Numerical Studies for Linear and Non-Linear Fractional order Differential Equations and their Optimal Control

It’s well known that fractional calculus is an extension of the classical calculus, since,in the fractional case we can dene the derivative and integral with any non intergerorder by using one of the famous denitions such as, Riemann-Liouville, Grunwald-Letnikov and Caputo denitions [66]. Fractional-order derivatives and integrals provideuseful tools for description many natural phenomenons [66].The main objective of this thesis is to develop new ecient numerical methodsand supporting analysis, based on the Shifted Chebyshev Polynomials, Shifted LegendrePolynomials, Variational Iteration Method (VIM), Pade approximation VIM andCrank-Nicolson nite dierence method for solving fractional Logistic dierential equationwith two dierent delays, fractional diusion equation, fractional Riccati dierentialequation, fractional Logistic equation, linear and non-linear system of fractionaldierential equations, the fractional order SIRC model associated with the evolution ofinuenza A disease in human population and solving the fractional dierential equationgenerated by optimization problem are implemented. Also, we introduced Legendrecollocation method for solving a wide class of fractional optimal control problems.One important contribution of this thesis is the demonstration of how to choose differentapproximation techniques for dierent fractional derivatives. Another objectiveof this thesis is to derive the analytical solutions for some fractional systems dierentialin one and two dimensions. By doing so, we can ascertain the accuracy of our proposednumerical methods.ivThe stability and convergence of our proposed numerical methods are also investigated.Numerical experiments are carried out in support of our theoretical analysis.We also emphasise that the numerical methods, we develop are applicable for manyother types of fractional dierential equations.This thesis is presented by publications. Our original contribution to the literatureis listed in six published papers. This thesis consists of six chapters:Chapter one.This chapter consists of 8 sections. In this chapter, we give the denitions andresults which we use throughout this thesis.Chapter two.In this chapter, the approximate formula of the Caputo fractional derivative usingChebyshev polynomials series is introduced. Special attentions are given to study theconvergence analysis and estimate the upper bound of the error of the proposed formula.This formula is implemented to obtain approximate solutions of some models whichrepresented by fractional dierential equations such as, logistic dierential equationwith two dierent delays, Riccati dierential equation, linear and non-linear system ofdierential equations, the order SIRC model associated with the evolution of inuenzaA disease in human population.Chapter three.In this chapter, the approximate formula of the Caputo fractional derivative usingLegendre polynomials series is introduced. Special attention is given to study theconvergence analysis and to estimate the upper bound of the error of the proposedformula. This formula is implemented to obtain the approximate solutions of somemodels which represented by fractional dierential equations such as, fraction diusionequation, fractional wave equation, logistic dierential equation with two dierent devlays, linear and non-linear system of dierential equations.Chapter four.In this chapter, numerical studies for the fractional order dierential equations usingdierent classes of nite dierence methods (FDM) such as the Crank-NicolsonFDM (C-N-FDM) using the Grunwald-Letnikov^s denition are presented. A discreteapproximation to the fractional derivative D is applied. C-N-FDM is used to studysome real life models problems, such as, the fractional Riccati dierential equation,fractional logistic equation, time fractional diusion and time fractional wave equations.The obtained results from the proposed method are compared with the resultusing Pade-variational iteration method (Pade-VIM) and VIM respectively. Stabilityanalysis and the truncation error are presented for time fractional diusion and timefractional wave equations. In the end linear and nonlinear system of the fractionaldierential equation are solved using C-N-FDM.Chapter ve.In this chapter, Chebyshev, Legendre collocation method and nite dierence methodare implemented to solve the fractional dierential equation generated by optimizationproblem. Also, we introduced Legendre collocation method for solving a wide class offractional optimal control problems. In the end of this chapter, we apply the classicalcontrol theory to a fractional diusion equation in a bounded domain. Some theoremsand lemmas to study the existence and the uniqueness of the solution of the fractionaldiusion equation in a Hilbert space are given in [62].Chapter six.Conclusions and Future Research.