This module introduces and develops the fundamental aspects of reaction-diffusion theory. The module begins with the prerequisites of linear diffusion theory and reaction dynamics theory. These processes are then combined in the derivation of the scalar reaction – diffusion PDE. The Cauchy, Dirichlet and Neumann problems are considered. The question of existence, uniqueness and continuous dependence are addressed in detail. Maximum principles, comparison theorems and invariant set theorems are established in generality. These fundamental concepts are then used to study equilibrium states and steady states in reaction-diffusion theory. Bifurcations are discussed and examples given for both equilibrium and steady states. Liapunov and asymptotic stability is defined, and studied through the development of linearized theory. This is then put on a rigorous basis though the establishment of corresponding linearization theorems. The theory is illustrated with detailed examples from applications in autocatalytic chemistry and population dynamics.

Learning Outcomes

By the end of the module students should be able to:

At the end of this module, students should have a full appreciation of linear diffusion theory and reaction kinetic theory, and how these two processes combine in reaction-diffusion theory.

Detailed questions concerning the Cauchy, Dirichlet and Neumann problems in reaction-diffusion theory should be understood, and the use of comparison theorems, maximum principles and invariant set theorems appreciated.

Students should have the ability to apply the general theory developed to study both equilibrium states and steady states, and determine their respected temporal stability properties.

There should also be an appreciation of applying the general theory developed to specific examples arising in applications.

Assessment

27691-01 : Raw Module Mark : Coursework (0%)
27691-03 : Final Module Mark : Coursework (100%)

Assessment Methods & Exceptions

90% on one three hour examination; 10% from coursework and/or class tests.