Approximation of Caputo Fractional Derivative and Numerical Solutions of Fractional Differential Equations

Abstract

In this paper, we consider an approximation of the Caputo fractional derivative and its asymptotic expansion formula, whose generating function is the polylogarithm function. We prove the convergence of the approximation and derive an estimate for the error and order. The approximation is applied for the construction of finite difference schemes for the two-term ordinary fractional differential equation and the time fractional Black–Scholes equation for option pricing. The properties of the approximation are used to prove the convergence and order of the finite difference schemes and to obtain bounds for the error of the numerical methods. The theoretical results for the order and error of the methods are illustrated by the results of the numerical experiments.