Al-Nemrat, Asem Mustafa Moh’ad
(2019)
Modified Sumudu Transform
Analytical Approximate Methods For
Solving Boundary Value Problems.
PhD thesis, Universiti Sains Malaysia.
Abstract
In this study, emphasis is placed on analytical approximate methods. These
methods include the combination of the Sumudu transform (ST) with the homotopy
perturbation method (HPM), namely the Sumudu transform homotopy perturbation
method (STHPM), the combination of the ST with the variational iteration method
(VIM), namely the Sumudu transform variational iteration method (STVIM) and finally,
the combination of the ST with the homotopy analysis method (HAM), namely
the Sumudu transform homotopy analysis method (STHAM). Although these standard
methods have been successfully used in solving various types of differential equations,
they still suffer from the weakness in choosing the initial guess. In addition, they
require an infinite number of iterations which negatively affect the accuracy and convergence
of the solutions. The main objective of this thesis is to modify, apply and
analyze these methods to handle the difficulties and drawbacks and find the analytical
approximate solutions for some cases of linear and nonlinear ordinary differential
equations (ODEs). These cases include second-order two-point boundary value
problems (BVPs), singular and systems of second-order two-point BVPs. For the proposed
methods, the trial function was employed as an initial approximation to provide
more accurate approximate solutions for the considered problems. In addition, for the
STVIM method, a new algorithm has been proposed to solve various kinds of linear
and nonlinear second-order two-point BVPs. In this algorithm, the convolution theorem
has been used to find an optimal Lagrange multiplier.
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