Derya Altıntan, Ph.D.
Department of Scientific Computing
June 2011
Supervisor: Ömür Uğur (Institute of Applied Mathematics, Middle East Technical University, Ankara)
Abstract
It is obvious that differential equations can be used to model real-life problems. Although it is possible to obtain analytical solutions of some of them, it is in general difficult to find closed form solutions of differential equations. Finding thus approximate solutions has been the subject of many researchers from different areas.
In this thesis, we propose a new approach to Variational Iteration Method (VIM) to obtain the solutions of systems of first-order differential equations. The main contribution of the thesis to VIM is that proposed approach uses restricted variations only for the nonlinear terms and builds up a matrix-valued Lagrange multiplier that leads to the extension of the method (EVIM).
Close relation between the matrix-valued Lagrange multipliers and fundamental solutions of the differential equations highlights the relation between the extended version of the variational iteration method and the classical variation of parameters formula.
It has been proved that the exact solution of the initial value problems for (nonhomogenous) linear differential equations can be obtained by such a generalisation using only a single variational step.
Since higher-order equations can be reduced to first-order systems, the proposed approach is capable of solving such equations too; indeed, without such a reduction, variational iteration method is also extended to higher-order scalar equations. Further, the close connection with the associated first-order systems is presented.
Such extension of the method to higher-order equations is then applied to solve boundary value problems: linear and nonlinear ones. Although the corresponding Lagrange multiplier resembles the Green’s function, without the need of the latter, the extended approach to the variational iteration method is systematically applied to solve boundary value problems, surely in the nonlinear case as well.
In order to show the applicability of the method, we have applied the EVIM to various real-life problems: the classical Sturm-Liouville eigenvalue problems, Brusselator reaction-diffusion, and chemical master equations. Results show that the method is simple, but powerful and effective.
Keywords: Variational Iteration Method, Lagrange Multipliers, Restricted Variations, Fundamental Matrix