Chapter 4: Analytical Solutions of Fractional PDEs, in the Book: Fractional Modelling of Fluid Flow and Transport Phenomena
El-Amin, Mohamed F.
El-Amin, Mohamed F.
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2024-07-31
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This 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.
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Fractional Modelling of Fluid Flow and Transport Phenomena