find solutions of the laminar heat transfer problem in a rotating disk, the steady flow of a Reiner-Rivlin fluid with Joule heating and viscous dissipation and the classical von Kάrmάn equations for boundary layer flow induced by a rotating disk. In Chapter 5 solutions of steady two-dimensional flow of a viscous incompressible fluid in a rectangular domain bounded by two permeable surfaces and the MHD viscous flow problem due to a shrinking sheet with a chemical reaction, were solved using the new methods.

Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2012.