Thesis (M. Ed.)--University of Hong Kong, 1990. / Includes bibliographical references (leaf 71-75).

Thesis (M. Ed.)--University of Hong Kong, 1996. / Includes bibliographical references (leaf 113-123).

Thesis (Ph. D.)--University of Missouri-Columbia, 2002. / Typescript. Vita. Includes bibliographical references (leaves 325-335). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2002. / Typescript. Vita. Includes bibliographical references (leaves 325-335). Also available on the Internet.

Thesis (M.Ed.)--University of Hong Kong, 1990. / Includes bibliographical references (leaves 71-75). Also available in print.

Thesis (M.Ed.)--University of Hong Kong, 1996. / Includes bibliographical references (leaves 113-123). Also available in print.

Thesis (M.Ed.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 129-133). Also available in print.

Thesis (M.A.)--Brigham Young University. Dept. of Mathematics Education, 2005. / Includes bibliographical references (p. 107-110).

<p> We present a detailed analysis of the application of a multi-scale Hierarchical Reconstruction method for solving a family of ill-posed linear inverse problems. When the observations on the unknown quantity of interest and the observation operators are known, these inverse problems are concerned with the recovery of the unknown from its observations. Although the observation operators we consider are linear, they are inevitably ill-posed in various ways. We recall in this context the classical Tikhonov regularization method with a stabilizing function which targets the specific ill-posedness from the observation operators and preserves desired features of the unknown. Having studied the mechanism of the Tikhonov regularization, we propose a multi-scale generalization to the Tikhonov regularization method, so-called the Hierarchical Reconstruction (HR) method. First introduction of the HR method can be traced back to the Hierarchical Decomposition method in Image Processing. The HR method <i> successively</i> extracts information from the previous hierarchical residual to the current hierarchical term at a <i>finer</i> hierarchical <i> scale.</i> As the sum of all the hierarchical terms, the hierarchical sum from the HR method provides an reasonable approximate solution to the unknown, when the observation matrix satisfies certain conditions with specific stabilizing functions. When compared to the Tikhonov regularization method on solving the same inverse problems, the HR method is shown to be able to decrease the total number of iterations, reduce the approximation error, and offer self control of the approximation distance between the hierarchical sum and the unknown, thanks to using a ladder of <i>finitely many</i> hierarchical scales. We report numerical experiments supporting our claims on these advantages the HR method has over the Tikhonov regularization method.</p>

<p> The solution for sequential Caputo linear fractional differential equations with variable coefficients of order q, 0 &lt; q &lt; 1 can be obtained from symbolic representation form. Since the iterative method developed in Chapter 2 is time-consuming even for the simple linear fractional differential equations with variable coefficients, the direct numerical approximation developed in Chapter 3 is very useful tool when computing the linear and non-linear fractional differential equations of a specific type. This direct numerical method is useful in developing the monotone method and the quasilinearization method for non-linear problems. As an application of this result, we have obtained the numerical solution for a special Ricatti, type of differential equation which blows up in finite time. The generalized monotone iterative method with coupled lower and upper solutions yields monotone natural sequence which converges uniformly and monotonically to coupled minimal and maximal solutions of Caputo fractional boundary value problem. We obtain the existence and uniqueness of sequential Caputo fractional boundary value problems with mixed boundary conditions with the Green's function representation.</p>