Fractional Calculus - Science topic

Dear Abhijeet Das

In the context of tumor growth, fractal calculus and fractional calculus offer different perspectives.

Fractal calculus focuses on the geometrical complexity of the tumor, especially its irregular and self-similar structures, such as the vascular networks and the tumor boundary. Tumors often grow in a way that resembles fractals, with patterns that repeat at different scales. Fractal calculus helps to characterize these complex, irregular shapes and quantify the heterogeneity of the tumor, which can be important for understanding its aggressiveness, potential for invasion, and ability to metastasize.

On the other hand, fractional calculus is more concerned with the dynamic and temporal aspects of tumor growth. It helps model processes that involve long-range interactions, memory effects, and non-local behaviors. These are common in tumor biology, where growth and spread don't just depend on local conditions but are influenced by the surrounding microenvironment, past interactions, and diffusion processes. Fractional calculus can capture these history-dependent effects and model complex processes like nutrient diffusion, drug delivery, and metastatic spread, which can't be accurately described using traditional integer-order calculus.

Together, these two approaches offer a complete understanding of tumor growth. Fractal calculus helps explain the spatial irregularities of the tumor, while fractional calculus models the dynamic, long-range interactions and memory effects that govern tumor behavior and its response to treatment.

Best Regards....

Typical bad papers in Fractional Calculus often have one or more of the following issues:

Is this special issue still open for submission?

A priori the answer is yes, like any mathematical tool. The question is where does it provide "best" help? In those local processes (velocity, acceleration, ...) ordinary calculus, together with the new local differential operators, provides better results; On the other hand, fractional calculus is better in processes in which past history is significant (population growth, epidemic, tumors, ...). I must clarify something: fractional calculus IS NOT A GENERALIZATION of ordinary calculus. On several occasions I have clarified it because it seems to be a recurring error, there is no way in which the fractional derivative is reduced to the ordinary one, therefore it is not a generalization, IT IS AN EXTENSION. Some of the new local differential operators, IF THEY ARE A GENERALIZATION of the ordinary derivative, can be for a value of the order, for example, in the Khalil conformable derivative if α is 1, it reduces to the ordinary derivative, the same thing happens with our derivative N, if the kernel is 1, we have the ordinary derivative.

If you have time, try to have a look at

and

Dear Prof. Jocelyn Sabatier

If you remember this question that I asked before,

There are nearly 1100 reads and some answers, but not a single exact answer for what I was looking for.

Let us take the case of mathematical modelling. Many of the researchers write that we should prefer fractional-order models over integer-order models because fractional-order model outputs are better than integer-order outputs. Then, they solve the model and plot several meaningless graphs (50% times), choosing many random fractional orders without any real data. It is not true all the time. Just recently I published the following paper where it is clearly visible that random choose of fractional-orders is inaccurate: https://www.researchgate.net/publication/376236178_Forecasting_of_HIVAIDS_in_South_Africa_using_1990_to_2021_data_novel_integer-_and_fractional-order_fittings

There should be a separate parameter estimation for each fractional order case (if you see Table 5).

Also, those who blindly claim that fractional-order models are more accurate than integer-order models, should not forget a famous Quote of Prof. George E.P. Box (1919-2013) "All models are approximations. Essentially, all models are wrong but some are useful."

To be very honest, I do not have an answer to your question. If I find something, I will come back here.

Some basic references attached (for an undergraduate course).

1. Flexibility

2. Noise control

We will be glad to recieve your works

Generally, all of the definitions can be used every fields. Especially, Caputo Fabrizio fractional operator are applied to different types of singular and nonsingular operator. However, we must know that, function is need to be absolutely continuous every point. If you interested in fractional differential equations, I recommend two different sources. First source, "An introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications 1st Edition - October 21, 1998. Author: Igor Podlubny". Second source " Fractional Calculus: Models and Numerical Methods (Complexity, Nonlinearity and Chaos) by Kai Diethelm(Author), Dumitru Baleanu(Author), Enrico Scalas(Author)". These two different sources mention that Definition of Grünwal-Letnikov fractional operator, Rieman-Liouville fractional derivative, Caputo definition. You can look them. My regards, Vaijanath Laxmanrao Chinchane

see this article

Yes I am

I recommended the following book:

Kindergarten of Fractional Calculus

This book presents a simplified deliberation of fractional calculus, which will appeal not only to beginners, but also to various applied science mathematicians and engineering researchers. The text develops the ideas behind this new field of mathematics, beginning at the most elementary level, before discussing its actual applications in different areas of science and engineering.

Hi

There is a good Matlab program named FDE12 for solving FDE's.

please see

It is not correct ...

There is a paper In title Fractional modeling and optimal control analysis of rabies virus under the convex incidence rate I think you will find useful

The answer is if the kernel of a given operator corresponds to the relaxation function of the modeled process. Then, it can be used as kernel, single kernels, or a series of kernels (thus leading to a series of fractional differential operators).

Some people believe that changing the operator, that can in a blind manner fit better experimental data.

Using the replacement technology in creating fractional-order models, actually, no-realistic models are created, but surrogate models (look for literature on surrogate modeling). Since there is an additional parameter, that is the fractional-order (commonly denoted as "alpha", appearing in the final solution, then changing it, one can obtain a variety of solutions of the surrogate solutions. For some "alpha" the surrogate solution may match experimental data, for others not, but matching does not mean that we have a solution corresponding to the physics, since the very beginning, when the replacement was done, the surrogate model disappeared from the physics and the rule of modelling are violated.

Thus, which kernel is better, is an unphysical question. Modelli, especially fractional-order modelling needs deep knowledge of the physics of dissipative processes, not only replacement and calculations with results that cannot be interpreted.

Some articles supporting my words are attached

Please note that at

cutoff frequency the

gain is .707 of the max.

This is due to the fact that

(.707)^2 is 1/2 and this is the

half power point.

This and more is in my book

The American University Laboratories For Electrical Engineering Part 1

Dear Farid Chabane first you should understand the differences between Caputo model and Riemann–Liouville model for that I refer you to

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

The usual derivative is a local derivative in the sense that the value of the derivative at one point only depends on the value of the function in a neighborhood of that point. But the fractional derivative is not a local operator and cannot be due to some general theoretical result .

So the definition depends on the domain of definition of the functions under scrutiny. This is not the same definition if we are looking at functions defined on RR or on [0,1][0,1] or on [0,∞)[0,∞) and of course the derivative of say sinsin is not the same in these three cases. Same for the derivative of the constant function.

Please, you nay check

for more discussion

You may go through the research article entitled Time Nonlocal Six-phase-lag Generalized Theory of Thermoelastic Diffusion with Two-temperature

I suggest for you the following papers Dr Sina Etemad

[springeropen.com] New applications of the variational iteration method-from differential equations to q-fractional difference equations

You can also read the following paper

On solutions of variable-order fractional differential equations

DOI:10.11121/ijocta.01.2017.00368

I recommended the following two papers

SIR epidemic model of childhood diseases through fractional operators with Mittag-Leffler and exponential kernels

Jena, Rajarama Mohan, Chakraverty, Snehashish, Baleanu, Dumitru

Journal:

Mathematics and Computers in Simulation

Year:

2020

and

A new fractional model for vector-host disease with saturated treatment function via singular and non-singular operators

Khan, Muhammad Farooq, Alrabaiah, Hussam, Ullah, Saif, Khan, Muhammad Altaf, Farooq, Muhammad, Mamat, Mustafa bin, Asjad, Muhammad Imran

Journal:

Alexandria Engineering Journal

Year:

2020

Best regards

Dear friends

Thank you very much for your kind messages. I will contact with you soon. Thanks.

Hi Dr Sina Etemad . Fractional (fractional-order) derivative is a generalization of integer-order derivative and integral. ... The kernel function of fractional derivative is called memory function, but it does not reflect any physical process. See the link: https://www.nature.com/articles/srep03431

Dear Prof. Muhammad Ali

Thank you very much.

Examples on practical applications in fractional boundary value problems;

Fractional differential equations /models are applied in different fields of studies: mathematics, physics, chemistry, biology, medicine,…

They are used to describe different physical process such as dynamics of blood flow, wave motions, elasticity,…

For more information, read attached files!

You can see the following links:

check Wilfredo Palma's book

Fractional derivative operators are unbounded when mapping a function space into itself. Integral operators with non-singular kernels (and some with singular kernels) are bounded operators. That said, many diffusive-like processes are best represented by integral operators with non-singular kernels. I would argue that in many cases we use gradient, divergence, Laplacian in our models because the analysis is simpler when we deal with differential operators. You will see this approximation by differential operators if you go back to work of VanderWaals.

see the following paper

In this paper, a fractional relaxation model is studied to determine the effect of heat transfer and magnetic field on the blood flow. The flow is due to an oscillating periodic pressure gradient and body acceleration. We apply Laplace transform as well as finite Hankel transform to obtain the closed-form solutions of the velocity and temperature distributions of the fractional time partial differential equations. The effect of the fluid flow parameters is shown graphically with changes in the ordinary model as well as the fractional parameters. The analysis shows that a fractional derivative is an excellent tool that gives a remarkable change in controlling temperature and blood flow. The analysis depicts graphically, that the presence of a strong applied (exterior) magnetic field, reduces the temperature and blood flow velocities, which is appropriate to avoid tissue damage during treatment. Besides, it is seen that some of the aforementioned parameters influenced the fluid flow profiles in an increasing and decreasing fashion which is interpreted as useful to the study.

See this paper

About applications of conformable fractional derivative in quantum Mechanics.

Good luck

I prefer, instead of discussing the advantages of one operator over another, to consider that both are tools that can be useful and effective in problems of different nature.

Sure, In fractional calculus we use length as a measure but we know that we can not consider it for fractals for example, for example, the Cantor set has zero length or the Koch curve has infinity length then one must use proper measure for fractal (e.g Hausdorff measure).

A number of papers handle this problem, for example, see

Existence and approximations of solutions for time-fractional Navier-stokes equations

Peng, Li, Debbouche, Amar, Zhou, YongJournal:Mathematical Methods in the Applied SciencesYear:2018

An analysis and comparison of the time accuracy of fractional-step methods for the Navier–Stokes equations on staggered grids

S. Armfield, R. StreetJournal:International Journal for Numerical Methods in FluidsYear:2002

Hello, Try this book:

FRACTIONAL DIFFERENTIAL EQUATIONS An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications

If you want the PDF version send me your email and I will send it to you.

Fractional calculus has been applied to signal and image processing for almost two decades. For instance, it was successfully used for noise filtering and pattern recognition. Some books on this topic are:

Ostalczyk, P. Discrete Fractional Calculus: Applications In Control And Image Processing (Series In Computer Vision Book 4), World Scientific, 2015.

Das, S., Pan, I. Fractional Order Signal Processing: Introductory Concepts and Applications (SpringerBriefs in Applied Sciences and Technology), Springer, 2012.

Sheng, H., Chen, YQ, Qiu, TS. Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications (Signals and Communication Technology), Springer, 2012.

Cabral, T.M., Rangayyan, R.M. Fractal Analysis of Breast Masses in Mammograms (Synthesis Lectures on Biomedical Engineering), Morgan & Claypool Publishers, 2012.

Fractional calculus has also been applied to neural networks, machine learning and artificial intelligence.

Dear colleagues

Interesting question and I want to add a reply, about the Leibnitz Rule. You can build local derivatives that do not satisfy it! Conclusion: in my opinion there is no clear classification in this regard. I prefer a broader sentence: fractional derivatives are not derivatives, local generalized derivatives are not fractional.

Thank you very much Dr. Abdelkader Mohamed Elsayed and Dr., Rashid Nasrolahpour ! I greatly appreciate that!

Fractals never produce discomfort in me, quite the opposite. I think human imagination is wired to recognize patterns, thereby gaining some security and using limited brain cells to store the information. It would be nice if fractals (in the sense of the definition of that word, not just "looks like") existed in nature but they do not, as becomes clear upon sufficiently microscopic resolution.

Thank you very much Dr. P.K. Karmakar !!

Dear Dr. Sarah Deif

Greeting

you can see the following links

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Best Wishe

Thank you Dr. !!

I would like to refer to a great book published recently on Springer:

This book is a great help for me and many researchers who are working actively on exploring the open problems in the fractional calculus of variations.

Fractional Calculus is useful in Stochastic process applicable in Finance, financial risk management, dynamic economic systems, study of volatility in financial time series so.

You can find so many examples in the following book:

-An introduction to the fractional calculus and fractional differential equations, KENNETH S. MILLER

if you need the book kindly send me a message otherwise enjoy the book ;

For those who are interested on this topic, I have some applications with the psi-Caputo fractional derivative:

R. Almeida, A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. (44) 460-481 (2017)

R. Almeida, What is the best fractional derivative to fit data? Appl. Anal. Discrete Math. (11) 358-368 (2017)

R. Almeida, A. B. Malinowska and M. T. T. Monteiro, Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Math. Meth. Appl. Sci. (41), 336-352 (2018)

Dear colleague

Actually the question is very general, it depends if you are talking about global or local derivatives. In both there are some advances, for example with Francisco Martínez and Pshtiwan Mohammed we have worked for Laplace Transformation in the local non conformable case. In the local case the definition (and of course its practical use) depends on the kernel.

Thanks

I think it is worthwhile to mention the Indian mathematician Tilak Raj Pabhakar who proposed and studied a generalization to three parameters of the Mittag-Leffler function, see [T.R.Prabhakar, A singular integral equation with a generalized Mittag–Leffler function in the kernel, Yokohama Math. J., 19(1):7–15, 1971]. Although its work was not specifically devoted to fractional calculus, the now so-called Prabhakar function allows to solve in the time-domain fractional differential equations involved by the Havriliak-Negami model, a dielectric model which needs a special fractional derivative with two fractional parameters.

144 computational-mathematics-PhD positions in Germany

The fractional order derivative comes into the picture when there is a shape changing space. In terms of mechanical engineering, it is same as when there are visco-elastic or plasto-elastic material. These materials are purely nonlinear in nature and there mathematical modeling involves the generalized version of ordinary differentiation (integer order differentiation), called as fractional order derivatives.

One of the best physical applications is related to human respiratory airways. The airway tubes are visco-elastic in nature and the complete tracheo-bronchial tree is bifurcating , dichotomous and self similar in nature . The mathematical model to study the pressure variations due to dynamic breathing patterns will involve fractional order derivative itself.

The classical derivatives are in local nature, i.e., using classical derivatives we can describe changes in a neghberhood of a point but using fractional derivatives we can describe changes in an interval. Namely,fractional derivative is in nonlocal nature. This property makes these derivatives suitable to simulate more physical phenomena such as earthquake vibrations, polymers and etc.. To more details, I suggest "podluny" and "Diethelm" books.

Dear Alexander, take a look at this table

Hope you enjoy.

Generally, the definition of the fractional power of an operator in an abstract space follows via Dunford's integral. Indeed, the case of fractional derivative of a function is similar.

The derivative is well-known, and you can define the derivative on some space as operator, you need then a norm.

For the fractional Leibniz' rule see the attachments. The first reference is from the book of Igor Podlubny (1999) : Fractional Differential Equations. See also chapter 4 of the paper of Ehsan Azmoodeh : Riemann-Stieltjes integrals with respect to function Brownian motion and applications. You can see there also the fractional integration by parts.

Dear Christohe Ndjatchi,

You can read the artical:

An efficient method for solving Bratu equations

· May 2006

· Applied Mathematics and Computation 176(2):704-713

· DOI:

· 10.1016/j.amc.2005.10.021

Please see the attached file

I have some works on the non spline function

See the papers.

The geometrical and physical meaning of ordinary derivative is simple and intuitive: For smooth function f which is differentiable at x, the local behavior of f around point x can be approximated by using the ordinary derivative: f(x + h) ≈ f(x) + hf′(x). The ordinary derivative gives the linear approximation of smooth function. Here we expect the fractional derivative to have the similar geometrical meaning.

We hope for non-differentiable functions, the fractional derivative could give some kind approximation of its local behavior. A SIMPLE DEFINITION DIRECTLY FROM GEOMETRICAL MEANING: We expect that the fractional derivative could give nonlinear (power law) approximation of the local behavior of non-differentiable functions f(x + h) ≈ f(x) + h^α f^(α)(x)/Γ(α + 1) in which the function f is not differentiable because df ≈ (dx)^α so the classical derivative df/dx will diverge. Note that the purpose of adding the coefficient Γ(α + 1) is just to make the formal consistency with the Taylor series.

Fractional dynamics is the field of study in physics and mechanics investigating the behavior of objects and systems that are characterized by power law non-locality, power law long-term memory or fractal properties by using integrations and differentiation of non integer orders, i.e., by utilizing methods of the fractional calculus.

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. These definitions, however, are non-local in nature, which makes them unsuitable for investigating properties related to local scaling or fractional differentiability. Recently, the concept of local fractional derivatives have gained relevance, namely because they kept some of the properties of ordinary derivatives, although they loss the memory condition inherent to the usual fractional order derivatives. The best local fractional order derivative definition is not unique. Similarly to what happens to the classical definitions of fractional operators, the best choice depends on the experimental data that fits better in the theoretical model, and because of this we find already a vast number of definitions for local fractional derivatives.

The second order derivative of a function f in calculus is the derivative of the derivative of f. Roughly speaking, the second order derivative measures how the rate of change of a quantity is itself changing. For instance, the second order derivative of the position of a vehicle with respect to time is the instantaneous acceleration of the vehicle, or the rate at which the velocity of the vehicle is changing with respect to time.

May I suggest my two books of which the second eDition are planned in a few years

1) R. Gorenflo, A.A Kilbas, F. Mainardi and S.V. Rogosin: “Mittag-Leffler

Functions. Related Topics and Applications”, Springer, Berlin (2014), pp. XII+ 420 ISBN 978-3-662-43929-6, Springer Monographs in Mathematics. See:

http://www.springer.com/mathematics/analysis/book/978-3-662-43929-2nd edition in prepoaration

2). F. Mainardi: "Fractional Calculus and Waves in Linear Viscoelasticity", Imperial College Press, London (2010), pp. 340, ISBN 978-1-84816-329-4. 2nd edition inpreparation. See: http://www.worldscientific.com/worldscibooks/10.1142/p614

Well, pleased to see the fractional order idea is used for additional benefit. It will be nice to do a fair comparison so that, it could be claimed that, using fractional order control strategy is the only pathway to do good tracking. Is this true?

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. Geometrical meaning of ordinary derivative: The geometrical meaning of ordinary derivative is simple and intuitive: For smooth function f which is differentiable at x, the local behavior of f around point x. A simple definition directly from geometrical meaning: One can expect that the fractional derivative could give nonlinear (power law) approximation of the local behavior of non-differentiable functions.

A two-parameter function of the Mittag-Leffler type is given by

$$E_{\alpha,\beta}(z)=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(\alpha k+\beta)},~~(\alpha>0,\beta>0).$$

For special choices of the values of the parameters $\alpha, \beta,$ 

$$E_{1,1}(z)=e^z$$

$$E_{1,2}(z)=\frac{e^{z-1}}{z}$$

$$E_{2,1}(z^2)=coshz$$

$$E_{1,2}(z)=\frac{sinhz}{z}$$

Only the right hand sides of the two are similar but not the same. The appearances look similar but with different behavings. There are several natural phenomena which are similar but totally different.

In regards to the General Leibniz rule and fractional derivatives, please see the attached article.

Presents a systematic treatment of fuzzy fractional differential equations as well as newly developed computational methods to model uncertain physical problems

Complete with comprehensive results and solutions, Fuzzy Arbitrary Order System: Fuzzy Fractional Differential Equations and Applications details newly developed methods of fuzzy computational techniquesneeded to model solve uncertainty. Fuzzy differential equations are solved via various analytical andnumerical methodologies, and this book presents their importance for problem solving, prototypeengineering design, and systems testing in uncertain environments.

In recent years, modeling of differential equations for arbitrary and fractional order systems has been increasing in its applicability, and as such, the authors feature examples from a variety of disciplines to illustrate the practicality and importance of the methods within physics, applied mathematics, engineering, and chemistry, to name a few. The fundamentals of fractional differential equations and the basic preliminaries of fuzzy fractional differential equations are first introduced, followed by numerical solutions, comparisons of various methods, and simulated results. In addition, fuzzy ordinary, partial, linear, and nonlinear fractional differential equations are addressed to solve uncertainty in physical systems. In addition, this book features:

Basic preliminaries of fuzzy set theory, an introduction of fuzzy arbitrary order differential equations, and various analytical and numerical procedures for solving associated problems

Coverage on a variety of fuzzy fractional differential equations including structural, diffusion, and chemical problems as well as heat equations and biomathematical applications

Discussions on how to model physical problems in terms of nonprobabilistic methods and provides systematic coverage of fuzzy fractional differential equations and its applications

Uncertainties in systems and processes with a fuzzy concept

Fuzzy Arbitrary Order System: Fuzzy Fractional Differential Equations and Applications is an ideal resource for practitioners, researchers, and academicians in applied mathematics, physics, biology, engineering, computer science, and chemistry who need to model uncertain physical phenomena and problems. The book is appropriate for graduate-level courses on fractional differential equations for students majoring in applied mathematics, engineering, physics, and computer science.

1.  Formulation of the problem requires additional assumption that  $f'(x)$  exists at every $x\ge x_0$. Otherwise it may happen, that  $X_n$  equals the "bad" value  of $ x $ with positive probability so that  $f'(X_n)$ would be undefined. Alternatively, one can assume that  $f'(X_n) :=  f_-' (X_n)$,  since the left derivatiwe for convex  decreasing functions always exists and is finite except perhaps at the initial point. Then the sufficient assumption is that  $f'(x_0)$ is finite.

2.  Accordingly, below I am solving the problem with  $f'(X_n)$  replaced by  $g(X_n)$,  where $ g$  is a positive decreasing function defined on $ [x_0, \infty)$,  with $x_0\ge 0$  and  $ \lim g(C) = 0$, as  $C \to \infty$. Let us name

$ Y_n :=  g(X_n) X_n  1\{ X_n > x_0\} $

The integrals from the definition of  uniform integrability of the sequence  $Y_n$, $n=1,2,...$, can be estimated as follows, for   C>x_0 :

$  \int_\{|Y_n \ge C\}  \, Y_n  \, dP 

\le 

\int_\{X_n \ge  C/ g(x_0) \} \,  g(C)\,  X_n \, dP

\le

g(C) \,  E\{ X_n}   \to  0$ ,  as   $  C \to \infty $

Hopefully, it helps.

Best regards, Joachim

I am also ready to collaborate. My e-mail address is: msenol@nevsehir.edu.tr

Dear Sarah,

I think that you have to impose some extra conditions on f, k and even to the notion of the fractional derivative to obtain the corresponding fractional differential equation (FDE). Even in the simplest cases it is not clear what is the corresponding FDE. Let us consider a couple of examples.

First, take k(x,t)=1 and f is differentiable. Then differentiating

(1) D^\alpha y(x)=f(x)+\int_0^x k(x,t)y(t)dt

yields

(2) DD^\alpha y(x)=f'(x)+y(x).

But since the fractional integration and differentiation does not commute in general, DD^\alpha is not necessarily a fractional derivative of order \alpha+1. If D^\alpha denotes the Riemann-Liouville fractional derivative, then (2) corresponds to FDE:

(3) D^(\alpha+1) y(x)-y(x)=f'(x).

However, if D^\alpha denotes e.g. the Caputo fractional derivative, then DD^\alpha is not necessarily D^(1+\alpha).

Second, take k(x,t)=c(x-t)^(\beta-1) for some constant c. If c=1/\Gamma(\beta), then the integral term in (1) corresponds to the Riemann-Liouville fractional integral of order \beta, which is denoted as I^\beta y. If the fractional derivative D^\beta f exists and D^\beta is the left inverse of I^\beta, then (1) converts into

(4) D^\beta D^\alpha y(x)=D^\beta f(x)+y(x).

But for the same reason as in the first case, D^\beta D^\alpha is not D^(\alpha+\beta) in general.

Hopefully you will find this useful in your further considerations.

Best regards, Jukka

Dear Sara,

First of all, let me congratulate you for your achievement.

Do not bother about applications. Go on your research,

But if you want to have an application, you can see

 Dear Aldo,

I would say that the question is an open problem. If the question is considered in the scale of Lp-spaces, then it follows from the identity (see Theorem 6.1 in Samko-Kilbas-Marichev, as you obviously know)

D^\alpha I^\alpha f=f 

that the kernel is trivial in the image of Lp, 1<p<1/\alpha, under the fractional integration I^\alpha, which is a subspace of Lq with q=p/(1-\alpha*p). But the kernel considered as a subspace of continuous functions is unknown, I would say. So, if you have a manuscript on the result, go ahead and publish it! :)

Best regards, Jukka