Chapter 2: Fundamentals of Fractional Calculus
El-Amin, Mohamed F.
El-Amin, Mohamed F.
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2024-07
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Abstract
This 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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Fractional Modelling of Fluid Flow and Transport Phenomena