Caputo Fractional Derivative

For a function whose th derivative is suitably integrable, the left-sided Caputo fractional derivative of noninteger order , where is a positive integer and , is defined by

where is a Riemann-Liouville operator of fractional integration and is the gamma function. For integer order , the Caputo fractional derivative is defined to be the ordinary th derivative.

Unlike the Riemann-Liouville fractional derivative, which applies ordinary differentiation after fractional integration, the Caputo fractional derivative applies fractional integration after ordinary differentiation. In particular, the Caputo fractional derivative of a constant function is zero. For , its Laplace transform is

where . Thus, initial conditions for differential equations involving Caputo derivatives can be expressed using values of ordinary integer-order derivatives (Podlubny 1999).

For lower terminal , the Caputo fractional derivative is implemented in the Wolfram Language as CaputoD[f, x, alpha].