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

Please see the attached file.

Hello Mahdi Saedshoar Heris

Refer to the attached PDF for an illustration. Fractional calculus is used in design of PID controllers.

The following some papers in the topic

(1)[IEEE Proceedings of the 2004 IEEE International Conference on Solid Dielectrics, 2004. ICSD 2004. - Toulouse, France (July 5-9, 2004)] Proceedings of the 2004 IEEE International Conference on Solid Dielectrics, 2004. ICSD 2004. - Application of fractional calculus to modelling of relaxation phenomena of organic dielectric materials

Reyes-Melo, M.E., Martinez-Vega, J.J., Guerrero-Salazar, C.A., Ortiz-Mendez, U.

2

(2)Application of fractional calculus to the modeling of the complex rheological behavior of polymers: From the glass transition to flow behavior. I. The theoretical model

M. E. Reyes-Melo, V. A. González-González, C. A. Guerrero-Salazar, F. García-Cavazos, U. Ortiz-Méndez

3

(3)Application of fractional calculus to the modeling of dielectric relaxation phenomena in polymeric materials

E. Reyes-Melo, J. Martinez-Vega, C. Guerrero-Salazar, U. Ortiz-Mendez

4

(4)Application of fractional calculus to the modeling of the complex magnetic susceptibility for polymeric-magnetic nanocomposites dispersed into a liquid media

M. E. Reyes-Melo, M. A. Garza-Navarro, V. A. González-González, C. A. Guerrero-Salazar, J. Martínez-Vega, U. Ortiz-Méndez

5

(5)Application of fractional calculus to viscoelastic behaviour modelling and to the physical ageing phenomenon in glassy amorphous polymers

M. Alcoutlabi, J.J. Martinez-Vega

6

(5)AIP Conference Proceedings [AIP NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2012: International Conference of Numerical Analysis and Applied Mathematics - Kos, Greece (19–25 September 2012)] - Applications of fractional calculus to epidemiological models

Skwara, Urszula, Martins, José, Ghaffari, Peyman, Aguiar, Maíra, Boto, João, Stollenwerk, Nico

This special issue may help

Article Fractional Calculus and Its Applications in Applied Mathemat...

See the attached files.

1. Fractional calculus models of complex dynamics in biological tissues by Richard L.Magin

2. Application of fractional calculus in modeling and solving the bioheat equation by R. Magin , Y. Sagher and S. Boregowda

3. On some applications of the space-time fractional derivative by Wasan Ajeel AhmoodEmail and Adem Kılıçman

4. Application of fractional calculus to the modeling of the complex rheological behavior of polymers: From the glass transition to flow behavior. I. The theoretical model by Reyes-Melo, González-González, Guerrero-Salazar, García-Cavazos and Ortiz-Méndez

5. Fractional-order circuit models of the human body impedance for compliance tests against contact currents by De Santis, Martynyuk Lampasi, M. Fedula and Ortigueira

Hi

Thanks a lot

Prof. Manuel Duarte Ortigueira

Prof. Dejenie A. Lakew

Prof. Amaechi J. Anyaegbunam

Prof. Maged Gumaan Bin-Saad

Prof. Nisar K S

Prof. Boddu Muralee Bala Krushna,

For useful information.

Thanks a lot Dr. Alfonso Jaimes-Nájera

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

Thanks a lot Prof. Mainardi

Thank you for your answers.

Hi colleague,

Your question on "...physical phenomena" covers a number of topics. The applications of fractional calculus to fluid flow and particle movement in porous media such as soils and other geological strata are among these topics. In these areas, I published a few papers:

1. Flow in soils with swelling and non-swelling properties, which are not saturated by fluid:

https://doi.org/10.1016/j.jhydrol.2014.09.021

2. Flow in aquifers, which are saturated by fluid:

https://doi.org/10.1016/j.jhydrol.2017.01.015

Dr. Ninghu Su

Thank you very much

Dear Prof. Su,

I hope this paper can help you.

Best wishes,

HongGuang Sun

1. Fractional calculus models of complex dynamics in biological tissues by Richard L.Magin

2. Application of fractional calculus in modeling and solving the bioheat equation by R. Magin , Y. Sagher and S. Boregowda

3. On some applications of the space-time fractional derivative by Wasan Ajeel AhmoodEmail and Adem Kılıçman

4. Application of fractional calculus to the modeling of the complex rheological behavior of polymers: From the glass transition to flow behavior. I. The theoretical model by Reyes-Melo, González-González, Guerrero-Salazar, García-Cavazos and Ortiz-Méndez

5. Fractional-order circuit models of the human body impedance for compliance tests against contact currents by De Santis, Martynyuk Lampasi, M. Fedula and Ortigueira

Hi, if you have interest, please have a look at our paper on Fractional Plasticity Modelling of Soils, where the clear Physical Meaning of the Fractional Order was given.

Regards,

Yifei Sun

Thanks a lot Prof. Sun

Hello

1- Fractional Euler-Bernoulli beam theory based on the fractional strain-displacement relation and its application in free vibration, bending and buckling analyses of micro/nano beams

2- The analysis of non-linear free vibration of FGM nano-beams based on the conformable fractional non-local model

3- A study of critical point instability of micro and nano beams under a distributed variable-pressure force in the framework of the inhomogeneous non-linear nonlocal theory

4- Linear And Non-Linear Free Vibration Of Nano Beams Based On A New Fractional Non-Local Theory

5- A new fractional nonlocal model and its application in free vibration of Timoshenko and Euler-Bernoulli beams

Thanks a lot Dr. Rahimi

Thanks

For applications of locally defined fractional differential operators see my recent paper in J. PhysA. (2018) at

http://iopscience.iop.org/article/10.1088/1751-8121/aad345/pdf

A very close but not final preprint can be found here:

Article On generalized and fractional derivatives and their applicat...

Thanks a lot Prof. Mingarelli

Good evening. I want to contribute with the attached file. Published in New Trends in NanoTechnology and Fractional Calculus Applications. This article contains, in addition to a good theoretical foundation, a certain number of examples of the application of fractional calculus to physics. I hope to be helpful

Thanks a lot Prof. Hernandez

Dear Dr Mahdi Saedshoar Heris

The following files can help you:

https://pdfs.semanticscholar.org/db42/15b5ce6ac702e44a645eaa827ad122819699.pdf.

&

https://www.researchgate.net/deref/https%3A%2F%2Fpdfs.semanticscholar.org%2Fd003%2F4719a7dce8a949e3212e759f00626c992dba.pdf

I. Podlubny, Fractional Order System and Fractional Order Controllers, UEF-03-94, Inst. Exp. Phys, Slovak Acad. Sci., Kosice, 1994

I. Podlubny, Fractional Differential Equations , Academic Press, San Diego, 1999, 62-88

Best Regards

Thanks a lot dr. Abdallah

Example of modeling of biological phenomena:

https://www.researchgate.net/profile/Carla_Pinto4/contributions?ev=prf_act

Thanks a lot Prof. Pinto

May I also add the following (ordered with increasing impact and range of application):

(1) Fractional Calculus by Adam Loverro http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=304F1A4581D9686D683E3FBE04E60DDD?doi=10.1.1.363.9101&rep=rep1&type=pdf

(2) "Nonlocal diffusion and applications" by Claudia Bucur and Enrico Valdinoci https://arxiv.org/pdf/1504.08292.pdf

(3) Fractional-in-time semilinear parabolic equations and applications by Ciprian Gal, Mahamadi Warma Article Fractional-in-time semilinear parabolic equations and applications

Cheers,

JS

Thanks a lot Dr Shomberg

Article A new collection of real world applications of fractional ca...

Thanks Dr. Yang

Dear Mahdi,

In Applications of fractional calculus in solving Abel-type integral equations: Surface-volume reaction problem, Appl. Math. Comput. (2017), you may see a fractional integral naturally appears in a surface-volume reaction problem.

Thank you.

Thanks a lot Dr. Katugampola

The Laplace Transformation Method.

Thanks a lot Elsaid Elashry