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完整後設資料紀錄
DC 欄位 | 值 | 語言 |
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dc.contributor.advisor | 劉進賢,楊照彥 | |
dc.contributor.author | Yu-Wen Wang | en |
dc.contributor.author | 王昱文 | zh_TW |
dc.date.accessioned | 2021-06-08T03:14:48Z | - |
dc.date.copyright | 2017-02-21 | |
dc.date.issued | 2017 | |
dc.date.submitted | 2017-02-10 | |
dc.identifier.citation | [1] Roos HG, Stynes M, Tobiska L. Numerical Methods for Singularly Perturbed Differential Equations. Springer-Verlag, Berlin, 1996.
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Numerical solutions of linear and nonlinear singularerturbation problems.Comput. Math. Appl. 55 (2008) 2574-2592. [17] Liu CS. The Lie-group shooting method for solving nonlinear singularly perturbed boundary value problems, Commun. Nonlinear Sci. Numer. Simulat. 17 (2012) 1506-1521. [18] Khuri SA, Sayfy A. Self-adjoint singularly perturbed boundary value problems: an adaptive variational approach, Math. Meth. Appl. Sci. 36 (2013) 1070-1079. [19] Dogan N, Erturk VS, Akin O. Numerical treatment of singularly perturbed two-point boundary value problems by using differential transformation method, Disc. Dyna. Nat. Soc., Article ID 579431 (2012) 10 pages. [20] Liu CS. Solving an inverse Sturm-Liouville problem by a Lie-group method. Boundary Value Probl. Article ID 749865 (2008), 18 pages. [21] Dong L, Alotaibi A, Mohiuddine SA, Atluri SN. Computational methods in engineering: a variety of primal & mixed methods, with global & local interpolations, for well-posed or ill-posed BCs. CMES: Comput. Model. Eng. Sci. 99 (2014) 1-85. [22] Liu CS. An equilibrated method of fundamental solutions to choose the best source points for the Laplace equation. Eng. Anal. Bound. Elem. 36 (2012) 1235-1245. [23] Reddy YN, Chakravarthy PP. An initial-value approach for solving singularly perturbed two-point boundary value problems. Appl. Math. Comput. 155 (2004)95-110. [24] Ilicasu FO, Schultz DH. High-order finite-difference techniques for linear singular perturbation boundary value problems. Comput. Math. Appl. 47 (2004) 391-417. [25] Varner TN, Choudhury SR. Non-standard difference schemes for singular perturbationproblems revisited. Appl. Math. Comput. 92 (1998) 101-123.17 [26] Kadalbajoo MK, Aggarwal VK. Fitted mesh B-spline collocation for solving self-adjointsingularly perturbed boundary value problems. Appl. Math. Comput. 161 (2005) 973-987. [27] Morton K.W. Numerical Solution of Convection-Diffusion Problems. Academic Chap-man & Hall, New York, 1996. [28] Roos HG, Stynes M, Tobiska L. Robust Numerical Methods for Singularly Perturbed Differential Equations. Springer-Verlag, Berlin, 2008. [29] Clavero C, Gracia JL. High order methods for elliptic and time dependent reaction-diffusion singularly perturbed problems. Appl. Math. Comput. 2005;168:1109-27. [30] Morton KW. The convection-diffusion Petrov-Galerkin story. IMS J. Numer. Anal.2010;30:231-40. [31] Demkowicz L, Gopalakrishnan J. A class of discontinuous Petrov-Galerkin methods. II. Optimal test functions. Numer. Meth. Part. Diff. Eqs. 2011;27:70-105. [32] Kast SM, Dahm JPS, Fidkowski KJ. Optimal test functions for boundary accuracy in discontinuous finite element methods. J. Comput. Phys. 2015;298:360-86.7 [33] Kadalbajoo MK, Gupta V. A brief survey on numerical methods for solving singularly perturbed problems. Appl. Math. Comput. 2010;217:3641-716. [34] Hemker PW. A numerical study of stiff two-point boundary problems. Ph.D Thesis,Mathematisch Centrum, Amsterdam, 1977. [35] Liu CS. An equilibrated method of fundamental solutions to choose the best source points for the Laplace equation. Eng. Anal. Bound. Elem. 2012;36:1235-45. [36] Liu CS. Optimally scaled vector regularization method to solve ill-posed linear problems. Appl. Math. Comput. 2012;218:10602-16. [37] Liu CS. A two-side equilibration method to reduce the condition number of an ill-posed linear system. Comput. Model. Eng. Sci. 2013;91:17-42. [38] Liu CS. A multiple/scale/direction polynomial Trefftz method for solving the BHCP in high-dimensional arbitrary simply-connected domains. Int. J. Heat Mass Transfer 2016;92:970-8. [39] Liu CS, Atluri SN. Numerical solution of the Laplacian Cauchy problem by using a better post conditioning collocation Trefftz method. Eng. Anal.Bound. Elem. 2013;37:74-83. [40] Liu CS. Solving third-order singularly perturbed problems: exponentially and polynomially fitted trial functions. J. Math. Research 2016;8(2):16-24. [41] Egger H, Schoberl J. 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[47 Liu CS, Atluri SN:A fictitious time integration method (FTIM) for solving mixed complementarity problems with applications to non-linear optimization. CMES: Computer Modeling in Engineering Sciences, vol.34,No.2, pp.155-178,2008. | |
dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/20999 | - |
dc.description.abstract | 二階線性奇異攝動常微分方程式轉化為奇異線性型拋物線偏微分方程式。由格林第二恆等式所推導的邊界積分方法得到伴隨Trefftz試函數,導出符合伴隨性質的控制方程式和伴隨邊界條件的譜函數後,我們使用弱形式積分方程式方法求解奇異解。該方法與指數擬合試函數是為了用來自動滿足邊界條件。對於在解線性和非線性變係數奇異問題之內部邊界層也非常準確。
奇異對流擴散方程和橢圓型奇異反應擴散方程式若使用一般數值方法會使得問題變病態。於是我們使用配點法來提高精準度。當奇異解以一組二維指數試函數方程式形式表示,並滿足邊界條件與控制方程式我們可以得到一個小尺度的線性系統來求展開係數。在數值算例中證實了配點法求解高度奇異橢圓型問題非常精確有效。 | zh_TW |
dc.description.abstract | Second-order linear singularly perturbed ordinary differential equation is transformed into a singular linear parabolic type partial differential equation. Then, the Green’s second identity is employed to derive a boundary integral equation in terms of the adjoint Trefftz test functions. After deriving the closed-form spectral functions, we develop a weak-form integral equation method (WFIEM) to find the singular solution. The WFIEM together with the exponentially fitted trial functions, which are designed to satisfy the boundary conditions automatically, can provide accurate solutions of the highly singular problem, even for the time-varying equation with the internal boundary layer.
We develop a collocation method (CM) to solve the singular convection- diffusion equation and singular reaction-diffusion equation of elliptic type, which are too ill-posed to be solved by the conventional numerical method. However, when the singular solution is expressed in terms of 2D exponential trial functions, and after collocating points to satisfy the boundary conditions and the governing equation, we can obtain a small scale linear system, which is solved to determine the expansion coefficients. The numerical algorithm CM is effective and accurate in solutions of highly singular elliptic type problems, and the numerical examples confirm these assessments. | en |
dc.description.provenance | Made available in DSpace on 2021-06-08T03:14:48Z (GMT). No. of bitstreams: 1 ntu-106-R03543024-1.pdf: 37770644 bytes, checksum: 5308c7555586506dd19e197d9a332722 (MD5) Previous issue date: 2017 | en |
dc.description.tableofcontents | 口試委員審定書…………………………………………………………………………………#
誌謝 I 摘要 II ABSTRACT III 圖目錄 VII 第一章 研究動機與文獻回顧 1 1.1 前言 1 1.2 文獻回顧 2 1.3 研究動機與目的 2 1.4 論文架構 3 第二章 理論基礎 5 2.1 奇異攝動問題 5 2.2 對流擴散問題 5 2.3 擬時間積分法(FTIM) 5 2.4 共軛梯度法(CONJUGATE GRADIENT METHOD) 6 2.5 邊界積分法(BOUNDARY INTEGRAL EQUATION METHOD,BIEM) 6 2.5 格林第二恆等式(GREEN’S SECOND IDENTITY) 7 第三章 以指數擬合試函數解線性與非線性奇異攝動問題 8 3.1 將ODE轉成PDE的過程: 8 3.2 邊界積分方法(BIEM): 10 3.3 試函數 的推導: 11 3.4 弱形式方法(WFIEM): 13 3.5 時間獨立系統的奇異攝動問題: 16 3.6 弱形式積分方法的數值運算方法: 17 3.7 共軛梯度法(CONJUGATE GRADIENT METHOD,CGM) 18 3.8 非線性奇異攝動問題的數值方法: 19 第四章 以指數型試函數用於配點法解奇異橢圓方程式 21 4.1 變數變換 21 4.2 配點法的數值算法 22 4.3 以共軛梯度法求解未知係數 24 第五章 以指數試函數解奇異型常微分方程式的數值算例 26 5.1 數值算例一 26 5.2 數值算例二 27 5.3 數值算例三 28 5.4 數值算例四 29 5.5 數值算例五 30 5.6 數值算例六 31 第六章 以指數擬合試函數解橢圓型偏微分方程式數值算例 46 6.7 數值算例七 46 6.8 數值算例八 47 6.9 數值算例九 48 參考文獻 65 | |
dc.language.iso | zh-TW | |
dc.title | 以指數試函數解奇異常微分方程及橢圓偏微分方程 | zh_TW |
dc.title | Solving singular ODE and elliptic PDE
by exponential trial functions | en |
dc.type | Thesis | |
dc.date.schoolyear | 105-1 | |
dc.description.degree | 碩士 | |
dc.contributor.oralexamcommittee | 郭仲倫,張致文 | |
dc.subject.keyword | 奇異攝動問題,弱形式積分方程式方法,伴隨Trefftz試函數,譜函數,指數擬合試函數,奇異橢圓型方程式,配點法, | zh_TW |
dc.subject.keyword | Singularly perturbed problem,Weak-form integral equation method,Adjoint Trefftz test functions,Spectral functions,Exponentially fitted trial functions,Singular elliptic equation,Collocation method, | en |
dc.relation.page | 68 | |
dc.identifier.doi | 10.6342/NTU201603511 | |
dc.rights.note | 未授權 | |
dc.date.accepted | 2017-02-10 | |
dc.contributor.author-college | 工學院 | zh_TW |
dc.contributor.author-dept | 應用力學研究所 | zh_TW |
顯示於系所單位: | 應用力學研究所 |
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