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    政大機構典藏 > 理學院 > 應用數學系 > 學位論文 >  Item 140.119/147039
    Please use this identifier to cite or link to this item: https://nccur.lib.nccu.edu.tw/handle/140.119/147039


    Title: 二階線性常微分方程式不適定性問題之探討
    A Study on Ill-Posed Problem of Second Order Linear Ordinary Differential Equation
    Authors: 陳劭瑜
    Chen, Shao-Yu
    Contributors: 曾正男
    Tzeng, Jeng-Nan
    陳劭瑜
    Chen, Shao-Yu
    Keywords: 不適定性問題
    常微分方程式
    邊界值問題
    有限差分法
    Ill-Posed Problem
    Ordinary Differential Equation
    Boundary Value Problem
    Finite Difference Method
    Date: 2023
    Issue Date: 2023-09-01 15:26:09 (UTC+8)
    Abstract: 微分方程被廣泛運用於工程、社會科學及生物統計等領域,然在現實之應用過程中,可能因方程式較為複雜,以及不易取得合適之初始條件或邊界條件等因素,以致除無法推導其解析解外,亦無法順利求取其數值近似解。
    本研究旨在探討二階線性常微分方程之邊界值問題,並以當無法透過邊界條件求得未知待定係數之情形為例進行討論。而對於處理此類不適定性問題,藉由增加或改變原題目條件之方式,使方程式的解變為唯一解屬常見之一種方法。
    本研究提出改變原題目之方式為提高方程式中導數之階數,以及使有限差分法所建構出之矩陣A轉變為奇異矩陣。該方法係期望透過選擇不同奇異矩陣所得之數值解,以及其Null Space之基底向量,辨識所研究之問題屬適定性問題或是不適定性問題。倘為不適定性問題,則持續探討其解之唯一性,以及奇異矩陣之Null Space的基底向量與通解中特徵函數之關聯性等事宜。
    Differential equations are widely used in fields such as engineering, social science, and biostatistics. However, when they are put into use in practice, due to factors such as the complexity of equations and the difficulty in obtaining appropriate initial conditions or boundary conditions, it is possible that no analytic expression can be derived as a result, and it is also likely that no numerical approximate solution can be obtained successfully.
    This study aims to investigate the boundary value problem of second-order linear ordinary differential equations. An example was made and discussed where unknown undetermined coefficients cannot be obtained through boundary conditions. To address this sort of ill-posed problems, one common method is to add conditions, so as to make the solution of the equation the unique solution.
    In the study, the method adopted to change the condition of the original problem is by increasing the order of derivatives in the equation, and by transforming matrix A constructed by the finite difference method into a singular matrix. By employing this method, it is hoped that one could use the numerical solutions obtained from choosing different Singular Matrices as well as the basis vectors of the Null Space of those Singular Matrices to determine whether the problem under discussion is a well-posed or an ill-posed one. If it is an ill-posed problem, then one could keep exploring areas such as the uniqueness of that solution, as well as the correlation between the basis vectors of the Null Space of the singular matrices and the characteristic functions in general solutions.
    Reference: [1]趙英宏, 趙元和 (2008)。 數值分析。 台北市:五南圖書出版股份有限公司.
    [2]Ben Noble, James W. Daniel (2002). Applied Linear Algebra (3rd ed.). Pearson
    [3]Cristiane Aparecida Pendeza Martinez, Andre´ Lu´Is Machado Martinez, Glaucia Maria Bressan ∗ , Emerson Vitor Castelani And Roberto Molina De Souza. (2019). Multiple Solutions For A Fourth Order Equation With Nonlinear Boundary Conditions: Theoretical And Numerical Aspects. Differential Equations & Applications, Vol. 11, 335-348.
    [4]David Kincaid, Ward Cheney. (2002). Numerical Analysis: Mathematics of Scientific Computing (3rd ed.). American Mathematical Society
    [5]David Poole. (2014). Linear Algebra: A Modern Introduction (4th ed.). Cengage Learning
    [6]Edward M. Landesman, Magnus R. Hestenes. (1991). Linear Algebra for Mathematics, Science, and Engineering. Pearson College Div
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    [11]J. Baumeister† and A. Leit˜ao‡. On iterative methods for solving ill-posed problems modeled by partial differential equations. arXiv:2011.14441, Nov 2020.
    [12]Lin Li, Ping Lin , Xinhui Si, Liancun Zheng. (2017). A numerical study for multiple solutions of a singular boundary value problem arising from laminar flow in a porous pipe with moving wall. journal of computational and applied mathematics, Vol. 313, 536-549.
    [13]R. Byron Bird, Warren E. Stewart, Edwin N. Lightfoot. (2001). Transport Phenomena (2nd ed.). Wiley
    [14]Richard L. Burden, J. Douglas Faires. (2010). Numerical Analysis (9th ed.). Cengage Learning
    [15]Ron Larson, Robert P. Hostetler, Bruce H. Edwards. (1997). Calculus (6th ed.). Houghton Mifflin
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    [19]Zohreh Ranjbar. (2010). Numerical Solution of Ill-posed Cauchy Problems for Parabolic Equations. Linköping Studies in Science and Technology. Dissertations, No. 1300
    Description: 碩士
    國立政治大學
    應用數學系
    108751015
    Source URI: http://thesis.lib.nccu.edu.tw/record/#G0108751015
    Data Type: thesis
    Appears in Collections:[應用數學系] 學位論文

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