There are many fractional derivative operators such as Caputo operator, Riemann-Liouville operator, Hadamard operator, etc.
what is the better fractional derivative operator to study fractional boundary value problems and why?
Hello, Sabri
The both Riemann-Liouville and Caputo fractional derivative are best for the study of boundary value problems and number of research paper is available on google.
For physical problems where you mostly have integer order derivatives as the boundary conditions, Caputo operator is the best! But if you have problems where fractional derivatives at the boundary conditions are available, then go for RL derivative. However, under zero initial boundary condition, both Caputo and RL derivative are equivalent.
For modeling of the "physical" systems Caputo derivative is the optimum choice! Do not go for RL derivative if you do not have fractional values at the initial boundary conditions.
Fractional calculus is a generalization of ordinary calculus, where derivatives and integrals of an arbitrary real order are defined. These fractional operators may model more efficiently certain real world phenomena, especially when the dynamics is affected by constraints inherent to the system. There exist several definitions for fractional derivatives and fractional integrals like the Riemann-Liouville, Caputo, Hadamard, Riesz, Griinwald-Letnikov. Some of the usual features concerning the differentiation of functions fail like the Leibniz rule, the chain rule, the semigroup property, to name a few. The derivative of a function f(x) at a point x is a local property only when a is an integer. It is not the case for non-integer power derivatives. In other words, it is not correct to say that the fractional order derivative at x of a function depends only on values of f very near x , in the way that integer-power derivatives certainly do. Therefore, it is expected that the theory involves some sort of boundary conditions, involving information on the function. One of the most impressing difference between the two operators Riemann-Liouville and Caputo is that the Caputo derivative of a constant is zero (Heavi side unit step function), whereas in the cases of a finite value of the lower terminal in the Riemann-Liouville fractional order derivative of a constant C is not equal to zero. In particular, the Riemann-Liouville fractional order derivative of a constant "C" is equal to zero if lower terminal is " -infinity". Since the following reason is observed myself: In 1967, M. Caputo was introduced the Caputo fractional order derivative in his paper. In contrast to the Riemann Liouville fractional order derivative, when solving differential equations using Caputo's definition, it is not necessary to define the fractional order initial conditions. Among all definitions Riemann-Liouville fractional order derivative is most useful.
Maham Ali Most likely you will have to integral transform the FDE to solve it. If you look at the Laplace transform of a Caputo derivative (attached), you can see that it requires integer-derivative evaluation of the function at the initial conditions, which is available for physical problems. For example, the distance covered by a ball can be estimated if its initial position (0th derivative), initial velocity (1st order derivative) and acceleration (2nd order derivative) are available, which are known available values and have physical interpretation in the real world. But if you look at the Laplace transform of a RL derivative (attached), it requires fractional order derivative of the function evaluated at its initial conditions. Are fractional order derivatives of distance and its physical meaning available to us? I guess not! But in case they are available, which is quite possible in a given "mathematical" problem, RL derivative can be used since you now have the necessary information available to you. I hope I am clear.
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