BOUNDARY VALUE PROBLEMS FOR FRACTIONAL ORDER DIFFERENTIAL EQUATIONS: EXISTENCE THEORY AND NUMERICAL SIMULATIONS

Abstract

The study of fractional calculus has been initiated in the Seventeenth century and has received much attention in the last few decades. Because of the fractional order derivatives, scientists have developed excellent approach for the description of memory and hereditary properties of different problems in science and engineering. Therefore, we see the applications of fractional calculus in the fields such as; signal processing, diffusion process, physics, fluid mechanics, bioscience, chemistry, economics, polymer rheology and many others. In this thesis, we are concerned with the existence and uniqueness of positive solutions for different classes of boundary value problems for fractional differential equations (FDEs). We also study numerical solutions of FDEs and for some classes exact analytical solutions of local FDEs. Existence and uniqueness theory for positive solutions is developed for following classes of bound- ary value problems (BVPs) for FDEs: Class of two point BVPs for FDEs, class of three point BVPs for FDEs, Class of multi point BVPs for FDEs, a general class of BVPs with p-Laplacian operator, BVPs for coupled systems of FDEs, BVPs for coupled system of fractional order differential integral equations, BVPs for coupled system of fractional order q–difference equations, and BVPs for coupled systems of hybrid FDEs. For numerical solutions, Bernstein polynomials (BPs) are used and operational matrices (OM) for fractional order integrations and differentiations are developed. Based on these OM, numerical schemes for numerical solutions are developed for the following classes of of FDEs; fractional partial differential equations, coupled systems of FDEs, optimal control problems. We also use B-Spline functions and develop operational matrices of B-Spline functions for the numerical solution of a coupled system of FDEs. We also study exact solutions of some local FDEs, we use different mathematical methods for differ- ent local fractional (LF) problems. In this work, we produce iterative techniques for the approximation of solutions of different problems in LF calculus and the efficiency of the schemes are tested by many examples.