Approximations of fractional integrals and Caputo fractional derivatives

In this paper we propose two algorithms for numerical fractional integration and Caputo fractional differentiation. We present a modification of trapezoidal rule that is used to approximate finite integrals, the new modification extends the application of the rule to approximate integrals of arbitrary order α > 0. We then, using the new modification derive an algorithm to approximate fractional derivatives of arbitrary order α > 0, where the fractional derivative based on Caputo definition, for a given function by a weighted sum of function and its ordinary derivatives values at specified points. The study is conducted through illustrative examples and error analysis.