There are different types of a fractional derivative. But most of them using Caputo fractional derivative, why? what are the advantages of Caputo fractional derivative compare with all other fractional derivatives?
The fractional derivatives have several different kinds of definitions, among which the Riemann–Liouville fractional derivative and the Caputo fractional derivative are two of the most important ones in applications [2]. The Risez ctional derivative is a linear representation of the left Riemann–Liouville fractional derivative and right Riemann–Liouville fractional derivative. A close relationship exists between the Riemann–Liouville fractional derivative and the Caputo fractional derivative. The Riemann–Liouville fractional derivative can be converted to the Caputo fractional derivative under some regularity assumptions of the function [1,2]. In fractional partial differential equations, the time-fractional derivatives are commonly defined using the Caputo fractional derivatives. The main reason lies in that the Riemann–Liouville approach needs initial conditions containing the limit values of Riemann–Liouville fractional derivative at the origin of time t = 0, whose physical meanings are not very clear. However, in cases with the time-fractional Caputo derivative, the initial conditions take the same form as that for integer-order differential equations, namely, the initial values of integer-order derivatives of functions at the origin of time t = 0 [1,2,3].
References
[1] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[2] Z. Odibat, Computing eigenelements of boundary value problems with fractional derivatives, Appl. Math. Comput. 215 (2009) 3017–3028.
[3] M. Stojanovi´ c, Numerical method for solving diffusion-wave phenomena, J. Comput. Appl. Math. 235 (2011) 3121–3137.
the Caputo fractional derivative of constant is 0, like Fréchet differentiation in R ((if the order was natural number ))
Not needed for mentioning fractional initial value - Intersting - Following !
Caputo fractional operator handle initial and boundary value problem and is more flexible for analysis
The Caputo derivative is the most appropriate fractional operator to be used in modeling real world problem. The Caputo derivative is of use to modeling phenomena which takes account of interactions within the past and also problems with non-local properties.
- Badges
- Science topic
- Similar topics
- Mathematics