Wavelets operational methods for fractional differential equations and systems of fractional differential equations
In this thesis, new and effective operational methods based on polynomials and wavelets for the solutions of FDEs and systems of FDEs are developed. In particular we study one of the important polynomial that belongs to the Appell family of polynomials, namely, Genocchi polynomial. This polynomia...
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| Format: | Thesis |
| Language: | English English |
| Published: |
2017
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| Subjects: | |
| Online Access: | http://eprints.uthm.edu.my/291/1/24p%20ABDULNASIR%20ISAH.pdf http://eprints.uthm.edu.my/291/2/ABDULNASIR%20ISAH%20WATERMARK.pdf |
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| Summary: | In this thesis, new and effective operational methods based on polynomials and
wavelets for the solutions of FDEs and systems of FDEs are developed. In particular
we study one of the important polynomial that belongs to the Appell family of
polynomials, namely, Genocchi polynomial. This polynomial has certain great
advantages based on which an effective and simple operational matrix of derivative
was first derived and applied together with collocation method to solve some singular
second order differential equations of Emden-Fowler type, a class of generalized
Pantograph equations and Delay differential systems. A new operational matrix of
fractional order derivative and integration based on this polynomial was also
developed and used together with collocation method to solve FDEs, systems of
FDEs and fractional order delay differential equations. Error bound for some of the
considered problems is also shown and proved. Further, a wavelet bases based on
Genocchi polynomials is also constructed, its operational matrix of fractional order
derivative is derived and used for the solutions of FDEs and systems of FDEs. A
novel approach for obtaining operational matrices of fractional derivative based on
Legendre and Chebyshev wavelets is developed, where, the wavelets are first
transformed into corresponding shifted polynomials and the transformation matrices
are formed and used together with the polynomials operational matrices of fractional
derivatives to obtain the wavelets operational matrix. These new operational matrices
are used together with spectral Tau and collocation methods to solve FDEs and
systems of FDEs. |
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