A Caputo fractional derivative of a function with respect to another function

作者：Almeida, Ricardo^*

来源：Communications in Nonlinear Science & Numerical Simulation, 2017, 44: 460-481.

DOI：10.1016/j.cnsns.2016.09.006

• Fractional calculus
• Semigroup law
• Numerical methods
• Population growth model

In this paper we consider a Caputo type fractional derivative with respect to another function. Some properties, like the semigroup law, a relationship between the fractional derivative and the fractional integral, Taylor's Theorem, Fermat's Theorem, etc., are studied. Also, a numerical method to deal with such operators, consisting in approximating the fractional derivative by a sum that depends on the first-order derivative, is presented. Relying on examples, we show the efficiency and applicability of the method. Finally, an application of the fractional derivative, by considering a Population Growth Model, and showing that we can model more accurately the process using different kernels for the fractional operator is provided.

• 出版日期2017-3