Abstract
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.