Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Reproducing Kernel Method for Fractional Derivative with Non-local and Non-singular Kernel Read chapter Read first chapter

2019 | OriginalPaper | Chapter

On the Atangana–Baleanu Derivative and Its Relation to the Fading Memory Concept: The Diffusion Equation Formulation

Author : Jordan Hristov

Published in: Fractional Derivatives with Mittag-Leffler Kernel

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

The constructions of physically adequate forms of the diffusion equation with implementation of the Atangana–Baleanu derivative with Mittag-Leffler exponential kernel have been discussed. The specific form of the corresponding Atangana–Baleanu integral relates it directly to the fading memory concept, following the Boltzmann linear superposition principle with the standard Riemann-Liouville integral as the time-fading term. This approach relates the new fractional operators with non-singular kernel to the classical Riemann-Liouville integral. Using the concept of the fading memory and the specific form of the Atangana–Baleanu integral three forms of the diffusion equation have been investigated. The adequate definition of the flux to gradient relationship has been the main focus of the study resulting in two physically adequate formulations of the diffusion equation. The direct (formalistic) fractionalization of the classical diffusion equation results in physically inadequate relationships.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Development and Elaboration of a Compound Structure of Chaotic Attractors with Atangana–Baleanu Operator
next chapter Numerical Solutions and Pattern Formation Process in Fractional Diffusion-Like Equations
Literature
1.
Atangana, A.: Non validity of index law in fractional calculus: a fractional differential operator with Markovian and non-Markovian properties. Phys. A 505, 688–706 (2018)MathSciNetCrossRef
2.
Atangana, A., Baleanu, D.: New fractional derivatives with non-local and non-singular kernel: theory and application to Heat transfer model. Therm. Sci. 20, 763–769 (2016)CrossRef
3.
Atangana, A., Gómez-Aguilar, J.F.: Fractional derivatives with no-index law property: application to chaos and statistics. Chaos Solitons Fractals 114, 516–535 (2018)MathSciNetCrossRef
4.
Atangana, A., Gómez-Aguilar, J.F.: Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena. Eur. Phys. J. Plus 133, 1–21 (2018)CrossRef
5.
Baleanu, D., Fernandez, A.: On some new properties of fractional derivatives with Mittag-Leffler kernel. Commun. Nonlinear Sci. Numer. Simul. 59, 444–462 (2018)MathSciNetCrossRef
6.
Boltzmann, L.: Zur Theorie der Elastischen Nachwirkung. Sitzungsber. Akad. Wiss. Wien. Mathem.- Naturwiss. 70, 275–300 (1874)
7.
Caputo, M.: Linear model of dissipation whose Q is almost frequency independent-II. Geophys. J. R. Astron. Soc. 13, 529–539 (1969)CrossRef
8.
Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1, 73–85 (2015)
9.
Caputo, M., Fabrizio, M.: Applications of new time and spatial fractional derivatives with exponential kernels. Progr. Fract. Differ. Appl. 2, 1–11 (2016)CrossRef
10.
Coleman, B., Gurtin, M.E.: Equipresence and constitutive equations for rigid heat conductors. Z. Angew. Math. Phys. 18, 188–208 (1967)MathSciNetCrossRef
11.
12.
Coronel-Escamilla, A., Gómez-Aguilar, J.F., Baleanu, D., Córdova-Fraga, T., Escobar-Jiménez, R.F., Olivares-Peregrino, V.H., Qurashi, M.M.A.: Bateman-Feshbach tikochinsky and Caldirola-Kanai oscillators with new fractional differentiation. Entropy 19(2), 1–21 (2017)CrossRef
13.
Fa, K.S., Lenzi, E.K.: Anomalous diffusion, solutions, and the first passage time: Influence of diffusion coefficient. Phys. Rev. E 71, 1–8 (2005)CrossRef
14.
Fernandez, A., Baleanu, D.: The mean value theorem and Taylor’s theorem for fractional derivatives with Mittag-Leffler kernel. Adv. Diff. Eqs. 1, 1–18 (2018)MATH
15.
Fidley, W.N., Lai, J.S., Onaran, K.: Creep and Relaxation of Nonlinear Viscoelastic Materials. North-Hollad, Amsterdam (1976)
16.
Gómez-Aguilar, J.F.: Analytical and numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations. Phys. A: Stat. Mech. Appl. 494, 52–75 (2018)MathSciNetCrossRef
17.
Gómez-Aguilar, J.F., Atangana, A.: New insight in fractional differentiation: power, exponential decay and Mittag-Leffler laws and applications. Eur. Phys. J. Plus 132(1), 1–13 (2017)CrossRef
18.
Gómez-Aguilar, J.F., Escobar-Jiménez, R.F., López-López, M.G., Alvarado-Martínez, V.M.: Atangana-Baleanu fractional derivative applied to electromagnetic waves in dielectric media. J. Electromagn. Waves Appl. 30(15), 1937–1952 (2016)CrossRef
19.
Gómez-Aguilar, J.F., Atangana, A., Morales-Delgado, J.F.: Electrical circuits RC, LC, and RL described by Atangana-Baleanu fractional derivatives. Int. J. Circ. Theor. Appl. 1, 1–22 (2017)
20.
Goodman, T.R.: Application of Integral Methods to Transient Nonlinear Heat Transfer. Advances in Heat Transfer, vol. 1, pp. 51–122. Academic Press, San Diego (1964)
21.
22.
Gurtin, M.E., Pipkin, A.C.: A general theory of heat conduction with finite wave speeds. Arch. Rational Mech. Anal. 31, 113–126 (1968)MathSciNetCrossRef
23.
Havlin, S., Ben, Avraham D.: Diffusion in disordered media. Adv. Phys. 36, 695–798 (1987)CrossRef
24.
Hristov, J.: Approximate solutions to time-fractional models by integral balance approach. In: Cattani, C., Srivastava, H.M., Yang, X.-J. (eds.) Fractional Dynamics, pp. 78–109. De Gruyter Open, Berlin (2015)
25.
Hristov, J.: Transient heat diffusion with a non-singular fading memory: from the Cattaneo constitutive equation with Jeffrey’s kernel to the Caputo-Fabrizio time-fractional derivative. Therm. Sci. 20, 765–770 (2016)
26.
Hristov, J.: Integral solutions to transient nonlinear heat (mass) diffusion with a power-law diffusivity: a semi-infinite medium with fixed boundary conditions. Heat Mass Transf. 52, 635–655 (2016)CrossRef
27.
Hristov, J.: Steady-state heat conduction in a medium with spatial non-singular fading memory: derivation of Caputo-Fabrizio space-fractional derivative with Jeffrey’s kernel and analytical solutions. Therm. Sci. 21, 827–839 (2017)CrossRef
28.
Hristov, J.: Double integral-balance method to the fractional subdiffusion equation: approximate solutions, optimization problems to be resolved and numerical simulations. J. Vib. Control 23, 2795–2818 (2017)MathSciNetCrossRef
29.
Hristov, J.: Space-fractional diffusion with a potential power-law coefficient: transient approximate solution. Progr. Fract. Differ. Appl. 3, 119–139 (2017)
30.
Hristov, J.: Transient space-fractional diffusion with a power-law superdiffusivity: approximate integral-balance approach. Fundam. Inform. 151, 371–388 (2017)MathSciNetCrossRef
31.
Hristov, J.: Fractional derivative with non-singular kernels: from the Caputo-Fabrizio definition and beyond: appraising analysis with emphasis on diffusion models. In: Bhalekar, S. (ed.) Frontiers in Fractional Calculus, pp. 269–342. Bentham Science Publishers, Sharjah (2017)
32.
Hristov, J.: Integral-balance solution to nonlinear subdiffusion equation. In: Bhalekar, S. (ed.) Frontiers in Fractional Calculus, pp. 71–106. Bentham Science Publishers, Sharjah (2017)
33.
Hughes, B.D.: Random Walks and Random Environments, vol. 1. Random Walks. Oxford University Press, New York (1995)MATH
34.
Le Vot, F., Abad, E., Yuste, S.B.: Continuous time random walk model for anomalous diffusion in expanding media. Phys. Rev. E 1–11 (2017)
35.
Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010)CrossRef
36.
Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)MathSciNetCrossRef
37.
Metzler, R., Klafter, J.: The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A Math. Gen. 37, 161–208 (2004)MathSciNetCrossRef
38.
Miller, R.K.: An integrodifferential equation for rigid heat conductors with memory. J. Math. Anal. Appl. 66, 313–332 (1978)MathSciNetCrossRef
39.
Nachlinger, R.R., Weeler, L.: A uniqueness theorem for rigid heat conductors with memory. Q. Appl. Math. 31, 267–273 (1973)MathSciNetCrossRef
40.
Nunciato, J.W.: On heat conduction in materials with memory. Q. Appl. Math. 29, 187–273 (1971)
41.
Nunciato, J.W.: On uniqueness in the linear theory of heat conduction with finite wave speds. SIAM J. Appl. Math. 25, 1–4 (1973)MathSciNetCrossRef
42.
Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of Their Applications. Academic Press, San Diego, Calif, USA (1999)MATH
43.
Saad, K.M., Gómez-Aguilar, J.F.: Analysis of reaction diffusion system via a new fractional derivative with non-singular kernel. Phys. A: Stat. Mech. Appl. 509, 703–716 (2018)MathSciNetCrossRef
44.
Scott-Blair, G.W.: Analytical and integrative aspects of the stress-strain-time problem. J. Sci. Instrum. 21, 80–84 (1944)CrossRef
45.
46.
Tateishi, A.A., Ribeiro, H.V., Lenzi, E.K.: The role of fractional time-derivative operators on anomalous diffusion. Front. Phys. 1, 1–16 (2017)
47.
Volterra, V.: Theory of functional, English edn. Blackie and Son, London (1930)MATH
48.
Yuste, S.B., Lindenberg, K.: Comments on first passage time for anomalous diffusion. Phys. Rev. E 1–8 (2004)
Metadata
Title
On the Atangana–Baleanu Derivative and Its Relation to the Fading Memory Concept: The Diffusion Equation Formulation
Author
Jordan Hristov
Copyright Year
2019
Book
Fractional Derivatives with Mittag-Leffler Kernel

Print ISBN: 978-3-030-11661-3

Electronic ISBN: 978-3-030-11662-0

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-11662-0_11