Fractional derivatives, fractional differential equations, and their numerical approximation

Abstract

Fractional calculus has been around since the development of modern calculus. However, it has been left mainly unexplored until the end of the 20th century. This thesis will provide a survey of fractional calculus and the preliminary concepts like exponentiation beyond integers to include real exponents like π, square root of 2, and other irrational numbers, and the power rule for a monomial of degree n with rational and real exponents. Some of the more commonly used fractional derivatives such as Riemann-Liouville and Caputo's definition are introduced along with their properties. The thesis concludes by examining various methods for solving fractional differential equations including an algorithm developed by Odibat and Momani from their paper "An Algorithm for the Numerical Solution of Differential Equations of Fractional Order."