22846

ﮐﺎرﺑﺮد ﺗﻮاﺑﻊ ﺧﺎص در ﺣﻞ ﻋﺪدي ﻣﺴﺎﺋﻞ ﻣﺒﺘﻨﯽ ﺑﺮ ﻣﻌﺎدﻻت اﻧﺘﮕﺮاﻟﯽ

دكتري

رياضي كاربردي

1393

1399/8/27

دكتر خسرو مالك نژاد

دكتر جليل رشيدي نيا

رياضي

ﻣﻌﺎدﻻت اﻧﺘﮕﺮال ﯾكي از ﺷﺎﺧﻪ ﻫﺎي ﻣﻬﻢ و ﺟﺬاب رﯾﺎضي اﺳﺖ. اﯾﻦ ﻧﻮع از ﻣﻌﺎدﻻت در زﻣﯿﻨﻪ ﻫﺎي ﻣﺨﺘﻠﻒ ﻋﻠﻮم و ﻣﻬﻨﺪسي ﮐﺎرﺑﺮد ﻫﺎي ﻓﺮاواني دارد. ﻫﻤﭽﻨﯿﻦ ارﺗﺒﺎط ﻧﺰدﯾك ﺑﯿﻦ ﻣﻌﺎدﻻت اﻧﺘﮕﺮال و ﻣﺒﺎﺣﺚ دﯾگري از رﯾﺎﺿﯿﺎت ﻣﺎﻧﻨﺪ ﺟﺒﺮ ﺧﻄي، ﻧﻈﺮﯾﻪ ﻋﻤﻠگرﻫﺎ و ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ وﺟﻮد دارد. ﻋﻠي اﯾﺤﺎل ﻫﺪف اﯾﻦ رﺳﺎﻟﻪ اراﺋﻪ روش ﻫﺎي ﻋﺪدي ﻣﻄﻠﻮب، ﺑﻤﻨﻈﻮر ﺣﻞ ﺑﺮﺧي ﻣﻌﺎدﻻت اﻧﺘﮕﺮاﻟي ﻣي ﺑﺎﺷﺪ. در ﻓﺼﻞ اول، ﻣﺒﺎﻧي اﯾﻦ ﻧﻮع ﻣﻌﺎدﻻت و ﺑﺪﺧﯿﻤي ﺑﻌﻀي از آﻧﻬﺎ ﺑﯿﺎن ﺧﻮاﻫﺪ ﺷﺪ. ﻓﺼﻞ دوم، ﺑﻪ ﺑﺮرﺳي ﺑﺮﺧي ﺗﻮاﺑﻊ ﺧﺎص ﮐﻪ ﺑﻌﻨﻮان ﺗﻮاﺑﻊ ﭘﺎﯾﻪ اي ﺑيﺎر ﮔﯿﺮي ﺧﻮاﻫﻨﺪ ﺷﺪ، ﻣي ﭘﺮدازد. در ﻓﺼﻞ ﺳﻮم، ﺑﺮاي ﻣﺤﺎﺳﺒﻪ ﺟﻮاب ﺗﻘﺮﯾﺒﯽ ﻣﻌﺎدﻟﻪ اﻧﺘﮕﺮال ﻣﻨﻔﺮد ﺑﺮ روي ﺑﺎزه ﻧﺎﻣﺘﻨﺎﻫي، از ﯾي روش ﻣﺎﺗﺮﯾﺴي ﻋﻤﻠي ﺑﺮ اﺳﺎس ﺗﻮاﺑﻊ ﭘﺎﯾﻪ اي ﻻﮔﺮ اﺳﺘﻔﺎده ﻣي ﺷﻮد. ﺑﻤﻨﻈﻮر اﺟﺘﻨﺎب از ﺗﻮاﺑﻊ ﻻﮔﺮ اﺳﺘﻔﺎده ﻣي ﮐﻨﯿﻢ. ﻣﺰﯾﺖ اﺻﻠي اﯾﻦ روش دﻗﺖ ﺧﻮبx از ﻧﻮﺳﺎﻧﺎت ﺷﺪﯾﺪ، ﺑﻪ ازاي ﻣﻘﺎدﯾﺮ ﺑﺰرگ و ﻫﺰﯾﻨﻪ ﻣﺤﺎﺳﺒﺎت ﮐﻢ ﺑﻮده، ﮐﻪ دﻟﯿﻞ آن ﺧﻮاص ﻣﻄﻠﻮب ﺗﻮاﺑﻊ ﻻﮔﺮ) ﺗﺎﺑﻊ ﭘﺎﯾﻪ اي 2آن ﺑﺎ اﺣﺘﺴﺎب و ﻣﺎﺗﺮﯾﺲ ﻋﻤﻠﯿﺎﺗي دوآل آن ﻣي ﺑﺎﺷﺪ. ﻣﻘﺪار دﻗﯿﻖ اﯾﻦ ﻣﺎﺗﺮﯾﺲ ﺑﺮاﺑﺮ ﻣﺎﺗﺮﯾﺲ واﺣﺪ اﺳﺖ. ﮐﻪ اﯾﻦ اﻣﺮ ﻣﻮﺟﺐ ﺳﺎدﮔي در ﻓﺮآﯾﻨﺪ ﺗﻘﺮﯾﺐ ﺷﺪه و ﺧﻄﺎ ﻫﺰﯾﻨﻪ ﻣﺤﺎﺳﺒﺎت را ﮐﺎﻫﺶ ﻣي دﻫﺪ و دﻗﺖ روش را ﺑﻬﺒﻮد ﻣي ﺑﺨﺸﺪ. در اداﻣﻪﻫﻤيﺮاﯾﯽوﭘﺎﯾﺪاريروشﻣﻮردﺑﺮرﺳيﻗﺮارﮔﺮﻓﺘﻪوﺳﺮاﻧﺠﺎمﻣﺜﺎل ﻫﺎيﻋﺪديﺑﺮايﻣﻘﺎﯾﺴﻪﺑﺎﺳﺎﯾﺮروش ﻫﺎ و ﮐﺎراﯾﯽ روش ﭘﯿﺸﻨﻬﺎدي اراﺋﻪ ﺷﺪه اﺳﺖ و ﺳﺮاﻧﺠﺎم در ﻓﺼﻞ ﭼﻬﺎرم، ﯾي اﻟيﻮرﯾﺘﻢ ﻋﺪدي ﻋﻤﻠي ﺑﺮ اﺳﺎس يﺗﻮاﺑﻊ ﻣﻮﺟي ﻟﮋاﻧﺪر دو ﺑﻌﺪي ﺑﺮاي ﻣﺤﺎﺳﺒﻪ ﺟﻮاب ﺗﻘﺮﯾﺒﯽ دﺳﺘﮕﺎه ﻣﻌﺎدﻻت اﻧﺘﮕﺮال وﻟﺘﺮا‐ﻓﺮدﻫﻠﻢ ﻏﯿﺮ ﺧﻄ دو ﺑﻌﺪي و دﺳﺘﮕﺎه ﻣﻌﺎدﻻت اﻧﺘﮕﺮال وﻟﺘﺮا ﻏﯿﺮ ﺧﻄي دو ﺑﻌﺪي اراﺋﻪ ﺧﻮاﻫﺪ ﺷﺪ. ﻣﺰﯾﺖ اﺻﻠي اﯾﻦ روش دﻗﺖ يﺑﺎﻻ و ﮐﺎراﯾﯽ در اﻧﺠﺎمﻣﺤﺎﺳﺒﺎت اﺳﺖ. ﮐﻪ اﯾﻦ اﻣﺮ ﭘﯿﺎﻣﺪ اﺳﺘﻔﺎده ازﺗﻮاﺑﻊ ﻣﻮﺟك ﻟﮋاﻧﺪرﻣي ﺑﺎﺷﺪ. وﯾﮋﮔي اﺻﻠي اﯾﻦ ﺗﻮاﺑﻊ ﭘﺎﯾﻪ اي ﺗﻮاﻧﺎﯾﯽ آن در اﻓﺮاز ﻧﻘﺎط ﻏﯿﺮ ﻫﻤﻮار ﺑﻪ زﯾﺮ ﺑﺎزه ﻫﺎي ﮐﻮﭼي اﺳﺖ، ﺗﺎ ﺗﻘﺮﯾﺐ ﺑﻬﺘﺮي ﺑﺪﺳﺖ آﯾﺪ. در اداﻣﻪ ﻫﻤيﺮاﯾﯽ روش ﻣﻮرد ﺑﺮرﺳي ﻗﺮار ﮔﺮﻓﺘﻪ و ﺳﺮاﻧﺠﺎم ﻣﺜﺎل ﻫﺎي ﻋﺪدي ﺑﺮاي ﻣﻘﺎﯾﺴﻪ ﺑﺎ ﺳﺎﯾﺮ روش ﻫﺎ .و ﮐﺎراﯾﯽ روش ﭘﯿﺸﻨﻬﺎدي اراﺋﻪ ﺷﺪه اﺳﺖ

1399/09/23

Application of special functions in the numerical solution of the problems based on integral equations

11/17/2020 12:00:00 AM

Integral equations are one of the important branches of mathematics, which have many applications in the field of sciences and engineering. Also, there are a close connection between integral equations and other topics of mathematics such as linear algebra, operator theory and differential equations. The main motivation of this thesis is to present optimal numerical methods for solving some integral equations. The outline of this thesis is as follows. In chapter 1 the basis of these equations and some of the ill-posedness will be discussed. Chapter 2 discusses some specific functions that will be used as basis functions. In chapter 3 presents a practical matrix method based on the Laguerre functions to approximate the solution of a singular integral equation on the infinite interval. The Laguerre functions which are obtained from the Laguerre polynomials are used to avoid fluctuations for large values. The main charactristic of the scheme is good accuracy with two basis functions and less computational cost which are the consequences of the Laguerre functions properties and its dual operational matrix. This matrix is equal to the identity matrix which simplify the approximation procedure and reduce the computational error of the scheme. In this technique, dual operational matrix, matrix forms and collocation method are formulated to convert singular integral equation into a matrix equation. The convergence analysis and the stability of the proposed method are proven in detail. Some numerical examples with comparison illustrate the efficiency of the scheme. Finally, chapter 4 develops a practical numerical algorithm based on two- dimensional Legendre wavelet functions for calculating approximate solution of a nonlinear system of two-dimensional Volterra–Fredholm and Volterra integral equations. The main characteristic of this approach is high accuracy and computational efficiency of performing which are the consequences of Legendre wavelet properties. The main benefit of this basis functions are their ability to detect singularities and their efficiency in dealing with non-sufficiently smooth function in comparison with Legendre polynomials and they minimize the error. The convergence analysis and error bound of the proposed Legendre wavelet method are investigated. Numerical examples confirm that the Legendre wavelet collocation method is accurate, reliable for solving nonlinear system of two-dimensional integral equations.