Full metadata record
| DC Field | Value | Language |
|---|---|---|
| dc.contributor | Department of Applied Mathematics | en_US |
| dc.contributor.advisor | Zhou, Zhi (AMA) | en_US |
| dc.creator | Zhang, Zhengqi | - |
| dc.identifier.uri | https://theses.lib.polyu.edu.hk/handle/200/12625 | - |
| dc.language | English | en_US |
| dc.publisher | Hong Kong Polytechnic University | en_US |
| dc.rights | All rights reserved | en_US |
| dc.title | Inverse problems of time-fractional differential equations : analysis and numerical methods | en_US |
| dcterms.abstract | This thesis is devoted to theoretical and numerical analysis of inverse problems of fractional differential equations, which have drawn much attention over the past decades due to the maybe ”mild” ill-posedness of fractional derivatives. | en_US |
| dcterms.abstract | In recent years, some numerical algorithms and mathematical analysis are provided and tested. However, in most of these works, only some convergence results and semidiscrete numerical analysis are analyzed. Then our aim is to give a thorough numerical analysis of inverse problems, where numerical estimates are provided including noise level, regularization and discretization parameters. The numerical estimates provide a balancing way to choose regularization and discretization parameter from the noise level. Therefore, we could use a relevant coarser grid to obtain some optimal convergent results. | en_US |
| dcterms.abstract | After a background and preliminary introduction in Chapter 1 and Chapter 2, firstly in Chapter 3 we focus on the backward subdiffusion problem, with the application of quasi-boundary regularization method, piecewise-linear finite element method and convolution quadrature, we show a total error estimate based on smoothing properties of (discrete) solution operators, and nonstandard error estimate for the direct problem in terms of problem data regularity. Next in Chapter 4 when the backward subdffision model includes a time-dependent coefficient, we use a perturbation argument of freezing the diffusion coefficients. Similarly, we apply a quasi-boundary value method and a fully discrete method consisting of finite element method in space and backward Euler convolution quadrature in time. An a priori error estimate is established. Based on the motivation in subdiffusion we extend our idea to fractional-wave equation in Chapter 5 where we want to simultaneously determine two initial conditions based on two different observations. After a new proposed quasi-boundary value method and a classical fully discrete method in space and time, a conditional a priori error estimate is shown. On the other hand, we focus on the inverse potential problem in Chapter 6, to recover potential in a fractional differential equation, with the severely ill posed nature, we construct a monotone operator one of whose fixed points is the unknown potential. The uniqueness of the identification is theoretically verified. Moreover, we show a conditional stability in Hilbert spaces under some suitable conditions on the problem data. Next, a completely discrete scheme is developed by using Galerkin finite method in space and finite difference method in time. A discrete fixed point iteration is constructed and a thorough numerical analysis is given. Lastly in Chapter 7, we summarize our work and mention possible future research topics. | en_US |
| dcterms.abstract | In each chapter, various numerical experiments are provided to support our obtained numerical error estimates. By a balancing choice of parameters, we would obtain an optimal convergence rates, which is strongly supported by our numerical experiments. | en_US |
| dcterms.extent | xi, 141 pages : color illustrations | en_US |
| dcterms.isPartOf | PolyU Electronic Theses | en_US |
| dcterms.issued | 2023 | en_US |
| dcterms.educationalLevel | Ph.D. | en_US |
| dcterms.educationalLevel | All Doctorate | en_US |
| dcterms.LCSH | Inverse problems (Differential equations) -- Numerical solutions | en_US |
| dcterms.LCSH | Numerical analysis | en_US |
| dcterms.LCSH | Fractional differential equations | en_US |
| dcterms.LCSH | Hong Kong Polytechnic University -- Dissertations | en_US |
| dcterms.accessRights | open access | en_US |
Copyright Undertaking
As a bona fide Library user, I declare that:
- I will abide by the rules and legal ordinances governing copyright regarding the use of the Database.
- I will use the Database for the purpose of my research or private study only and not for circulation or further reproduction or any other purpose.
- I agree to indemnify and hold the University harmless from and against any loss, damage, cost, liability or expenses arising from copyright infringement or unauthorized usage.
By downloading any item(s) listed above, you acknowledge that you have read and understood the copyright undertaking as stated above, and agree to be bound by all of its terms.
Please use this identifier to cite or link to this item:
https://theses.lib.polyu.edu.hk/handle/200/12625