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Fractional Differential Equations - Science topic
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Questions related to Fractional Differential Equations
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.
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.
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???
I want to factorize the plant tf as Gp+ and Gp- to implement in an IMC controller.
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
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.
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 ?
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?
What are the most recommended publishers for submitting a book in the field of Iterative Methods for solving fractional Differential Equations?
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?
explore modelling using fractional differential equations.
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?
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
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
For example, in terms of ordinary differential equation we have ordinary derivative in boundary condition.
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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 ?
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.
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
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?
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.
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?
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.
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!!
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?
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!!
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
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
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!!
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?
Can we provide a geometrical and physical interpretations for fractional derivatives like as integer order derivatives?
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?
I did not find any research papers in this regard.
Explicit numerical schemes for fractional partial differential equations.
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?
In ordinary calculus integration represent the area in that direction. What is the geometrical representation of fractional integral and fractional derivative?
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
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?
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.
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.
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
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
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?
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?
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?
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.
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.
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
For example: D^\alpha x(t)=-Cx(t)+AF(x(t))+BF(x(t-\tau(t)))+I(t). Thanks!
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.
$ 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
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:
$y(0)=y_0, D^{\frac{1}{n}}y(0)=y_1, …, D^{\frac{m-1}{n}}y(0)=y_{m-1}?$
Does anyone know of any direct algorithm to evaluate the inverse Mittag-Leffler function on the real axis?
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.
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.
In my opinion since the utilization of Zadeh's Extension Principle is quite difficult in practice, we prefer the idea of using level sets.
I am interested to know present research of Fractional Calculus through nanotechnology.
Do both require knowledge of fractional order calculus?
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?
How to evaluate the integral term in fractional differential equations.
D_q(t) (f.g) = D_q(t) f .g + f.D_q(t)g
I think we should treat it by numerical quadrature.
How to simulate fractional powers of s=jw in MATLAB/Simulink??
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?
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?
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?