Applications of Novel Approaches to Backward Heat Conduction Problems and Groundwater Pollution Source Identification Problems
博士 === 國立臺灣海洋大學 === 系統工程暨造船學系 === 97 === This dissertation investigates two inverse problems of parabolic equations using four different numerical methods. These problems are ill-posed because the solution、if it exists、does not depend continuously on the measured data. Even many researchers have pro...
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| Format: | Others |
| Language: | en_US |
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2009
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| Online Access: | http://ndltd.ncl.edu.tw/handle/82818224428386201487 |
| Summary: | 博士 === 國立臺灣海洋大學 === 系統工程暨造船學系 === 97 === This dissertation investigates two inverse problems of parabolic equations using four different numerical methods. These problems are ill-posed because the solution、if it exists、does not depend continuously on the measured data. Even many researchers have proposed lots of methods to overcome these ill-posed problems; however、an effective numerical scheme to tackle these problems is still pending. The backward heat conduction problem (BHCP) and the groundwater pollution source identification problem governed by the backward advection-dispersion equation (ADE) are tackled by the backward group preserving scheme (BGPS)、the Lie-group shooting method (LGSM)、the quasi-boundary semi-analytical method、and the fictitious time integration method (FTIM)、respectively. First、these two ill-posed problems are analyzed by considering the stable semi-discretization numerical schemes and then、the resulting ordinary differential equations at the discretized spaces are numerically integrated towards the time direction by using the BGPS. Nevertheless、the LGSM is employed to find the unknown initial conditions. The key point is based on the erection of a one-step Lie group element G(T) and the formation of a generalized mid-point Lie group element G(r). Then、by imposing G(T) = G(r) we can seek the missing initial conditions through a minimum discrepancy of the target in terms of the weighting factor In addition、the Fourier series expansion technique is used to calculate the temperature field and the concentration field by using the quasi-boundary semi-analytical method. Then、we consider a direct regularization by adding an extra term or in the final time condition to obtain a second kind Fredholm integral equation for u(x、y、0) or C(x、y、0). The termwise separable property of the kernel function allows us to transform the backward problem into a two-point boundary value problem and therefore、a closed-form solution is derived. After that、by using the FTIM、we transform the original parabolic equation into another parabolic type evolution equation by introducing a fictitious time variable、and adding a fictitious viscous damping coefficient to enhance the stability of numerical integrations of the discretized equations by employing a group preserving scheme. Since four new computational algorithms are based on a concrete theoretical foundation、they offer effective approaches for solving these inverse problems. At last、several designed numerical examples especially with noisy data will be carefully discussed and validate these proposed methods.
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