Fractional differential equations with dependence on the   Caputo-Katugampola derivative
  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. 
Ricardo Almeida, Agnieszka B. Malinowska and Tatiana Odzijewicz
math.CA