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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. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2013.

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Thesis (M.S.)--University of Missouri--Rolla, 2007. / Vita. The entire thesis text is included in file. Title from title screen of thesis/dissertation PDF file (viewed February 4, 2008) Includes bibliographical references.

Thesis (M.S.)--Ohio University, June, 2001. / Title from PDF t.p.

Thesis (Ph. D.)--University of Adelaide, Dept. of Applied Mathematics, 1988. / Includes bibliographical references (leaves 44-45).

Thesis (M.S.)--Rochester Institute of Technology, 1994. / Typescript. Includes bibliographical references (leaves 104-105).

Thesis (M.S. in Physics) Naval Postgraduate School, March 1992. / Thesis Advisors: Denardo, B.C. ; Garrett, Steven L. "March 1992." Includes bibliographical references (p. 74-76). Also available in print.

Thesis (Ph. D.)--Drexel University, 2004. / Includes abstract. Includes bibliographical references (leaf 80).

Thesis (Ph. D.)--Rutgers University, 2009. / "Graduate Program in Mathematics." Includes bibliographical references (p. 84-85).