Chebyshev spectral pertutrbation based method for solving nonlinear fluid flow problems.
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Date
2014
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Abstract
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.
Description
M. Sc. University of KwaZulu-Natal, Pietermaritzburg 2014.
Keywords
Chebyshev systems., Nonlinear theories., Theses--Mathematics.