Extended Haar Wavelet Quasilinearization method for solving boundary value problems / Nor Artisham Che Ghani
Several computational methods have been proposed to solve single nonlinear ordinary differential equations. In spite of the enormous numerical effort, however yet numerically accurate and robust algorithm is still missing. Moreover, to the best of our knowledge, only a few works are dedicated to...
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Format: | Thesis |
Published: |
2018
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Online Access: | http://studentsrepo.um.edu.my/9039/1/Siti_Arthisham_Che_Ghani.pdf http://studentsrepo.um.edu.my/9039/6/artisham.pdf http://studentsrepo.um.edu.my/9039/ |
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Summary: | Several computational methods have been proposed to solve single nonlinear ordinary
differential equations. In spite of the enormous numerical effort, however yet
numerically accurate and robust algorithm is still missing. Moreover, to the best of our
knowledge, only a few works are dedicated to the numerical solution of coupled
nonlinear ordinary differential equations. Hence, a robust algorithm based on Haar
wavelets and the quasilinearization process is provided in this study for solving both
numerical solutions; single nonlinear ordinary differential equations and systems of
coupled nonlinear ordinary differential equations, including two of them are the new
problems with some additional related parameters. In this research, the generation of
Haar wavelets function, its series expansion and one-dimensional matrix for a chosen
interval 0, B is introduced in detail. We expand the usual defined interval 0, 1 to
0, B because the actual problem does not necessarily involve only limit B to one,
especially in the case of coupled nonlinear ordinary differential equations. To achieve
the target, quasilinearization technique is used to linearize the nonlinear ordinary
differential equations, and then the Haar wavelet method is applied in the linearized
problems. Quasilinearization technique provides a sequence of function which
monotonic quadratically converges to the solution of the original equations. The highest
derivatives appearing in the differential equations are first expanded into Haar series.
The lower order derivatives and the solutions can then be obtained quite easily by using
multiple integration of Haar wavelet. All the values of Haar wavelet functions are
substituted into the quasilinearized problem. The wavelet coefficient can be calculated
easily by using MATLAB software. The universal subprogram is introduced to calculate
the integrals of Haar wavelets. This will provide small computational time. The initial approximation can be determined from mathematical or physical consideration. In the
demonstration problem, the performance of Haar wavelet quasilinearization method
(HWQM) is compared with the existing numerical solutions that showed the same basis
found in the literature. For the beginning, the computation was carried out for lower
resolution. As expected, the more accurate results can be obtained by increasing the
resolution and the convergence are faster at collocation points. For systems of coupled
nonlinear ordinary differential equations, the equations are obtained through the
similarity transformations. The transformed equations are then solved numerically. This
is contrary to Runge-Kutta method, where the boundary value problems of HWQM
need not to be reduced into a system of first order ordinary differential equations.
Besides in terms of accuracy, efficiency and applicability in solving nonlinear ordinary
differential equations for a variety of boundary conditions, this method also allow
simplicity, fast and small computation cost since most elements of the matrices of Haar
wavelet and its integration are zeros, it were contributed to the speeding up of the
computation. This method can therefore serve as very useful tool in many physical
applications. |
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