Numerical determination of potentials in conservative systems.

• Other Authors: Chan, Yuet Tai.
• Format: Others
• Language: English
• Published: 1999
• Subjects: Inverse problems (Differential equations) > Numerical solutions
• Online Access: http://library.cuhk.edu.hk/record=b5890086; http://repository.lib.cuhk.edu.hk/en/item/cuhk-322633

Chan Yuet Tai. === Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. === Includes bibliographical references (leaves 107-111). === Chapter 1 --- Introduction to Sturm-Liouville Problem --- p.1 === Chapter 1.1 --- What are inverse problems? --- p.1 === Chapter 1.2 --- Introductory background --- p.2 === Chapter 1.3 --- The Liouville transformation --- p.3 === Chapter 1.4 --- The Sturm-Liouville problem 一 A historical look --- p.4 === Chapter 1.5 --- Where Sturm-Liouville problems come from? --- p.6 === Chapter 1.6 --- Inverse problems of interest --- p.8 === Chapter 2 --- Reconstruction Method I --- p.10 === Chapter 2.1 --- Perturbative inversion --- p.10 === Chapter 2.1.1 --- Inversion problem via Fredholm integral equation --- p.10 === Chapter 2.1.2 --- Output least squares method for ill-posed integral equations --- p.15 === Chapter 2.1.3 --- Numerical experiments --- p.17 === Chapter 2.2 --- Total inversion --- p.38 === Chapter 2.3 --- Summary --- p.45 === Chapter 3 --- Reconstruction Method II --- p.46 === Chapter 3.1 --- Computation of q --- p.47 === Chapter 3.2 --- Computation of the Cauchy data --- p.48 === Chapter 3.2.1 --- Recovery of Cauchy data for K --- p.51 === Chapter 3.2.2 --- Numerical implementation for computation of the Cauchy data . --- p.51 === Chapter 3.3 --- Recovery of q from Cauchy data --- p.52 === Chapter 3.4 --- Iterative procedure --- p.53 === Chapter 3.5 --- Numerical experiments --- p.60 === Chapter 3.5.1 --- Eigenvalues without noised data --- p.64 === Chapter 3.5.2 --- Eigenvalues with noised data --- p.69 === Chapter 4 --- Appendices --- p.79 === Chapter A --- Tikhonov regularization --- p.79 === Chapter B --- Basic properties of the Sturm-Liouville operator --- p.80 === Chapter C --- Asymptotic formulas for the eigenvalues --- p.86 === Chapter C.1 --- Case 1: h ≠ ∞ and H ≠ ∞ --- p.87 === Chapter C.2 --- Case 2: h= ∞ and H ≠∞ --- p.90 === Chapter C.3 --- Case 3: h = ∞ and H = ∞ --- p.91 === Chapter D --- Completeness of the eigenvalues --- p.92 === Chapter E --- d'Alembert solution formula for the wave equation --- p.97 === Chapter E.1 --- "The homogeneous solution uH(x,t)" --- p.98 === Chapter E.2 --- "The particular solution up(x, t)" --- p.99 === Chapter E.3 --- "The standard d'Alembert solution u(x,t)" --- p.101 === Chapter E.4 --- Applications to our problem --- p.101 === Chapter F --- Runge-Kutta method for solving eigenvalue problems --- p.104 === Bibliography --- p.107