E-Thesis 570 views 125 downloads
Self-similar fast reaction limit of reaction diffusion systems with nonlinear diffusion / YINI DU
Swansea University Author: YINI DU
DOI (Published version): 10.23889/SUthesis.60766
Abstract
In this thesis, we present an approach to characterising fast-reaction lim-its of systems with nonlinear diffusion, when there are either two reaction-diffusion equations, or one reaction-diffusion equation and one ordinary dif-ferential equation on unbounded domains. Here, we replace the terms of the...
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Swansea
2022
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| Institution: | Swansea University |
| Degree level: | Doctoral |
| Degree name: | Ph.D |
| Supervisor: | Crooks, Elaine ; Mercuri, Carlo |
| URI: | https://cronfa.swan.ac.uk/Record/cronfa60766 |
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| Abstract: |
In this thesis, we present an approach to characterising fast-reaction lim-its of systems with nonlinear diffusion, when there are either two reaction-diffusion equations, or one reaction-diffusion equation and one ordinary dif-ferential equation on unbounded domains. Here, we replace the terms of the form uxx in usual reaction-diffusion equation, which represent linear diffusion, by terms of form φ(u)xx, representing nonlinear diffusion. For appropriate initial data, in the fast-reaction limit k → ∞, spatial segregation results in the two components of the original systems each converge to the positive and negative points of a self-similar limit profile f(η), where η = √xt , that satisfies one of four ordinary differential systems. The existence of these self-similar solutions of the k → ∞ limit problems is proved by using shooting methods which focus on a, the position of the free boundary which separates the regions where the solution is positive and where it is negative, and γ, the derivative of −φ(f) at η = a. The position of the free boundary gives us intuition how one substance penetrates into the other, so for specific forms of nonlinear diffusion, the relationship between the given form of the nonlinear diffusion and the position of the free boundary is also studied. |
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| Item Description: |
ORCiD identifier: https://orcid.org/0000-0002-9765-1314 |
| Keywords: |
Nonlinear diffusion; Reaction diffusion problem; Fast reaction; Free boundary; Self-similar solution |
| College: |
Faculty of Science and Engineering |