Introduction to Fractional Calculus

Abstract

This chapter presents basic definitions and characteristics of fractional calculus. Definitions and characteristics of the Riemann–Liouville fractional integral and derivative, Caputo fractional derivative, Grünwald–Letnikov fractional derivative, Riesz fractional derivative, modified Riemann–Liouville derivative, and local fractional derivative have been primarily explored here. However, the Riemann–Liouville definitions of fractional differentiation play a significant role in developing fractional calculus. Because the derivative of constant is not zero, the Riemann–Liouville definition is not widely applicable. In such scenarios, the Caputo fractional derivative is more suited than the standard Riemann–Liouville derivative for applications in various engineering problems. It may be noted that the Riemann–Liouville technique is challenging to understand physically, while the Caputo approach employs integer-order initial conditions, which are easier to apply in real-world applications. This chapter also covers the Euler psi function, incomplete gamma function, beta function, incomplete beta function, Wright function, Mellin-Ross function, error function, hypergeometric functions such as Gauss, Kummer, and extended hypergeometric functions, and the H -function.