Witelski, Thomas P. (1995) Problems in nonlinear diffusion. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-10252007-112314
A variety of effects can occur from different forms of nonlinear diffusion or from coupling of diffusion to other physical processes. I consider two such classes of problems; first, the analysis of behavior of diffusive solutions of the generalized porous media equation, and second, the study of stress-driven diffusion in solids. The porous media equation is a nonlinear diffusion equation that has applications to numerous physical problems. By combining classical techniques for the study of similarity solutions with perturbation methods, I have examined some new initial-boundary value problems for the porous media equation, including "stopping" and "merging" problems. Using matched asymptotic expansions and boundary layer analysis, I have shown that the initial deviations from similarity solution form in these problems are asymptotic beyond all orders. Applications of these studies to the Cahn-Hilliard and Fisher's equations are also considered. In my examination of stress-driven diffusion, I consider models for the behavior of systems in the emerging technological field of viscoelastic diffusion in polymer materials. Using asymptotic analysis, I studied some of the non-traditional effects, shock formation in particular, that occur in initial-boundary value problems for these models. Phase-interface traveling waves for "Case II" diffusive transport were also studied, using phase plane techniques.
|Item Type:||Thesis (Dissertation (Ph.D.))|
|Degree Grantor:||California Institute of Technology|
|Division:||Engineering and Applied Science|
|Major Option:||Applied And Computational Mathematics|
|Thesis Availability:||Restricted to Caltech community only|
|Defense Date:||21 April 1995|
|Default Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Imported from ETD-db|
|Deposited On:||25 Oct 2007|
|Last Modified:||26 Dec 2012 03:06|
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