$Solving partial and fraction differential equations using corrected fourier series method$

Solving partial and fraction differential equations using corrected fourier series method

Partial differential equations (PDE) are often used to construct models of the most basic theories in physics and engineering. Our goal here is to solve the PDEs problem by using Fourier series method that always been used. However, the truncated Fourier series will cause the Gibbs phenomenon. To el...

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Main Author:	Zainal, Nor Hafizah
Format:	Thesis
Language:	English
Published:	2014
Subjects:	Differential equations
Online Access:	http://psasir.upm.edu.my/id/eprint/70539/1/FS%202014%2071%20IR.pdf
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id	my-upm-ir.70539
record_format	uketd_dc
spelling	my-upm-ir.705392022-06-17T02:03:29Z Solving partial and fraction differential equations using corrected fourier series method 2014-11 Zainal, Nor Hafizah Partial differential equations (PDE) are often used to construct models of the most basic theories in physics and engineering. Our goal here is to solve the PDEs problem by using Fourier series method that always been used. However, the truncated Fourier series will cause the Gibbs phenomenon. To eliminate this phenomenon, the corrected Fourier series which consists of a uniformly convergent Fourier series and a correction function will be used. The correction function here is referred to the algebraic polynomials and Heaviside step functions. The Fourier series remains uniformly convergent until its -th derivative without Gibbs oscillation if the order of polynomial in correction functions not exceed -th order which the Gibbs oscillation of the Fourier series will be terminated until its -th derivative. In this study, we use the corrected Fourier series to solve partial differential equations and fractional partial differential equations. The theory of derivatives and integrals of fractional (non-integer) order was started over 300 years ago. In recent years, fractional calculus have been attract in various research due to its extensive application in engineering and science. We solve this problem by using corrected Fourier series method with modified Riemann-Liouville derivatives. The fractional derivatives are described in Riemann-Liouville sense. For the case PDEs, we compared the result with classical Fourier series method and exact solution. There is some case that classical Fourier series method cannot solve at a certain point. Meanwhile, corrected Fourier series method gives the solution at that point. For the case that cannot solve by using classical Fourier series method, we can solve by using corrected Fourier series method. For the fractional PDEs, there is no exact solution for order α as a non-integer number. Thus, we compared the result with others method which is variational iteration method (VIM) and homotopy method. The Maple software is used for all calculation in this study. Differential equations 2014-11 Thesis http://psasir.upm.edu.my/id/eprint/70539/ http://psasir.upm.edu.my/id/eprint/70539/1/FS%202014%2071%20IR.pdf text en public masters Universiti Putra Malaysia Differential equations Kilicman, Adem
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English
advisor	Kilicman, Adem
topic	Differential equations
spellingShingle	Differential equations Zainal, Nor Hafizah Solving partial and fraction differential equations using corrected fourier series method
description	Partial differential equations (PDE) are often used to construct models of the most basic theories in physics and engineering. Our goal here is to solve the PDEs problem by using Fourier series method that always been used. However, the truncated Fourier series will cause the Gibbs phenomenon. To eliminate this phenomenon, the corrected Fourier series which consists of a uniformly convergent Fourier series and a correction function will be used. The correction function here is referred to the algebraic polynomials and Heaviside step functions. The Fourier series remains uniformly convergent until its -th derivative without Gibbs oscillation if the order of polynomial in correction functions not exceed -th order which the Gibbs oscillation of the Fourier series will be terminated until its -th derivative. In this study, we use the corrected Fourier series to solve partial differential equations and fractional partial differential equations. The theory of derivatives and integrals of fractional (non-integer) order was started over 300 years ago. In recent years, fractional calculus have been attract in various research due to its extensive application in engineering and science. We solve this problem by using corrected Fourier series method with modified Riemann-Liouville derivatives. The fractional derivatives are described in Riemann-Liouville sense. For the case PDEs, we compared the result with classical Fourier series method and exact solution. There is some case that classical Fourier series method cannot solve at a certain point. Meanwhile, corrected Fourier series method gives the solution at that point. For the case that cannot solve by using classical Fourier series method, we can solve by using corrected Fourier series method. For the fractional PDEs, there is no exact solution for order α as a non-integer number. Thus, we compared the result with others method which is variational iteration method (VIM) and homotopy method. The Maple software is used for all calculation in this study.
format	Thesis
qualification_level	Master's degree
author	Zainal, Nor Hafizah
author_facet	Zainal, Nor Hafizah
author_sort	Zainal, Nor Hafizah
title	Solving partial and fraction differential equations using corrected fourier series method
title_short	Solving partial and fraction differential equations using corrected fourier series method
title_full	Solving partial and fraction differential equations using corrected fourier series method
title_fullStr	Solving partial and fraction differential equations using corrected fourier series method
title_full_unstemmed	Solving partial and fraction differential equations using corrected fourier series method
title_sort	solving partial and fraction differential equations using corrected fourier series method
granting_institution	Universiti Putra Malaysia
publishDate	2014
url	http://psasir.upm.edu.my/id/eprint/70539/1/FS%202014%2071%20IR.pdf
_version_	1747812862512857088

Solving partial and fraction differential equations using corrected fourier series method

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