Numerical approximation of spike-type solutions to a one-dimensional sub-diffusive gierer-meinhardt model with controlled precision.

This thesis focuses on numerically solving the sub-diffusive Gierer-Meinhardt model with controlled precision. We start by defining and explaining basic concepts of reaction diffusion, highlighting the main differences between normal and anomalous diffusion. Sub-diffusion is modelled at continuum level by fractional derivatives replacing regular ones in PDEs. Therefore a crucial part of our study involves fractional calculus, which provides a solid framework for describing subdiffusive processes. We explore integer and fractional derivatives and integrals, their main properties, those that can be extended from classical to fractional calculus, and the reasons for limitations in some cases. We then delve into the well-known Gierer-Meinhardt model, a reaction-diffusion system used to describe pattern formation in biological systems. Leveraging the matched asymptotic expansion technique, which is applicable due to the asymptotic smallness of certain parameters in the system, we transform the differential Gierer-Meinhardt model into a differintegroalgebraic system.This differintegro-algebraic system contains a fractional operator denoted Dγt, which involves the integral of a complex function impossible to determine analytically. This operator depends on multiple parameters, and the number of subdivisions needed for numerical computation varies significantly with these parameters and the desired precision. To address this challenge, we have developed a program capable of precalculating the required number of subdivisions before computation, thus saving significant computation time. All elements in place, we use these tools to study the dynamics of the obtained spikes.