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Chapter 2: Fundamentals of Fractional Calculus, of the book: Fractional Modelling of Fluid Flow and Transport PhenomenaThis chapter presents the concepts of fractional calculus used in the field of fluid mechanics required in the rest of the book. The chapter begins with an overview and then introduces preliminary concepts crucial for understanding fractional calculus, including the Gamma and Beta functions, the Mittag-Leffler function, and various fractional operators. These foundational elements are essential for grasping the more complex aspects of fractional calculus in fluid mechanics. Moreover, the chapter examines different fractional derivative models, providing the basic definition of several key types. These include the Riemann-Liouville, Caputo, Grünwald-Letnikov, Caputo-Fabrizio, and Atangana-Baleanu fractional derivatives. Each model is explored, offering insights into their unique characteristics and applications. Also, a significant portion of the chapter is dedicated to the Laplace transform, which covers its definition, basic principles, and properties, along with a list of common Laplace transforms and techniques for applying the inverse Laplace transform. This comprehensive coverage equips readers with the tools to use the Laplace transform in various contexts of fractional calculus. The chapter ends with exercises designed to reinforce the concepts covered.
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Chapter 4: Analytical Solutions of Fractional PDEs, in the Book: Fractional Modelling of Fluid Flow and Transport PhenomenaThis 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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Chapter 3: Fundamentals of Fractional Modeling of Fluid Flow, of the Book: Fractional Modelling of Fluid Flow and Transport PhenomenaAs the primary focus of this book is employing fractional modeling in the study of fluid dynamics and transport phenomena, this chapter presents the basics of fractional modeling of fluid flow. It begins with exploring fractional differential equations and discusses their advantages and challenges. Subsequent sections focus on the derivations of fractional-order formulations for conserving mass and momentum. The chapter also introduces the derivation of the fractional energy conservation equation, including models for heat conduction, heat convection-conduction, and general transport phenomena. Additionally, the discussion extends to fluid flow in porous media, featuring adaptations of Darcy’s Law that incorporate time and space memory effects and address anomalous diffusion processes.