With materials accompaniment ( Video clips )

Thesis (Ph.D.) - Cairo University - Faculty of Science - Department of Mathematics

This thesis is a contribution on numerical simulations of some real life model problems. These models are the Hodgkin-Huxley model (HHM), the four years life cycle of a population of lemmings model, the enzyme kinetics with an inhibitor molecule model, the Chen system model, and the Cable equation for modeling neuronal dynamics. These models are introduced here as vari- able order fractional models (VOFMs). Moreover special attention is given to study the fractional order wave equation (FWE) and the variable order frac- tional wave equation (VOFWE), which appear in di{uFB00}erent {uFB01}elds of applied mathematics and theoretical physics. A class of numerical methods is used to study the above models. These methods are Adams-Bashforth-Moulton (ABM) method, {uFB01}nite di{uFB00}erence method (FDM), non-standard {uFB01}nite di{uFB00}er- ence (NSFD) method, Crank-Nicolson (C-N) method and a novel non-standard Crank-Nicolson (NSCN) method. Theorems with their proofs are presented to study the stability analysis, the convergence and the error analysis of the pro- posed methods. Numerical test examples for these models are presented. Some Matlab codes done by the author are included in this thesis

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