Abstract:
Fractional derivative of non-integer order has been an interesting research topic for several centuries. Fractional differential equations which are the generalization of differentialequations are successful models of real life events and have many applications, which arevery successful in describing natural and physical phenomenon like anomalous kinetics, transport, and chaos.In this thesis, we investigate the following items:1) We study a new fractional derivative "Katugampola fractional derivative", which obeys classical properties including: linearity, product rule, chain rule, Rolle's Theorem and the Mean Value Theorem. Also we derive Taylor's Theorem using a variation of constant formula for Katugampola fractional derivative. This is then employed to extend some recent and classical integral inequalities to Katugampola fractional derivative, including the inequalities of Steffensen, Chebyshev, Hermite-Hadamard, Ostrowski, and Grüss.2) We define Katugampola Fourier transform and obtain some properties of this transform, and find the relation between Katugampola Fourier transform and the usual Fourier transform. Also we give the inversion formula and the Convolution Theorem for Katugampola Fourier transform. Moreover we define infinite and finite Fourier sine and cosine transforms. Also we define Katugampola Laplace transform and obtain some properties of this transforms and find the relation between Katugampola Laplace transform and the usual Laplace transform. We also calculate Katugampola Laplace transform for type functions and give the Convolution Theorem for Katugampola Laplace transform.3) We discuss the analytical solution of the diffusion equations. The so-called Katugampolafractional diffusion equation with Katugampola fractional derivative and time-fractional diffusion equation with Katugampola fractional derivative. We use two ways to solve time- fractional diffusion equation. The first by Katugampola Laplace transform and finite Katugampola Fourier sine transform; and the second by using separation of variables method.4) We defined the so-called time-fractional telegraph equation with Katugampola fractional derivative then we derive the analytical solution for three basic problems. The whole-space domain and half-space domain problems are solved by applying the Katugampola Laplace and Katugampola Fourier transforms with respect to the variables and , respectively. The bounded space domain problem is also solved by the spatial Katugampola Sine transform and temporal Katugampola Laplace transform, whose solution is given in the form of a series.5) We defined the so-called time-fractional evolution equation with Katugampola fractional derivative then we derive the analytical solutions of the time-fractional wave equation, linearized time-fractional Burgers equation and linear time-fractional KdV equation.