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題名 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
作者 陳劭瑜
Chen, Shao-Yu
貢獻者 曾正男
Tzeng, Jeng-Nan
陳劭瑜
Chen, Shao-Yu
關鍵詞 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
日期 2023
上傳時間 1-Sep-2023 15:26:09 (UTC+8)
摘要 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.
參考文獻 [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
[7]Erwin Kreyszig. (1998). Advanced Engineering Mathematics (8th ed.). John Wiley & Sons
[8]Gilbert Strang. (2006). Linear Algebra and Its Applications (4th ed.). Cengage Learning
[9]Howard Anton, Chris Rorres. (2005). Elementary Linear Algebra with Applications (9th ed.). Wiley, INC.
[10]I.N. Herstein, David J. Winter. (1988). Matrix Theory and Linear Algebra. Macmillan Pub Co.
[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
[16]Salas, Einar Hille, Garrett Etgen. (1998) Calculus (8th ed.). John Wiley & Sons, INC.
[17]Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence (2002). Linear Algebra (4th ed.). Pearson
[18]Steven J. Leon. (2005). Linear Algebra with Applications (7th ed.). Pearson College Div
[19]Zohreh Ranjbar. (2010). Numerical Solution of Ill-posed Cauchy Problems for Parabolic Equations. Linköping Studies in Science and Technology. Dissertations, No. 1300
描述 碩士
國立政治大學
應用數學系
108751015
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0108751015
資料類型 thesis
dc.contributor.advisor 曾正男zh_TW
dc.contributor.advisor Tzeng, Jeng-Nanen_US
dc.contributor.author (Authors) 陳劭瑜zh_TW
dc.contributor.author (Authors) Chen, Shao-Yuen_US
dc.creator (作者) 陳劭瑜zh_TW
dc.creator (作者) Chen, Shao-Yuen_US
dc.date (日期) 2023en_US
dc.date.accessioned 1-Sep-2023 15:26:09 (UTC+8)-
dc.date.available 1-Sep-2023 15:26:09 (UTC+8)-
dc.date.issued (上傳時間) 1-Sep-2023 15:26:09 (UTC+8)-
dc.identifier (Other Identifiers) G0108751015en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/147039-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用數學系zh_TW
dc.description (描述) 108751015zh_TW
dc.description.abstract (摘要) 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。		zh_TW
dc.description.abstract (摘要) 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.		en_US
dc.description.tableofcontents 致謝 i
摘要 ii
Abstract iii
目錄 iv
表目錄 v
圖目錄 viii
第一章 緒論 1
第一節 適定性問題(Well-Posed Problem) 1
第二節 二階常微分方程式之應用示例 3
第三節 研究動機及目的 5
第二章 文獻探討 7
第一節 數值微分 7
第二節有限差分法(Finite Difference Method) 14
第三節 最陡下降法(Steepest Descent Method) 19
第四節 奇異值分解(Singular Value Decomposition) 28
第五節 最小平方法(Least Squares Method) 31
第三章 研究方法 36
第一節 研究發想 36
第二節 方法簡介 37
第三節 流程概述 45
第四章 實測分析 61
第一節 Well-Posed Problem之測試 61
第二節 Ill-Posed Problem之測試：題型I 76
第三節 Ill-Posed Problem之測試：題型II 98
第五章 結論及未來工作 152
第一節 結論 152
第二節 未來工作 155
參考文獻 156		zh_TW
dc.format.extent 10768418 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0108751015en_US
dc.subject (關鍵詞) 不適定性問題zh_TW
dc.subject (關鍵詞) 常微分方程式zh_TW
dc.subject (關鍵詞) 邊界值問題zh_TW
dc.subject (關鍵詞) 有限差分法zh_TW
dc.subject (關鍵詞) Ill-Posed Problemen_US
dc.subject (關鍵詞) Ordinary Differential Equationen_US
dc.subject (關鍵詞) Boundary Value Problemen_US
dc.subject (關鍵詞) Finite Difference Methoden_US
dc.title (題名) 二階線性常微分方程式不適定性問題之探討zh_TW
dc.title (題名) A Study on Ill-Posed Problem of Second Order Linear Ordinary Differential Equationen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) [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
[7]Erwin Kreyszig. (1998). Advanced Engineering Mathematics (8th ed.). John Wiley & Sons
[8]Gilbert Strang. (2006). Linear Algebra and Its Applications (4th ed.). Cengage Learning
[9]Howard Anton, Chris Rorres. (2005). Elementary Linear Algebra with Applications (9th ed.). Wiley, INC.
[10]I.N. Herstein, David J. Winter. (1988). Matrix Theory and Linear Algebra. Macmillan Pub Co.
[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
[16]Salas, Einar Hille, Garrett Etgen. (1998) Calculus (8th ed.). John Wiley & Sons, INC.
[17]Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence (2002). Linear Algebra (4th ed.). Pearson
[18]Steven J. Leon. (2005). Linear Algebra with Applications (7th ed.). Pearson College Div
[19]Zohreh Ranjbar. (2010). Numerical Solution of Ill-posed Cauchy Problems for Parabolic Equations. Linköping Studies in Science and Technology. Dissertations, No. 1300		zh_TW