  In this paper we present a new type of fractional operator, the
Caputo-Katugampola derivative. The Caputo and the Caputo-Hadamard fractional
derivatives are special cases of this new operator. An existence and uniqueness
theorem for a fractional Cauchy type problem, with dependence on the
Caputo--Katugampola derivative, is proven. A decomposition formula for the
Caputo-Katugampola derivative is obtained. This formula allows us to provide a
simple numerical procedure to solve the fractional differential equation.
