Asymptotic expansions and approximations for the Caputo derivative
Provides improved numerical methods for solving fractional differential equations, which are important in modeling anomalous diffusion and viscoelasticity.
The paper derives second-order approximations for the Caputo derivative by modifying weights of shifted Grünwald-Letnikov and L1 approximations, enabling second-order numerical solutions for fractional differential equations with arbitrary initial conditions.
In this paper we use the asymptotic expansions of the binomial coefficients and the weights of the L1 approximation to obtain approximations of order $2-α$ and second-order approximations of the Caputo derivative by modifying the weights of the shifted Grünwald-Letnikov difference approximation and the L1 approximation of the Caputo derivative. A modification of the shifted Grünwald-Letnikov approximation is obtained which allows second-order numerical solutions of fractional differential equations with arbitrary values of the solutions and their first derivatives at the initial point.