Research Papers

Energy Straggling Function by Fα-Calculus

[+] Author and Article Information
Saleh Ashrafi

Faculty of Physics,
University of Tabriz,
Tabriz 5166616471, Iran
e-mail: ashrafi@tabrizu.ac.ir

Ali Khalili Golmankhaneh

Faculty of Physics,
University of Tabriz,
Tabriz 5166616471, Iran
e-mail: a.khalili@tabrizu.ac.ir

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 26, 2016; final manuscript received December 25, 2016; published online May 4, 2017. Assoc. Editor: Haiyan Hu.

J. Comput. Nonlinear Dynam 12(5), 051010 (May 04, 2017) (4 pages) Paper No: CND-16-1304; doi: 10.1115/1.4035718 History: Received June 26, 2016; Revised December 25, 2016

In this manuscript, we have used the recently developed Fα-calculus to calculate the energy straggling function through the fractal distributed structures. We have shown that such a fractal structure of space causes the fractal pattern of the energy loss. Also, we have offered Fα-differential Fokker–Planck equation for thick fractal absorbers.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


Hilfer, R. , ed., 2000, Applications of Fractional Calculus in Physics, World Scientific, Singapore.
Carpinteri, A. , and Mainardi, F. , eds., 2014, Fractals and Fractional Calculus in Continuum Mechanics, Vol. 378, Springer, Berlin.
Sabatier, J. , Agrawal, O. P. , and Tenreiro Machado, J. A. , 2007, Advances in Fractional Calculus, Vol. 4, Springer, Berlin.
Ma, L. , and Li, C. , 2016, “ Center Manifold of Fractional Dynamical System,” ASME J. Comput. Nonlinear Dyn., 11(2), p. 021010. [CrossRef]
Velmurugan, G. , and Rakkiyappan, R. , 2016, “ Hybrid Projective Synchronization of Fractional-Order Chaotic Complex Nonlinear Systems With Time Delays,” ASME J. Comput. Nonlinear Dyn., 11(3), p. 031016. [CrossRef]
Baleanu, D. , Muslih, S. I. , Rabei, E. M. , Golmankhaneh, A. K. , and Golmankhaneh, A. K. , 2010, “ On Fractional Dynamics on the Extended Phase Space,” ASME J. Comput. Nonlinear Dyn., 5(4), p. 041011. [CrossRef]
Agrawal, O. P. , 2008, “ Fractional Optimal Control of a Distributed System Using Eigenfunctions,” ASME J. Comput. Nonlinear Dyn., 3(2), p. 021204. [CrossRef]
Özdemir, N. , and Iskender, B. B. , 2010, “ Fractional Order Control of Fractional Diffusion Systems Subject to Input Hysteresis,” ASME J. Comput. Nonlinear Dyn., 5(2), p. 021002. [CrossRef]
Murillo, J. Q. , and Yuste, S. B. , 2011, “ An Explicit Difference Method for Solving Fractional Diffusion and Diffusion-Wave Equations in the Caputo Form,” ASME J. Comput. Nonlinear Dyn., 6(2), p. 021014. [CrossRef]
Zaslavsky, G. M. , 1994, “ Fractional Kinetic Equation for Hamiltonian Chaos,” Phys. D, 76(1), pp. 110–122. [CrossRef]
Riewe, F. , 1997, “ Mechanics With Fractional Derivatives,” Phys. Rev. E, 55(3), p. 3581. [CrossRef]
Bouchaud, J.-P. , and Georges, A. , 1990, “ Anomalous Diffusion in Disordered Media: Statistical Mechanisms, Models and Physical Applications,” Phys. Rep., 195(4–5), pp. 127–293. [CrossRef]
Metzler, R. , Barkai, E. , and Klafter, J. , 1999, “ Anomalous Diffusion and Relaxation Close to Thermal Equilibrium: A Fractional Fokker–Planck Equation Approach,” Phys. Rev. Lett., 82(18), p. 3563. [CrossRef]
Klages, R. , Radons, G. , and Sokolov, I. M. , eds., 2008, Anomalous Transport: Foundations and Applications, Wiley, Hoboken, NJ.
Metzler, R. , Barkai, E. , and Klafter, J. , 1999, “ Anomalous Transport in Disordered Systems Under the Influence of External Fields,” Phys. A, 266(1), pp. 343–350. [CrossRef]
Metzler, R. , Glöckle, W. G. , and Nonnenmacher, T. F. , 1994, “ Fractional Model Equation for Anomalous Diffusion,” Phys. A, 211(1), pp. 13–24. [CrossRef]
Zaslavsky, G. M. , 2002, “ Chaos, Fractional Kinetics, and Anomalous Transport,” Phys. Rep., 371(6), pp. 461–580. [CrossRef]
Zaslavsky, G. M. , 2005, “ Hamiltonian Chaos and Fractional Dynamics,” Oxford University Press, Oxford, UK.
Zaslavsky, G. M. , 2007, The Physics of Chaos in Hamiltonian Systems, Imperial College Press, London, UK.
Ben-Avraham, D. , and Havlin, S. , 2000, Diffusion and Reactions in Fractals and Disordered Systems, Cambridge University Press, New York.
O'Shaughnessy, B. , and Procaccia, I. , 1985, “ Analytical Solutions for Diffusion on Fractal Objects,” Phys. Rev. Lett., 54(5), p. 455. [CrossRef] [PubMed]
Mainardi, F. , Pagnini, G. , and Gorenflo, R. , 2007, “ Some Aspects of Fractional Diffusion Equation of Single and Distributed Order,” Appl. Math. Comp., 187(1), pp. 295–305. [CrossRef]
Gorenflo, R. , Mainardi, F. , Moretti, D. , Pagnini, G. , and Paradisi, P. , 2002, “ Discrete Random Walk Models for Space-Time Fractional Diffusion,” Chem. Phys., 284(1–2), pp. 521–544. [CrossRef]
Gafiychuk, V. , and Datsko, B. , 2012, “ Different Types of Instabilities and Complex Dynamics in Reaction-Diffusion Systems With Fractional Derivatives,” ASME J. Comput. Nonlinear Dyn., 7(3), p. 031001. [CrossRef]
Parvate, A. , Satin, S. , and Gangal, A. D. , 2011, “ Calculus on Fractal Curves in Rn,” Fractals, 19(1), pp. 15–27. [CrossRef]
Golmankhaneh, A. K. , Golmankhaneh, A. K. , and Baleanu, D. , 2015, “ About Schrödinger Equation on Fractals Curves Imbedding in R3,” Int. J. Theor. Phys., 54(4), pp. 1275–1282. [CrossRef]
Golmankhaneh, A. K. , Golmankhaneh, A. K. , and Baleanu, D. , 2013, “ Lagrangian and Hamiltonian Mechanics on Fractals Subset of Real-Line,” Int. J. Theor. Phys., 52(11), pp. 4210–4217. [CrossRef]
Kolwankar, K. M. , and Gangal, A. D. , 1998, “ Local Fractional Fokker–Planck Equation,” Phys. Rev. Lett., 80(2), p. 214. [CrossRef]
Kolwankar, K. M. , and Gangal, A. D. , 1996, “ Fractional Differentiability of Nowhere Differentiable Functions and Dimensions,” Chaos: Interdiscip. J. Nonlinear Sci., 6(4), pp. 505–513. [CrossRef]
Kolwankar, K. M. , and Gangal, A. D. , 1997, “ Hölder Exponents of Irregular Signals and Local Fractional Derivatives,” Pramana, 48(1), pp. 49–68. [CrossRef]
Kolwankar, K. M. , and Gangal, A. D. , 1999, “ Local Fractional Calculus: A Calculus for Fractal Space-Time,” Fractals, Springer, Berlin, pp. 171–181.
Leo, W. R. , 1994, Techniques for Nuclear and Particle Physics Experiments, Springer-Verlag, Berlin.
Fano, U. , 1963, “Penetration of Protons, Alpha Particles, and Mesons,” Ann. Rev. Nucl. Sci., 13(1), pp. 1–66.
Vavilov, P. V. , 1957, “Ionization Losses of High-Energy heavy Particles,” Sov. Phys. JETP., 5(4), pp. 749–751.
Symon, K. R. , 1948, “Fluctuations in Energy Lost by High Energy Charged Particles in Passing through Matter,” Ph.D. thesis, Harvard University, Cambridge, MA.
Tsoulfanidis, N. , and Sheldon, L. , 2011, Measurement and Detection of Radiation, CRC Press, Boca Raton, FL.
Knoll, G. F. , 2010, Radiation Detection and Measurement, Wiley, Hoboken, NJ.
Payne, M. G. , 1969, “ Energy Straggling of Heavy Charged Particles in Thick Absorbers,” Phys. Rev., 185(2), p. 611. [CrossRef]
Landau, L. , 1944, “ On the Energy Loss of Fast Particles by Ionization,” J. Phys., 8(4), pp. 201–205.
Hille, E. , and Tamarkin, J. D. , 1929, “ Remarks on a Known Example of a Monotone Continuous Function,” Am. Math. Mon., 36(5), pp. 255–264. [CrossRef]




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In