Numerical studies for some mathematical models of real life problems / Muner Mustafa Abouhasan ; Supervised Laila F. Abdelal , Nasser H. Sweilam

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

This thesis is a contribution to numerical studies of mathematical models of some real life problems.These models are: the fractional financial model of awareness and trial advertising decisions, the fractional coupled nonlinear system of thermoelasticity, the fractional Schrödinger equation, the fractional Lévy-Feller{u2013}diffusion equation advection-dispersion equation. Some of these models are introduced here as variable order fractional models, such as variable order fractional Schrödinger equation and variable order fractional Lévy-Feller advection-dispersion equation. Some of the numerical methods which were used in this thesis are non-standard finite difference methods, spectral collocation method and a new weighted average non-standard finite difference method.Theorems with their proofs are presented to study the stability analysis and the error analysis of the proposed finite difference techniques. Also, theorems with their proofs for new formulas expressing explicitly any fractional order derivative of Legendre polynomials and Jacobi polynomials of any degree using the fractional Riesz-Feller derivative in terms of Jacobi polynomials are presented. The concept of the fractional derivative in this thesis is considered in the sense of Caputo fractional derivative for the time derivative and in the sense of Riesz-Feller for the spatial derivative. Numerical test examples for the proposed models are presented to show the accuracy, applicability, and efficiency of the proposed techniques

Issued also as CD