Wavelet methods for solving linear and nonlinear singular boundary value problems
In this thesis, wavelet analysis method is proposed for solving singular boundary value problems. Operational matrix of differentiation is introduced. Furthermore, product operational matrix is also presented. Many different examples are solved using Chebyshev wavelet analysis method to confirm t...
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| Main Author: | |
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| Format: | Thesis |
| Language: | English |
| Published: |
2014
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| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/70501/1/FS%202014%2064%20IR.pdf http://psasir.upm.edu.my/id/eprint/70501/ |
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| Summary: | In this thesis, wavelet analysis method is proposed for solving singular boundary
value problems. Operational matrix of differentiation is introduced. Furthermore,
product operational matrix is also presented. Many different examples are
solved using Chebyshev wavelet analysis method to confirm the accuracy and the
efficiency of wavelet analysis method.
An efficient and accurate method based on hybrid of Chebyshev wavelets and
finite difference methods is introduced for solving linear and nonlinear singular
ordinary differential equations such as Lane-Emden equations, boundary value
problems of fractional order and singular and nonsingular systems of boundary
and initial value problems. High-order multi-point boundary value problems are
also solved. The useful properties of Chebyshev wavelets and finite difference
method make it a computationally efficient method to approximate the solution
of nonlinear equations in a semi-infinite interval. The given problem is converted
into a system of algebraic equations using collocation points. The main advantage
of this method is the ability to represent smooth and especially piecewise smooth
functions properly. It is also clarified that increasing the number of subintervals
or the degree of the Chebyshev polynomials in a proper way leads to improvement
of the accuracy. Moreover, this method is applicable for solving problems on large
interval. Several examples will be provided to demonstrate the powerfulness of
the proposed method. A comparison is made among this method, some other
well-known approaches and exact solution which confirms that the introduced
method are more accurate and efficient. For future studies, some problems are
proposed at the end of this thesis. |
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