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Fractional Differential Equations - Science topic

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For the parameter estimation of epidemic models usually fminsearch, fmincon functions of Matlab are used. However, there is a lack of detailed informations regarding how to choose the initial values and coding for estimation. Due to this, as a beginner in this field, I'm suffering. Could anyone help me with this? I would like to get your suggestions and guidance for this issue.
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To a great extent, the initial values depend on your model boundary conditions. Preferably an initial value is taken when a significant step change in model outcome is observed.
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As you can see on my page, I developed several alternatives to fractional models to model fractional behaviours (Fractional models and fractional behaviours are two different concepts, the first one denotes a particular class of models, the second is a class of dynamical behaviours that can be generated and modelled by a wide variety of mathematical tools other than fractional calculus).
I would like to evaluate the efficiency of these models on real data. I thus look for proven fractional behaviours data. Not frequency data over 2 decades, not temporal data over a reduced time range and large sampling period, but data which are truly fractional and which can be demonstrated. Not this kind of data that can be capture as well with an interger first or second order model.
We can consider collaboration and joint publications with those who can provide me with such data, if they wish.
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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.
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Hi respected seniors and experts, I recently submitted a paper on the Caputo-Fabrizio operator but the paper was rejected with the comment "The Caputo-Fabrizio is not a fractional operator". But thousands of papers have already been published on the mentioned-above operator, does that mean all those articles are wrong???
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I want to factorize the plant tf as Gp+ and Gp- to implement in an IMC controller.
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Dear researchers
As we know, recently a new type of derivatives have been introduced which depend two parameters such as fractional order and fractal dimension. These derivatives are called fractal-fractional derivatives and divided into three categories with respect to kernel: power-law type kernel, exponential decay-type kernel, generalized Mittag-Leffler type kernel.
The power and accuracy of these operators in simulations have motivated many researchers for using them in modeling of different diseases and processes.
Is there any researchers working on these operators for working on equilibrium points, sensitivity analysis and local and global stability?
If you would like to collaborate with me, please contact me by the following:
Thank you very much.
Best regards
Sina Etemad, PhD
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Yes I am
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I am working on approximating solution of fractional integro- differential equation involving Caputo conformable derivative, so I am seeking on Matlab code helpful for this problem.
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Definitely, I ask myself whether it was useful to introduce fractional partial differential equations.
As based on fractional differential operators this class of equations also surfers from the same drawbacks as the one described in:
Fractional partial differential equations were introduced to model anomalous diffusion, i.e. phenomena that exhibits power law behaviours other than 1/2. But it was shown recently that these kinds of behaviours can also be obtained with classical partial differential equations with spatially variables coefficients:
Where it is difficult to propose a physical interpretation with the fractional partial differential equations, classical partial differential equations with spatially variables coefficients allow interpretations relating to the geometry of the systems studied.
What is your opinion ?
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It is not correct ...
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Dear researchers
As you know, nowadays optimal control is one of the important criteria in studying the qualitative properties of a given fractional dynamical system. What is the best source and paper for introducing optimal control analysis for fractional dynamical models?
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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
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Hello sirs, as for Matlab code for fractional differential equations, I would recommend FOTF toolbox, FOMCON toolbox and Prof Garrappa's Matlab implementation.
There are also some open source softwares focus on fractional differential equations I am working on:
Any suggestion and advice are welcomed😀
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What are the most recommended publishers for submitting a book in the field of Iterative Methods for solving fractional Differential Equations?
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Thank you very much
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Unlike the ordinary derivative, we have several definitions of the fractional derivatives. Do these definitions (up to constant"the definition constant") lead to the same answer? May be I should put like the following can we adjust the initial conditions or the model settings to reach this goal?
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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
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explore modelling using fractional differential equations.
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You may go through the research article entitled Time Nonlocal Six-phase-lag Generalized Theory of Thermoelastic Diffusion with Two-temperature
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There are many numerical techniques for obtaining approximate solutions of fractional order boundary value problems in which the order of differential equation is a fractional constant number. If we assume that the order of BVP is a continuous functions with respect to the time, then is there any numerical technique to obtain approximate solutions of a variable-order fractional BVP?
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You can also read the following paper
On solutions of variable-order fractional differential equations
DOI:10.11121/ijocta.01.2017.00368
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I am working on one article related to infectious disease modeling using Atangana-Baleanu derivative. I used Adams-Moulton technique that was used in the article titled “A fractional order HIV‐TB coinfection model with nonsingular Mittag‐Leffler Law”, the link: https://www.researchgate.net/publication/338678753
To find S(3), I need to know S(1) and S(2). S(1) is given as an initial value, but S(2) is not given, and I have no real data for my study! My question is : How do I substitute the value of S(2)?
I tried with another numerical method that used in this article “A fractional order HIV/AIDS epidemic model with Mittag-Leffler kernel” , the link: https://doi.org/10.1186/s13662-021-03264-5 , and I faced the same.
I tried with S(2)=S(1)+h*F(S1(1),S2(1),…,Sn(1)) I got result, but the difference between S(2) and S(3) is not acceptable, and I am not sure if it works with thos methods or works only with Adam-Bashforth method!
If anyone help me, I will appreciate that, thanks
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@Muhammad Ali, @Zoleikha Soori, @Bhuban Deuri, thank you all for your answers, I used Adam-Bashforth method for the numerical solution of my work
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Dear researchers
In two recent years, many researchers and mathematicians interested in studying different fractional models of COVID19 and other diseases. They analyzed these models by means of different numerical techniques and also, investigated some conditions confirming the existence of solutions. Most of them are modeled via fractional (singular or non-singular) operators such as Caputo operators, Caputo-Fabrizio operators, Atangana-Baleanu operators.
But, I would like to study these models with the help of fractional discrete operators such as q-difference operators, Delta/Nabla fractional operators. My information in this regard is not enough.
Could you please suggest some newly-published papers on the mathematical model of COVID-19 via discrete fractional operators?
Thank you very much.
Best regards
Sina Etemad
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Dear Prof. Muhammad Ali
Thank you very much.
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For example, in terms of ordinary differential equation we have ordinary derivative in boundary condition.
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Dear Dr. Sabermahani
Please, see the following paper in which there are boundary value conditions with fractional derivatives:
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In fractional calculus many fractional operators, but their applications area unknown. Can you help me to determined the real Phenomena which we can apply the Capotu conformable fractional derivatives ?
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See this paper
About applications of conformable fractional derivative in quantum Mechanics.
Good luck
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Nowadays, many researchers are devoting their research work to the non-singular kernel. Most of them have mentioned that "the main advantage of this kind of operators is that the singular power-law kernel is now replaced by a non-singular kernel," which is easier to use in theoretical analysis, numerical calculations, and real-world applications. But in my opinion, the singular power-law kernel is very easy to use in the mentioned above calculations and applications. Kindly share your thoughts.
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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.
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This is a question that I also placed at
but that I placed here also for more visibility.
Fractional models are they physically consistent?
Let consider a very simple fractional model described by the transfer function H(s) = 1/s^n. You can stay looking at the Laplace representation of such a system, or you can try look at this model from a different perspective.
1 – You can compute its impulse response using the Cauchy’s method involving a Bromwich-Wagner path and thus you obtain an integral whose Laplace transform is :
H(s)=int(x^(-n)/(s+x)dx) with x in [0, infinity]
See for instance [1][2][3].
2 – You can also split integral H(s) into two parts to obtain a diffusive representation [4] [5] and then, using spatial Fourier transform on this diffusive representation you get a diffusion equation with a distributed sensor, defined on an infinite spatial domain. (See [2] and [3] for computation details).
In the case 1) the model exhibits infinitely small and infinitely large time constant. Is it physically consistent to use a model with infinitely fast dynamics ? Also the infinitely large time constants are at the origin of the long memory and more exactly of the infinite memory. Is it physically consistent for a model to have an infinite memory?
In the case 2), the model definition on an infinite space domain is questionable. Is it physically consistent for a model to be define on this kind of space domain? Note that infinite domain generates the infinite memory previously highlighted.
This questioning on physical consistency is not limited to fractional integrator case but can be extended, with the same analysis tools to fractional partial or not differential equations [6], pseudo state space descriptions [7], …..
This is in my opinions the reasons that justify that it is impossible to gives a physical interpretation of fractional differentiation and fractional model, unless considering non-physical assumptions (infinite space). Otherwise, how to justify an infinite memory or infinitely fast time constants.
The kernel singularity is not the only problem that can be solved by the introduction of new kernel to produce fractional behaviours. As shown in [3] [8], Many other kernel exit, that also solve the singularity problem, by that also solve the consistency problem evocated here.
[1] - Sabatier J., Merveillaut M., Malti R., Oustaloup A. (2008), On a Representation of Fractional Order Systems: Interests for the Initial Condition Problem, 3rd IFAC Workshop on "Fractional Differentiation and its Applications" (FDA'08), Ankara, Turkey, November 5-7.
[2] - Sabatier J., Merveillaut M., Malti R., Oustaloup A. (2010), How to Impose Physically Coherent Initial Conditions to a Fractional System? Communications in Nonlinear Science and Numerical Simulation, Vol 15, No. 5.
[3] J. SABATIER - Non-singular kernels for modelling power law type long memory behaviours and beyond,Cybernetics and Systems, pp. 1-19, doi:10.1080/01969722.2020.1758470.
[4] - Matignon, D. Stability properties for generalized fractional differential systems. ESAIM Proc. 1998, 5, 145–158.
[5] - Montseny, G. Diffusive representation of pseudo‐differential time‐operators. ESAIM Proc. 1998, 5, 159–175.
[6] - Sabatier J., Farges C. (2018), Comments on the description and initialization of fractional partial differential equations using Riemann-Liouville's and Caputo's definitions, Journal of Computational and Applied Mathematics, Vol. 339, pp 30-39.
[7] - J. SABATIER, Fractional state space description: a particular case of the Volterra equation, Fractal and Fractional, Vol. 4, N° 23, doi:10.3390/fractalfract4020023
[8] - J. SABATIER, Introduction of new kernels and new models to solve the drawbacks of fractional integration/differentiation operators and classical fractional order models
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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).
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In application sometimes, one choose Caputo fractional derivative and in other situations one prefer Riemann-Liouville one. What are the main differences between them, and how to choose?
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Unfortunately, different from classical or integer-order derivative, there are several kinds of definitions for fractional derivatives. These definitions are generally not equivalent with each other.
The following papers are "on Riemann and Caputo fractional differences"
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The introduction of time-fractional derivatives in the Navier-Stokes equations goes back to Lions and was recently extensively studied by many authors. When motivating such problems authors tend generally to motivate fractional models instead. In some papers (the ones of Zhou and Peng for example) say that such equations can model anomalous diffusion in fractal media. What I am looking for is some concrete situations where time-fractional Navier-Stokes equations is the good way to model the fluid flow.
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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
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Recently, some new operators having the keyword "fractional" are proposed to define the non-integer order derivative.
For integer order derivatives, it is well known that the locality and validity of Libeniz rule are the main properties of integer order derivatives, but what is the characteristic property of a fractional derivative? When we can call an operator is "fractional" and what features should an operator have to call it a "derivative" operator?
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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.
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I am interested in collaborating with any researcher working on modelling corona virus using fractional derivatives. If you are a researcher or you have a related project, please feel free to let me know if you need someone to collaborate with you on this research study. If you know someone else working on this research project, please share my collaboration interest with him.her. I would be very happy to collaborate on this research project with other researchers worldwide.
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Thank you very much Dr. Abdelkader Mohamed Elsayed and Dr., Rashid Nasrolahpour ! I greatly appreciate that!
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In the literature, fractal derivatives provide many physical insights and geometrical interpretations, but I am wondering where we can apply this particular derivative appropriately. Please refer me to references or examples because I am very interested to learn more about new derivatives and their applications!! I greatly appreciate all the brilliant efforts in this discussion!!
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Thank you very much Dr. P.K. Karmakar !!
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I am still new to the topic, so I am searching for a numerical solver for fractional differential equations (FDEs) in the form of a MATLAB code or the like. I would like to test some simple FDEs at first in order to get a better understanding of the topic and then proceed with writing my own code.
So, is there any ready-made package that solves FDEs that you know of?
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Dear Dr. Sarah Deif
Greeting
you can see the following links
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Best Wishe
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There are many existing publishers that publish high quality books in mathematics, but my question here is: I want some suggestions about publishers who most likely publish books in the field of fractional calculus and fractional differential equations because I am interested in submitting a book proposal for a suggested publisher. Could you please share you information/knowledge about such recommended publishers in this specific field of research in mathematics? I would greatly appreciate your brilliant efforts and time!!
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Thank you Dr.
Mila Ilieva
!!
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All codes of above-mentioned programs for
1) Differential Equation DE
2) Fractional Differential Equation FDE
3) Integral Equation IE
4) Fractional Integral Equation FIE
5) Integro-Differential Equation IDE
6) Fractional Integro-Differential Equation FIDE
Please, if any one can help me to reach
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Hello Sir.
I will send you two files to give you an idea about solving differential equations with Matlab. You can find more information on this topic in Mathworks.
All the best.
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I am currently searching for any open position as a PhD research position in applied mathematics. I am actively working in the area of Fractional Differential Equations with Applications in Science and Engineering. I have attached my curriculum vitae (C.V.) (Please see the attached PDF file). If you know any available position related to applied mathematics or any person who are in search of candidates for this type of positions, please do let me know. I would also greatly appreciate if you could share my C.V. with your connections.
Thank you very much in advance!
With Kind Regards,
Mohammed K A Kaabar
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I am wondering about any possible suggestions of applications/phenomena from science and engineering where the fractional-order geometric calculus can be applied effectively. I would greatly appreciate your help by providing me with references, suggestions, or examples related to this topic of research!!
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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.
  1. Two-phase flow
  2. turbulent flow and separation flow application
  3. flow in the porous media application
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There are various methods that have been used in solving the fractional differential equations, but I am wondering what are the most powerful and efficient ones that can be applied effectively in solving the fractional differential equations?
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HAM and HPM coupled with the integral transformations like Laplace, Fourier, etc. Especially if you apply them to linear/nonlinear problems by considering the fractional operators without a singular kernel, I hope you will obtain great results.
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Can we provide a geometrical and physical interpretations for fractional derivatives like as integer order derivatives?
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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.
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Recently several operators are defined with having the keyword "fractional derivative".
My question is that what properties should an operator have to call it "fractional"? Additionally, when we can say an operator is a derivative?
what is the characteristic property for fractional derivatives?
Among the newly defined operators for derivatives with non-integer order which one is "fractional" and which one can be called a "derivative" operator?
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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.
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I did not find any research papers in this regard.
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Please see the attached file
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Explicit numerical schemes for fractional partial differential equations.
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You can read the Chapter 3 from the following book:
D. Baleanu, K. Diethelem, E. Scalas and JJ. Trujillo, Fractional Calculus: Models and Numerical Methods , Vol. 3. World Scientific, (2012) .
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I want to seek which discretization of RL-fractional derivative is better. In most of research articles people are using Caputo type derivative. Can I use Caputo type derivative discretization for RL-Fractional derivative?
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Dear Muhammad Ali,
Look into the following:
*Discretized Fractional Calculus by Ch. Lubich, SIAM J. Math. Anal., 17(3), 704–719.
*Fractional Calculus: Models and Numerical Methods by Dumitru Baleanu, Kai Diethelm, Enrico Scalas and J.J. trujillo. Vol. 3. World Scientific, 1-400(2012)
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In ordinary calculus integration represent the area in that direction. What is the geometrical representation of fractional integral and fractional derivative?
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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.
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Round of errors play important role in numerical methods. How we can investigate influence of these errors in finding collocation solutions of fractional differential equations
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In general, round-off errors have a relevant impact when either
1) trying to solve very ill conditioned linear systems
2) trying to achieve relative errors barely larger than the machine accuracy (one should not try to achieve less :-)) )
I am not an expert in FDEs, but these two general concepts should apply all the same. Therefore, as long as the linear system one gets have a condition number that is significantly smaller than the reciprocal of the machine accuracy AND the accuracy goals are not excessively ambitious, I would not expect an excessive impact of round-off on the numerical solutions of FDEs.
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1) E_a(z) is an entire function of order 1/a. Can you suggest a good reference about this topic?
2) How many derivatives do the functions E_a(z) and E_a(z^a) have at z=0?
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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 $\alpha, \beta,$ 
$$E_{1,1}(z)=e^z$$
$$E_{1,2}(z)=\frac{e^{z-1}}{z}$$
$$E_{2,1}(z^2)=coshz$$
$$E_{1,2}(z)=\frac{sinhz}{z}$$
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What will be a successful mixture? for what end? could you please steer me towards some good references that have applications, and solution by approximate analytical methods?
Many thanks and best regards
Sarmad.
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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.
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What did the fractional derivative add to pde in modeling and application wise? and what does it mean? please give a successful applied example.
Best regards
Sarmad.
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Dear Sarmad A. Jameel Altaie,
the following might be of use:
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Dear All,
Can I transform a linear fractional Volterra integro-differential equation into a fractional differential equation? If yes, then how?
The equations are written in the attached file.
Thank you very much in advance for your help.
Sarah
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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
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Dear all,
I have developed a new technique which solves linear integro-differential equations of fractional type. This includes Fredholm and Volterra equations.
I am looking for an application which can be modeled into such equation so I can apply my method. It can be any kind of application.
I also solve mixed system of equations e.g.1. A system of multiple Fredholm equations of different order of fractional derivatives (0, 1/2, 1, 3/2, etc..) or e.g. 2. A system of same or different kinds of Volterra equations. So, if there is an application to this kind of equation, it would be great!
I would appreciate your help. Please refer me to an article.
P.S. Only linear equations please.
Thank you very much in advance.
Sarah
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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
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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?
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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.
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There are many many papers studies the sufficient condition of existences of solution of fractional differential equations.
Why we are not find the exact solution direct? and, what are disciplinary which use the theory of existence and uniqueness solutions?
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If exact solutions can be found for a class of FDEs, there is no need to discuss the sufficient conditions for existences of solutions. But the fact is, it is so hard to get exact solutions of general FDEs. Then we consider the existence, uniqueness, numerical algorithms, ...
There are no special reasons different with ODEs or PDEs, I think.
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Is there any good review available describing solution methods especially exact or approximate analytical or closed-form solutions for PDEs with coefficients which are not constant. E.g. a beam equation with space varying coefficients?  
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I have a system of fractional-order differential equations in the sense of  Caputo’s derivatives. For this model, I obtained the equilibrium points and  proved that some equilibrium points are locally asymptotically stable. Now I want that study about the global stability of the equilibrium points.
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Hello Ehsan, 
You may have a look on this paper: 
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1.D_{0}^{\alpha}u_{0}(t)-a(t)u_{0}(t)f_{0}(t)
2.D_{0}^{\alpha}u_{1}(t)+4\pi n u_{1}(t)-a(t)u_{1}=f_{1}(t)
3.D_{0}^{\alpha}u_{2}+4\pi^{2}n^{2}u_{2}(t)+4\pi n u_{1}(t)-a(t)u_{2}(t)=f_{2}(t)
D_{0}^{\alpha} is riemann liouville derivative.
thinkyou.
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Use Fourier transform property. Or, if you want to do it numerically then Adomian decomposition method or Homotopy analysis method.
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the n-dimensional fractal set is a part of Rn. the hyperbolic iterated system construct the fractal set in standard coordinates.
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I want to define fractional transfer function k/(s+a)^alpha.
where, alpha is fractional order between 0 to 1.
Could you please help for the same?
Thanking You, 
Pritesh
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I got inverse Laplace transform of this form. It is k*  t^alpha-1 * e^(-a*t)/gamma(alpha) 
However, I intend to find a response of this system where the input signal is excited with PRBS signal. 
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For example: D^\alpha x(t)=-Cx(t)+AF(x(t))+BF(x(t-\tau(t)))+I(t).  Thanks!
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@Subrahamanyam Upadhyay.  Thanks a lot. Can you give me a link of the related paper?  Under Prof. Jukka T. Kemppainen and Dr. S. Bhalekar's help, we find the following paper. It gives me a lot of help.  Thanks again!
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I am a doing a research on analysis of hyperbolic paraboloid shell using levy type soluton which will be validated later by using ansys. Levy type method as suggested by apeland and popov is being used. Since its a eight order differential equation with boundary value problem having a problem in solving boundary condition. Can anyone suggest an easier method to solve the boundary conditions? or can anyone explain the method to use the tables given by Apeland and Popov?
P.S. Already tried spilines method.
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Hi,
Start studying below link...
All the best
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$ D_{0}^{\alpha}[u_{0}(t)-u_{0}(0)]=\dfrac{f_{0}(x)}{1-a(t)} $
$ D_{0}^{\alpha}[u_{1n}(t)-u_{1n}(0)]+4\pi^{2}n^{2}u_{1n}(t)+4\pi n u_{2n}(t)+a(t)u_{1n}(t)=f_{1n}(x)$
$ D_{0}^{\alpha}[u_{2n}(t)-u_{2n}(0)]-4\pi^{2}n^{2}u_{2n}(t)+a(t)u_{2n}(t)=f_{2n} $
  • thank you
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Dear collegue
Could you please give more information.
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For an initial value problem including fractional differential equations of order $\alpha, \ (0 <\alpha<1)$, like as classical case we impose only one initial condition. Now, if the order of derivative be a rational number $\alpha=\frac{m}{n}$, can we consider $m$ initial conditions:
$y(0)=y_0, D^{\frac{1}{n}}y(0)=y_1, …, D^{\frac{m-1}{n}}y(0)=y_{m-1}?$
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Dear Davoud,
I have to correct my answer a little bit. In general, if you have an equation of order m/n, you need N initial conditions, where N=Mm/n with M being the least common multiple of the denominators appearing in the multi-term fractional differential equation.
Suppose, e.g. that you have a multi-term equation of orders, say 2/3 and 1/2. Then the least common multiple of the denominators of the fractional orders is M=2*3=6, whence you will need M*m/n=6*2/3=4 initial conditions in this special case. I am sorry for the mistake I made above.
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Does anyone know of any direct algorithm to evaluate the inverse Mittag-Leffler function on the real axis?
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Hi Alfonso,
try this mathematica command:
Solve[MittagLefflerE[0.7, x] == 9, x]
it seems to work
{{x -> 1.54163}}
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I am asking about the stability of numerical methods. Specially pseudo spectral method.
when are solving fractional differential equations or for instance ordinary differential equations using Legendre operational matrices of integration and derivative. Then as clear that highest order derivative is assumed in product of two vectors coefficients vector and function vector, then operational matrices of derivative and integrals are used to obtain the lower order derivatives, and is then substituted in original equation, as a result we get system of algebraic equations,  
These algebraic equations are then solved to get approximation of solution differential equations. Clearly the use of large number of orthogonal polynomials will result in more accurate solution (as I experimentally observed over a large variety of problem).  
Now if some one is  asked that show that the algorithm is stable, and convergent. Then what does the term stable means here. In other words how can we theoretically prove that the algorithm is stable and convergent.  
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Thanks to all. For useful comments. The point which I want to discuss is now clear to me. Thanks Alfasno...., Thanks Wiwat....., Also thanks Demetris, Also many thanks Payam
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for example in dynamics first derivative is rate or velocity: dx/dt
or the second derivative is acceleration: d2x/dt2
but in some cases we see fractional differential equations such as( d^0.8x/dt^0.8). I mean the order is fractional and 0.8 can be another fractional number. in mathematics there is no problem with this but what is the physical meaning? this equations are frequently used in anomalous diffusion.
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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.
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In my opinion since the utilization of Zadeh's Extension Principle is quite difficult in practice, we prefer the idea of using level sets.
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If you like the ides of fuzzy control, you might also like the idea of multiple model adaptive control. We use nonparametric models of drug behavior. They have multiple discrete support points, which permit multiple model dosage design for  maximally precise dosage regimens. Each discrete support point is like a fuzzy point, but is quantitatively better. I would call multiple model control, which is widely used in the aerospace community, something like fuzzy but not fuzzy. That is the advantage, I think. You might go to www.lapk.org and look around.
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I am interested to know present research of Fractional Calculus through nanotechnology. 
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Dear friend,
Thanks for your answer.
Could you give me a idea.
How it will possible?
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Do both require knowledge of fractional order calculus?
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good to hear from U..........All the Best for Ur Future Endeavors.....
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There are many different approaches to discretization of fractional derivatives, but many of them do not preserve the fundamental properties of the systems described by continuous-time equations such as stability of the system. For example, the asymptotically stable continuous-time fractional system after discretization using shifted Grunwald-Letnikov formula becomes unstable.
Is there any known method of discretization that preserves the properties of the original system?
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Hello Krzysztof,
we can hardly say that there is the best way for such discretization in general. There are simply to many aspects (stability, accuracy, time efficiency, simplicity etc.). Nevertheless, you have mentioned the stability property. If you are looking for a simple method preserving stability of the underlying continuous system, maybe a discretization derived from implicit Euler method and Grunwald-Letnikov formulas could be of your interest. We have studied stability of this discretization in some of our papers for the case of Riemann-Liouville fractional derivative, but it can be utilized also for the Caputo one. Write me a message if you need help.
Best regards,
Tomas
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How to evaluate the integral term in fractional differential equations.
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The paper "Algorithms for the fractional calculus: A selection of numerical methods" by Diethelm et al (2005) in Computer methods in applied mechanics and engineering, vol 194, pp 743-773 gives a very good review. In the paper's own words, it gives the newcomer "the necessary tools required to work with fractional models in an efficient way" --- this paper would be a good place to start...good luck
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D_q(t) (f.g) = D_q(t) f .g + f.D_q(t)g
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No, you can't do that. Actually you cannot even do that if q(t) is a constant function, i.e. in the more standard case of a fractional derivative with non-variable order.
Just try an example: f(t) = t, g(t) = t, q(t) = q = constant with 0 < q <= 1 (using 0 as the starting point of the fractional operator).
In this example it doesn't matter if you use Riemann-Liouville or Caputo operators.
In either case the left-hand side of your equation is
D^q (t^2) = \Gamma(3) t^{2-q} / \Gamma(3-q)
and the right-hand side is
2 t D^q (t^1) = 2 t \Gamma(2) t^{1-q} / \Gamma(2-q)
= \Gamma(3) t^{2-q} / \Gamma(2-q)
so you can see that the fractions have the same numerator but different denominators (except, of course, for the non-fractional case q = 1 when \Gamma(3-q) = \Gamma(2) = 1 = \Gamma(1) = \Gamma(2-q)).
The correct formula is given, e.g., in my book "The analysis of fractional differential equations", Springer, Berlin, 2010. Theorem 2.18 gives the result that you need if your differential operators are of Riemann-Liouville type; Theorem 3.17 is the corresponding result if you want to use operators of Caputo type.
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I think we should treat it by numerical quadrature.
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This problem has an exact answer found after setting z=b+x, c=a-b and using the binomial expansion. Also it is one which meets the requirement that the integral equals x when c=0. Will type put the answer here plus give a jpg of the solution below-
Int{(a+x)/(b+x),x}=x+ncln(b+x)-Sum{C[n,k]c^k/[(k-1)*(b+x)^(k-1)],k=2..n}
where C[n,k]is the binomial coefficient.
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How to simulate fractional powers of s=jw in MATLAB/Simulink??
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Dear Das,
Maybe the following links may help you:
1 - Modeling and Simulation of Fractional Order Chaotic Systems Using Matlab/Simulink
2 - Fractional Derivatives, Fractional Integrals, and Fractional Differential Equations in Matlab
3 - MATLAB TOOLBOXES FOR FRACTIONAL ORDER CONTROL: AN OVERVIEW
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By G-L I mean Grunwald-Letnikov, I know both of them are equivalent for (alpha<0) and different for (alpha>0) but I can't realize what's the exact differences except the derivative of a constant is zero to Caputo type and non-zero to R-L type.
It is said to use R-L and Caputo for analytical solutions and G-L to numerical ones, and also in the 'Podlubny' book it's said that all the definitions are equivalent but as we all know R-L and Caputo are not equivalent. Someone can please explain the paradox for me?
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R-L and C are just two different operators that are realted to each other in a quite simple way. Quite a few details about this are given in my book, "The Analysis of Fractional Differential Equations" (Springer, Berlin, 2010). There you can also see the exact description of when they are equivalent. The most important difference between them is, of course, the structure of their kernels (i.e the set of functions that is mapped to zero). Depending on what you want from your operator, one of them or the other one may be the right choice for you.
GL is a nice basis for numerical work. It is also known to be equivalent to RL under suitable conditions (also given in my book).
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It is simple to calculate the Lyapunov spectra for ODE. Those in fractional-order can be estimated using time series of one of its system states using the method by Zeng, Eykholt, and Pielke (1991). Except from this twenty-two-year-old method, is there any improved or more efficient method for estimating the Lyapunov spectra of a fractional-order system?
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Hi
you can follow this paper
"On the bound of the Lyapunov exponents for the fractional differential
systems"
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I am searching for a new research application to fractional differential equations (fractional calculus). It could be in the field of science, engineering, finance or the like.
Any recommendations?
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Google "Fractance" which is an application of fractional derivative to electrical circuits.