Nowadays, we are using fractional derivatives to solve a wide variety of real-world problems. But when we look at the theory that deals with the criticism of fractional derivatives, it can be confusing (especially for me) to decide which derivative we should use to simulate our problems. If we look at the cases of some famous fractional derivatives like Caputo, Caputo-Fabrizio (CF), and Atangana-Baleanu (AB), these are enough to give you a sufficient amount of confusion.
In the list of recent works, Prof. Sabatier and Farges in ref. 10.3934/math.2021657 have justified that the Caputo and RL derivatives are not able to ensure a proper initialization when used in a model definition. In ref. https://doi.org/10.1515/fca-2019-0017, the authors have shown that CF and AB derivatives are not suitable to simulate the models with real data. Also, Dr. Angstmann et al. in https://doi.org/10.3390/math8112023 have proved the intrinsic discontinuities in the solutions of evolution equations of CF and AB type derivatives. Some other analysis in this field is given by Prof. Sabatier in ref. https://doi.org/10.3390/fractalfract4030040. Also, some true and false results on fractional derivatives are given in ref. Fractional Integrals and Derivatives: “True” versus “False” - Google Books. Prof. Tarasov in his study https://doi.org/10.1016/j.cnsns.2018.02.019 has suggested that the CF derivative cannot be considered a non-integer order derivative. Also, Prof. Diethelm with the team has announced that fractional derivatives with non-singular kernels should not be used ref https://doi.org/10.1515/fca-2020-0032. Many other studies are also present in this regard.
Now, my question is:
If we want to solve a mathematical model, especially an epidemiological model, then knowing the above discussion, which derivative should we use?
I am looking for answers with supporting arguments.
Straightforward answers could be 'choose any unstudied' for purely mathematical studies and 'choose the one that better fits your data' for experimental studies.
Of course keeping in mind all the mentioned issues such as dimentionality, initial conditions, both general and specific for derivatives with non-singular kernels, and the possibility to model fractional behaviours without fractional derivatives.
Dear Pushpendra Kumar
the following papers will give you examples of the Appropriate and proven method in this problem
1-A fractional epidemiological model for computer viruses pertaining to a new fractional derivative
Singh, Jagdev, Kumar, Devendra, Hammouch, Zakia, Atangana, Abdon
Journal:
Applied Mathematics and Computation
Year:
2018
2-An epidemiological MSEIR model described by the Caputo fractional derivative
Almeida, Ricardo, Brito da Cruz, Artur M. C., Martins, Natália, Monteiro, M. Teresa T.
Journal:
International Journal of Dynamics and Control
3-An epidemiological approach to insurgent population modeling with the AtanganaâBaleanu fractional derivative
Kolebaje, Olusola, Popoola, Oyebola, Khan, Muhammad Altaf, Oyewande, Oluwole
Journal:
Chaos, Solitons & Fractals
Year:
2020
I am recommending the tired paper.
Best luck
Sir, I think ,It is better if we take some model and apply all fractional derivatives on that model and obtained numerical solutions individually .After this, we obtained a graph for each fractional derivative .we can analyze the graph for each derivative .This might me give us a clue that which fractional derivative is better
I think that will be a good option but it is only applicable when you have the real data to compare.
Sir I have one more answer to your question is that we should study the properties of each derivative and compare the properties among one another and see which derivative satisfy a property in a better way. Also we should study which derivative has least number of drawbacks, so that fractional derivative I think is better.
If we are comparing properties of fractional derivatives then RL derivative gives the fractional derivative of a function even function is non-differentiable but has fractional order initial conditions which have no physical interpretation.
On the other hand Caputo derivative has the initial condition of integer order, but for Caputo derivative function should be differentiable.
But for the Jumarie derivative, the differentiability condition imposed by Caputo’s definition is not required. In addition, the fractional derivative of a constant function is equal to zero as well as initial condition is also of integer order. So, among RL, Caputo, Jumarie derivatives; Jumarie derivative has better properties.
I have already seen so many papers regarding this. The authors have compared the results but clearly did not mention which one is better. They have just compared and published the work. Also, particularly taking one method it is impossible to decide which one is better. Also, these kinds of operators have different kinds of properties. By checking properties, I think we can not tell about that. I think one option is to check some models using real data. But it is impossible to get real data for those kinds of models.
If we obtain real data for a model and compare the outcomes for different derivatives, we may determine which fractional derivative is superior. Is it feasible, then, that the derivative is superior for all models?
That's why I told that we have to check for some models. If similar kind of results are obtained for some models, then we can say that which one is better than other in general.
I think that similar kind of concept is used for changing integer order model to fractional order model directly. Otherwise how we can change a integer order to fractional order models directly?? Then that concept also should not feasible.
You can change the problem to fractional form of Caputo, Caputo-Fabrizio (CF), and Atangana-Baleneu (AB). Then you can solve the models and compare the results to know that which one is better!
Many articles have been published in recent years that examine the theory and applications of novel fractional-order derivatives developed by substituting the singular kernel of the Caputo or Riemann-Liouville derivative with a non-singular (i.e., bounded) kernel. Through careful mathematical reasoning, it will be demonstrated that these non-singular kernel derivatives have significant shortcomings that should restrict their utilization. They violate the fundamental principle of fractional calculus, and the value of the derivative at the initial time t = 0 is always zero, imposing an unnatural limitation on the differential equations and models that may employ these derivatives.
The problem with using fractional derivatives for epidemic mathematical models is that the dimensions of the fractional derivatives would not make sense (the dimensions on the left hand side would not be equal to those on the right hand side; on the LHS you have time raised to a certain parameter whereas on the RHS you have time raised to -1 or rather per unit time). Nonetheless, its not necessarily true that modelling with fractional derivatives does not, at all, make sense.
Reference: https://youtu.be/tx9rqd2beLc
(if you are in a rush, watch from the 48th minute)
The fractional operators are actually depends on kernel, basically kernels are the control convergence parameters, if kernel are singular then we cant attain a good convergence. The kernel of CF fractional derivative is constant and is different from the AB fractional derivative. So actually the AB kernel describe the complexity in very easy way and comprehensively.
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