An application of modern analytical solution techniques to nonlinear partial differential equations.
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Date
2014
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Abstract
Many physics and engineering problems are modeled by differential equations. In
many instances these equations are nonlinear and exact solutions are difficult to
obtain. Numerical schemes are often used to find approximate solutions. However,
numerical solutions do not describe the qualitative behaviour of mechanical systems
and are insufficient in determining the general properties of certain systems of
equations. The need for analytical methods is self-evident and major developments
were seen in the 1990’s. With the aid of faster processing equipment today, we are
able to compute analytical solutions to highly nonlinear equations that are more
accurate than numerical solutions.
In this study we discuss solutions to nonlinear partial differential equations with
focus on non-perturbation analytical methods. The non-perturbation methods of
choice are the homotopy analysis method (HAM) developed by Shijun Liao and the
variational iteration method (VIM) developed by Ji-Huan He. The aim is to compare the solutions obtained by these modern day analytical methods against each other
focusing on accuracy, convergence and computational efficiency.
The methods were applied to three test problems, namely, the heat equation, Burgers
equation and the Bratu equation. The solutions were compared against both the exact
results as well as solutions generated using the finite difference method, in some cases.
The results obtained show that the HAM successfully produces solutions which are
accurate, faster converging and requires less computational resources than the VIM.
However, the VIM still provides accurate solutions that are also in good agreement
with the closed form solutions of the test problems. The FDM also produced good
results which were used as a further comparison to the analytical solutions. The
findings of this study is in agreement with those published in the literature.
Description
Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2013.
Keywords
Differential equations, Partial., Nonlinear theories., Theses--Chemistry.