El-Amin, Mohamed F.2024-05-262024-05-262024-07-31http://hdl.handle.net/20.500.14131/1691This chapter introduces the analytical solutions of fractional partial differential equations (PDEs), focusing on their significance in modeling transport phenomena that exhibit anomalous diffusion or non-local dynamics. The chapter begins by exploring power-series methods for solving fractional differential equations (FDEs), illustrating the technique through examples such as gas flow in porous media and boundary-layer flow. It then transitions to the Adomian Decomposition Method (ADM), a semi-analytical approach that simplifies the solution of nonlinear, fractional-order differential equations. Through detailed examples, including the time-fractional convection-conduction equation, the time-fractional diffusion-reaction equation, time and time-space fractional advection-diffusion equation, the chapter showcases the versatility and efficiency of ADM in handling complex fractional PDEs. Finally, the diffusion equation with time Caputo-Fabrizio fractional derivative has been solved using Laplace transform method.Fractional Differential Equations, Power-Series Solutions, Adomian Decomposition Method (ADM), Gas Flow in Porous Media, Boundary-Layer Flow, Convection-Conduction Equation, Diffusion-Reaction Equation, Laplace transform method, Hyper-geometric functionMTH