I am working on the boundary problem with the p-Laplacian operator. I have used Caputo fractional derivative model in my first work, for that, I have questions in my mind, that are :
- Why did you select the Caputo model ? why didn't you select Riemann–Liouville model? what is the advantage to use them?.
-There are some approaches of fractional calculus that generalized the celebrated Liouville or Caputo models as Katugampola fractional calculus, the fractional calculus with respect to another function ..., if we use these approaches, are we get the same answers to the previous questions?
Dear Farid Chabane first you should understand the differences between Caputo model and Riemann–Liouville model for that I refer you to
https://www.researchgate.net/post/Can-anyone-name-the-differences-between-Riemann-Liouville-fractional-derivatives-and-Caputo-type-Which-of-them-are-equivalent-to-G-L-definition
In fact the different definitions of fractional calculus depends on the domain of definition of the functions under scrutiny ,functions defined on RR or on [0,1][0,1] or on [0,∞)[0,∞) and of course the derivative of say sin is not the same in these three cases.
In this regard I see that the use of the method of upper and lower solutions and fixed point theorems, the existence of solutions for a Riemann-Liouville fractional boundary value problem with the nonlinear term depending on fractional derivative of lower order is obtained under the classical Nagumo conditions in better than Caputo method
Dear Farid Chabane
In Riemann–Liouville model, solutions are practically useless, because there is no clear physical interpretation of this type of initial conditions. On the contrary, the Caputo fractional derivative is applicable, standard initial conditions in terms of derivative integer order are involved. These initial conditions have clear physical interpretation. On the other hand, the Caputo fractional derivative is more restrictive. Since it requires the existence of the n-th derivative of the function. Fortunately, most function that appear in the applications fullfill this requirement.
Please see attached file.
In my opinion you should use Caputo. It will be easier to interpret the boundary conditions. You will not necessarily arrive at the same result.
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