Abstract
The present chapter is a continuation of the work in [6]. We construct an approximation of order 2-\(\alpha \) of the Caputo derivative. The approximation is related to the midpoint sum of the integral in the definition of the Caputo derivative. The generating functions of the approximation involves the polylogarithm function of order \(\alpha \). The weights of the approximation have similar properties as the weights of the L1 approximation. We construct induced shifted approximations of order 2-\(\alpha \) of the Caputo derivative and the second-order shifted Grünwald difference formula approximations. The optimal shift values of the approximations are determined, where the approximations have second and third order accuracy respectively. We study also some applications of the approximations for the two-term equation’ numerical solutions. The developed reliable approximations will be important for applications in intelligent robot systems, control theory and signal processing.