### Abstract:

Fractional differential equations, which have derivatives of non-integer order, are very successful in describing natural and physical phenomenon like anomalous kinetics, transport, and chaos. To obtain the exact solutions of these equations, integral transforms are successful used . In this thesis , we investigate the following items : 1) we discuss the analytical solution of three kinds of the diffusion equation. The standard diffusion equation , the so-called time-fractional diffusion equation with Caputo sense and fractional diffusion equation with Caputo sense and Weyl derivative. 2) we defined the so-called time-fractional telegraph equation with Caputo sense and Weyl 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 Laplace and Fourier transforms in variables t and x , respectively. The bounded space domain problem is also solved by the spatial Sine transform and temporal Laplace transform, whose solution is given in the form of a series. 3) we defined the so-called time-fractional evolution equation with Caputo sense and Weyl derivative then we derive the fractional Green function to obtain the analytical solutions of the time-fractional wave equation , linearized time-fractional Burgers equation and linear time-fractional KdV equation .