Abstract:
The development of nonlinear science has grown an ever-increasing interest among
scientists and engineers for analytical asymptotic techniques for solving nonlinear
problems. Finding solutions to linear problems by means of computer is easier nowadays;
however, it is still difficult to solve nonlinear problems numerically or theoretically. The
reason is the use of iterative techniques in the various discretization methods or numerical
simulations to find numerical solutions to nonlinear problems. Almost all iterative
methods are sensitive to initial solutions; hence, it is very difficult to obtain converging
results in cases of strong nonlinearity.
The objective of this dissertation is to use Optimal Homotopy Asymptotic Method
(OHAM), a new semi-analytic approximating technique, for solving linear and nonlinear
initial and boundary value problems. The semi analytic solutions of nonlinear fourth
order, eighth order, special fourth order and special sixth order boundary-value problems
are computed using OHAM. Successful application of OHAM for squeezing flow is a
major task in this study.
This dissertation also investigates the effectiveness of OHAM formulation for Partial
Differential Equations (Wave Equation and Korteweg de Vries). OHAM is independent
of the free parameter and there is no need of the initial guess as there is in Homotopy
Perturbation Method (HPM), Variational Iteration Method (VIM) and Homotopy
Analysis Method (HAM). OHAM works very well with large domains and provides
better accuracy at lower-order of approximations. Moreover, the convergence domain can
be easily adjusted. The results are compared with other methods like HPM, VIM and
HAM, which reveal that OHAM is effective, simpler, easier and explicit.
