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# Semi-analytical solutions for delay partial differential equations

## Hassan Yahya Alfifi (2012)

### Abstract

Semi-analytical solutions for three delay partial differential equation models are presented in this thesis. The three models are: a class of generalised logistic equations, the Nicholson's blowflies equation and the Belousov-Zhabotinskii reaction equations, with delay feedback control. In each model considered in this thesis, one and two-dimensional geometries are considered. The Galerkin method is used to develop semi-analytical models, which results in the system of delay partial differential equation reducing to a system of delay ordinary differential equations. Semi-analytical results for steady-state solutions, transient solutions and the stability of the system are derived and compared with numerical solutions to verify their accuracy. In all cases good comparisons are found.

A class of generalised logistic partial differential equations is considered, for both distributed and point delays. Semi-analytical results for the stability of the system are derived with the critical parameter value, at which a Hopf bifurcation occurs, found. Bifurcation diagrams, transient solutions, Hopf bifurcation points and an asymptotic analysis for the periodic solution near the Hopf bifurcation point are all presented.

The second model is the diffusive Nicholson;s blowflies equation. Semi-analytical bifurcation diagrams, phase-plane maps and a classical period doubling route to chaos are drawn. Also, an asymptotic analysis for the periodic solution near the Hopf bifurcation point is examined, for the one-dimensional geometry.

The third model is the Belousov-Zhabotinskii equations model in a reaction-diffusion cell with feedback control. The feedback system consists of varying the concentrations in the boundary reservoirs in response to the concentrations in the centre of the cell. Non-smooth feedback control with delay is investigated. Steady state, transient solutions and Hopf bifurcation points are found. Also, theories regarding the stability and bifurcation of non-smooth ordinary differential systems are applied to this system. Examples of stable and unstable limit cycles are shown.