Homotopy Analysis And Legendre Multi-Wavelets Methods For Solving Integral Equations
Due to the ability of function representation, hybrid functions and wavelets have a special position in research. In this thesis, we state elementary definitions, then we introduce hybrid functions and some wavelets such as Haar, Daubechies, Cheby- shev, sine-cosine and linear Legendre multi wave...
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| Format: | Thesis |
| Language: | English English |
| Published: |
2009
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| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/12370/ http://psasir.upm.edu.my/id/eprint/12370/1/IPM_2009_12A.pdf |
| _version_ | 1848841826033729536 |
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| author | Vahdati, Saeed |
| author_facet | Vahdati, Saeed |
| author_sort | Vahdati, Saeed |
| building | UPM Institutional Repository |
| collection | Online Access |
| description | Due to the ability of function representation, hybrid functions and wavelets have a
special position in research. In this thesis, we state elementary definitions, then we
introduce hybrid functions and some wavelets such as Haar, Daubechies, Cheby-
shev, sine-cosine and linear Legendre multi wavelets. The construction of most
wavelets are based on stepwise functions and the comparison between two categories of wavelets will become easier if we have a common construction of them.
The properties of the Floor function are used to and a function which is one on the
interval [0; 1) and zero elsewhere. The suitable dilation and translation parameters
lead us to get similar function corresponding to the interval [a; b). These functions
and their combinations enable us to represent the stepwise functions as a function of
floor function. We have applied this method on Haar wavelet, Sine-Cosine wavelet,
Block - Pulse functions and Hybrid Fourier Block-Pulse functions to get the new
representations of these functions.
The main advantage of the wavelet technique for solving a problem is its ability
to transform complex problems into a system of algebraic equations. We use the Legendre multi-wavelets on the interval [0; 1) to solve the linear integro-differential
and Fredholm integral equations of the second kind. We also use collocation points
and linear legendre multi wavelets to solve an integro-differential equation which describes the charged particle motion for certain configurations of oscillating magnetic
fields. Illustrative examples are included to reveal the sufficiency of the technique.
In linear integro-differential equations and Fredholm integral equations of the second
kind cases, comparisons are done with CAS wavelets and differential transformation
methods and it shows that the accuracy of these results are higher than them.
Homotopy Analysis Method (HAM) is an analytic technique to solve the linear
and nonlinear equations which can be used to obtain the numerical solution too.
We extend the application of homotopy analysis method for solving Linear integro-
differential equations and Fredholm and Volterra integral equations. We provide
some numerical examples to demonstrate the validity and applicability of the technique. Numerical results showed the advantage of the HAM over the HPM, SCW,
LLMW and CAS wavelets methods. For future studies, some problems are proposed
at the end of this thesis. |
| first_indexed | 2025-11-15T07:49:24Z |
| format | Thesis |
| id | upm-12370 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| language | English English |
| last_indexed | 2025-11-15T07:49:24Z |
| publishDate | 2009 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-123702013-05-27T07:51:53Z http://psasir.upm.edu.my/id/eprint/12370/ Homotopy Analysis And Legendre Multi-Wavelets Methods For Solving Integral Equations Vahdati, Saeed Due to the ability of function representation, hybrid functions and wavelets have a special position in research. In this thesis, we state elementary definitions, then we introduce hybrid functions and some wavelets such as Haar, Daubechies, Cheby- shev, sine-cosine and linear Legendre multi wavelets. The construction of most wavelets are based on stepwise functions and the comparison between two categories of wavelets will become easier if we have a common construction of them. The properties of the Floor function are used to and a function which is one on the interval [0; 1) and zero elsewhere. The suitable dilation and translation parameters lead us to get similar function corresponding to the interval [a; b). These functions and their combinations enable us to represent the stepwise functions as a function of floor function. We have applied this method on Haar wavelet, Sine-Cosine wavelet, Block - Pulse functions and Hybrid Fourier Block-Pulse functions to get the new representations of these functions. The main advantage of the wavelet technique for solving a problem is its ability to transform complex problems into a system of algebraic equations. We use the Legendre multi-wavelets on the interval [0; 1) to solve the linear integro-differential and Fredholm integral equations of the second kind. We also use collocation points and linear legendre multi wavelets to solve an integro-differential equation which describes the charged particle motion for certain configurations of oscillating magnetic fields. Illustrative examples are included to reveal the sufficiency of the technique. In linear integro-differential equations and Fredholm integral equations of the second kind cases, comparisons are done with CAS wavelets and differential transformation methods and it shows that the accuracy of these results are higher than them. Homotopy Analysis Method (HAM) is an analytic technique to solve the linear and nonlinear equations which can be used to obtain the numerical solution too. We extend the application of homotopy analysis method for solving Linear integro- differential equations and Fredholm and Volterra integral equations. We provide some numerical examples to demonstrate the validity and applicability of the technique. Numerical results showed the advantage of the HAM over the HPM, SCW, LLMW and CAS wavelets methods. For future studies, some problems are proposed at the end of this thesis. 2009-12 Thesis NonPeerReviewed application/pdf en http://psasir.upm.edu.my/id/eprint/12370/1/IPM_2009_12A.pdf Vahdati, Saeed (2009) Homotopy Analysis And Legendre Multi-Wavelets Methods For Solving Integral Equations. PhD thesis, Universiti Putra Malaysia. Homotopy theory Wavelets (Mathematics) English |
| spellingShingle | Homotopy theory Wavelets (Mathematics) Vahdati, Saeed Homotopy Analysis And Legendre Multi-Wavelets Methods For Solving Integral Equations |
| title | Homotopy Analysis And Legendre Multi-Wavelets Methods For Solving Integral Equations |
| title_full | Homotopy Analysis And Legendre Multi-Wavelets Methods For Solving Integral Equations |
| title_fullStr | Homotopy Analysis And Legendre Multi-Wavelets Methods For Solving Integral Equations |
| title_full_unstemmed | Homotopy Analysis And Legendre Multi-Wavelets Methods For Solving Integral Equations |
| title_short | Homotopy Analysis And Legendre Multi-Wavelets Methods For Solving Integral Equations |
| title_sort | homotopy analysis and legendre multi-wavelets methods for solving integral equations |
| topic | Homotopy theory Wavelets (Mathematics) |
| url | http://psasir.upm.edu.my/id/eprint/12370/ http://psasir.upm.edu.my/id/eprint/12370/1/IPM_2009_12A.pdf |