Startseite Two occurrences of fractional actions in nonlinear dynamics
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Two occurrences of fractional actions in nonlinear dynamics

  • Rami Ahmad El-Nabulsi EMAIL logo
Veröffentlicht/Copyright: 12. August 2021
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International Journal of Nonlinear Sciences and Numerical Simulation
Aus der Zeitschrift International Journal of Nonlinear Sciences and Numerical Simulation Band 24 Heft 6

Abstract

Fractional theories have gained recently an increasing interest in dynamical systems since they offer some solutions to a number of puzzles in particular nonconservative and dissipative issues. Most of fractional dynamical theories are formulated by means of one occurrence of action that group kinetic energy and potential energy in one single package. In this work, we introduce a modified fractional dynamics based on the occurrence of two independent actions where fractional and nonfractional Euler–Lagrange equations are mixed together. We show that their combination divulge some properties that offer new insights in nonlinear dynamics. In particular, it was observed that a large family of solutions that could be used to model dissipative systems may be derived from the action with two occurrences of integrals. Moreover, damping systems may be obtained by means of simple Lagrangian functionals.

Keywords: action with two-occurrences of integrals; dissipation; nonlinear dynamics
MSC 2010: 49S05; 34C15
Corresponding author: Rami Ahmad El-Nabulsi, Faculty of Science, Research Center for Quantum Technology, Chiang Mai University, Chiang Mai 50200, Thailand; Department of Physics and Materials Science, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand; and Mathematics and Physics Divisions, Athens Institute for Education and Research, 8 Valaoritou Street, Kolonaki, 10671, Athens, Greece, E-mail: el-nabulsi@atiner.gr

Funding source: Chiang Mai University 10.13039/501100002842

Acknowledgments

The author is indebted to the group of anonymous referees for their useful comments and valuable suggestions.

  1. Author contribution: The author has accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: The author would like to thank Chiang Mai University for funding this research.

  3. Conflict of interest statement: The author declares that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

  4. Data availability statement: The author confirms the absence of sharing data.

Appendix A: Proof of Theorem 1

We give x(τ) an increment h(x) with x ɛ (τ) = x(τ) + ɛh(τ), ɛ ≪ 1 and h(x) is a differentiable function satisfying boundary conditions h(a) = h(t) = 0.

We let S 1 , ε = ∫ a t L 1 , ε ( x ̇ ε ( τ ) , x ε ( τ ) , τ ) d τ and S 2 , ε = ∫ a t L 2 , ε ( x ̇ ε ( τ ) , x ε ( τ ) , τ ) d τ . Having S ɛ = [S 1,ɛ , S 2,ɛ ] we have:

d S ε d ε = ∂ S ε ∂ ε ︸ = 0 s i n c e S d o e s n o t d e p e n d e x p l i c i t l y o n ε + ∂ S ε ∂ S 1 , ε d S 1 , ε d ε + ∂ S ε ∂ S 2 , ε d S 2 , ε d ε .

Here we have:

d S 1 , ε d ε = ∫ a t h ( τ ) ∂ L 1 ∂ x ε + d h d τ ∂ L 1 ∂ x ̇ ε d τ ,

d S 2 , ε d ε = 1 Γ ( α ) ∫ a t h ( τ ) ( t − τ ) α − 1 ∂ L 2 ∂ x ε + d h d τ ∂ ( t − τ ) α − 1 L 2 ∂ x ̇ ε d τ .

Using the following integration by parts for second terms of both integrals yield:

∫ a t d h d τ ∂ L 1 ∂ x ̇ ε d τ = h ( τ ) ∂ L 1 ∂ x ̇ ε a t ︸ = 0 h ( a ) = h ( τ ) = 0 − ∫ a b h ( τ ) d d t ∂ L 1 ∂ x ̇ ε d τ ,

∫ a t d h d τ ∂ ( t − τ ) α − 1 L 1 ∂ x ̇ ε d τ = h ( τ ) ( t − τ ) α − 1 ∂ L 1 ∂ x ̇ ε a t ︸ = 0 h ( a ) = h ( τ ) = 0 − ∫ a b h ( τ ) d d t ∂ ( t − τ ) α − 1 L 1 ∂ x ̇ ε d τ ,

we find:

d S 1 , ε d ε = ∫ a t h ( τ ) ∂ L 1 ∂ x ε − d d t ∂ L 1 ∂ x ̇ ε d τ ,

d S 2 , ε d ε = 1 Γ ( α ) ∫ a t h ( τ ) ( t − τ ) α − 1 ∂ L 2 ∂ x ε − d d t ( t − τ ) α − 1 ∂ L 1 ∂ x ̇ ε d τ .

After substituting into d S ε d ε , we get:

d S ε d ε = ∫ a t h ( τ ) ∂ S ε ∂ S 1 , ε ∂ L 1 ∂ x ε − d d t ∂ L 1 ∂ x ̇ ε + 1 Γ ( α ) ∂ S ε ∂ S 2 , ε ( t − τ ) α − 1 ∂ L 2 ∂ x ε − d d t ( t − τ ) α − 1 ∂ L 1 ∂ x ̇ ε d τ .

A necessary condition for S = [S 1, S 2] to have an extremum is that d S ε d ε = 0 . For ɛ = 0, we get:

∫ a t h ( τ ) ∂ S ε ∂ S 1 , ε ∂ L 1 ∂ x ε − d d t ∂ L 1 ∂ x ̇ ε + 1 Γ ( α ) ∂ S ε ∂ S 2 , ε ( t − τ ) α − 1 ∂ L 2 ∂ x ε − d d t ( t − τ ) α − 1 ∂ L 1 ∂ x ̇ ε d τ = 0 .

For any increment h, the previous integral must be equal to zero, and therefore, we get:

∂ L 1 ∂ x ε − d d t ∂ L 1 ∂ x ̇ ε ∂ L 2 ∂ x ε − d d t ∂ L 2 ∂ x ̇ ε + α − 1 t − τ ∂ L 2 ∂ x ̇ ε = − ( t − τ ) α − 1 Γ ( α ) ∂ S ε ∂ S 2 , ε ∂ S ε ∂ S 1 , ε .

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Appendix B: Proof of Theorem 2

Let x ɛ (τ) = x(τ) + ɛQ(τ), ɛ ≪ 1 where Q ( τ ) ∈ C 1 ( [ a , b ] ; R ) satisfying the boundary conditions Q(a) = Q(b) = 0 and that its left and right fractional derivatives are continuous. Using the chain rule, we calculate at the beginning the derivative of S = [S α , S β ] with respect to ɛ as follows:

d S d ε = ∂ S ∂ ε ︸ = 0 s i n c e S d o e s n o t d e p e n d e x p l i c i t l y o n ε + ∂ S ∂ S 1 d S 1 d ε + ∂ S ∂ S 2 d S 2 d ε .

Here:

d S 1 d ε = 1 Γ ( α ) ∫ a b Q ( τ ) ( t − τ ) β − 1 ∂ L 1 ∂ x ε + D τ α a Q ( τ ) ( t − τ ) β − 1 ∂ L 1 ∂ D τ α a x ε d τ ,

d S 2 d ε = 1 Γ ( β ) ∫ a b Q ( τ ) ( t − τ ) γ − 1 ∂ L 2 ∂ x ε + D τ β a Q ( τ ) ( t − τ ) γ − 1 ∂ L 2 ∂ D τ β a x ε d τ .

Using the integration by parts rule for fractional derivatives:

∫ a b D τ α a Q ( τ ) ( t − τ ) β − 1 ∂ L 1 ∂ D τ α a x ε d τ = Q ( τ ) I b 1 − α τ ( t − τ ) β − 1 ∂ L 1 ∂ D τ α a x ε a b ︸ = 0 Q ( a ) = Q ( b ) = 0 + ∫ a b Q ( τ ) D b α τ ( t − τ ) β − 1 ∂ L 1 ∂ D τ α a x ε d τ ,

∫ a b D τ β a Q ( τ ) ( t − τ ) γ − 1 ∂ L 2 ∂ D τ β a x ε d τ = Q ( τ ) I b 1 − β τ ( t − τ ) γ − 1 ∂ L 2 ∂ D τ β a x ε a b ︸ = 0 Q ( a ) = Q ( b ) = 0 + ∫ a b Q ( τ ) D b β τ ( t − τ ) γ − 1 ∂ L 2 ∂ D τ β a x ε d τ .

I b α t q = ∫ t b ( τ − t ) α − 1 q ( τ ) Γ ( α ) d τ , t < b , τ ∈ [ a , b ] is the right Riemann–Liouville fractional integral of order α, we can write:

d S 1 d ε = ∫ a b Q ( τ ) ( t − τ ) β − 1 ∂ L 1 ∂ x ε + τ D b α ( t − τ ) β − 1 ∂ L 1 ∂ D τ α a x ε d τ ,

d S 2 d ε = ∫ a b Q ( τ ) ( t − τ ) γ − 1 ∂ L 2 ∂ x ε + τ D b β ( t − τ ) γ − 1 ∂ L 2 ∂ a D τ β x ε d τ .

After substituting into d S d ε , we get:

d S d ε = ∂ S ∂ S 1 ∫ a b Q ( τ ) ( t − τ ) β − 1 ∂ L 1 ∂ x ε + τ D b α ( t − τ ) β − 1 ∂ L 1 ∂ D τ α a x ε d τ + ∂ S ∂ S 2 ∫ a b Q ( τ ) ( t − τ ) γ − 1 ∂ L 2 ∂ x ε + τ D b β ( t − τ ) γ − 1 ∂ L 2 ∂ a D τ β x ε d τ , = ∂ S ∂ S 1 ∫ a b Q ( τ ) ( t − τ ) β − 1 ∂ L 1 ∂ x ε + ( t − τ ) β − 1 D b α τ ∂ L 1 ∂ D τ α a x ε + τ D b α ( t − τ ) β − 1 ∂ L 1 ∂ D τ α a x ε d τ + ∂ S ∂ S 2 ∫ a b Q ( τ ) ( t − τ ) γ − 1 ∂ L 2 ∂ x ε + ( t − τ ) γ − 1 D b β τ ∂ L 2 ∂ a D τ β x ε + τ D b β ( t − τ ) γ − 1 ∂ L 2 ∂ a D τ β x ε d τ .

Using the following derivatives:

D b α τ ( t − τ ) β − 1 = ( − 1 ) β + 1 ( b − t ) β − α − 1 Γ ( 1 − α ) , β > α ,

D b β τ ( t − τ ) γ − 1 = ( − 1 ) γ + 1 ( b − t ) γ − β − 1 Γ ( 1 − β ) , γ > β ,

we find:

d S d ε = ∂ S ∂ S 1 ∫ a b Q ( τ ) ( t − τ ) β − 1 ∂ L 1 ∂ x ε + ( t − τ ) β − 1 D b α τ ∂ L 1 ∂ D τ α a x ε + ( − 1 ) β + 1 ( b − t ) β − α − 1 Γ ( 1 − α ) ∂ L 1 ∂ D τ α a x ε d τ + ∂ S ∂ S 2 ∫ a b Q ( τ ) ( t − τ ) α − 1 ∂ L 2 ∂ x ε + ( t − τ ) α − 1 D b β τ ∂ L 2 ∂ a D τ β x ε + ( − 1 ) γ + 1 ( b − t ) γ − β − 1 Γ ( 1 − β ) ∂ L 2 ∂ a D τ β x ε d τ .

A necessary condition for S = [S 1, S 2] to have an extremum is that d S d ε = 0 . For any increment Q we get:

∂ S ∂ S 1 ( t − τ ) β − 1 ∂ L 1 ∂ x ε + ( t − τ ) β − 1 D b α τ ∂ L 1 ∂ D τ α a x ε + ( − 1 ) β + 1 ( b − t ) β − α − 1 Γ ( 1 − α ) ∂ L 1 ∂ D τ α a x ε

+ ∂ S ∂ S 2 ( t − τ ) α − 1 ∂ L 2 ∂ x ε + ( t − τ ) α − 1 D b β τ ∂ L 2 ∂ a D τ β x ε + ( − 1 ) γ + 1 ( b − t ) γ − β − 1 Γ ( 1 − β ) ∂ L 2 ∂ a D τ β x ε = 0 ,

or

∂ L 1 ∂ x ε + τ D b α ∂ L 1 ∂ D τ α a x ε + ( − 1 ) β + 1 ( b − t ) β − α − 1 Γ ( 1 − α ) ( t − τ ) β − 1 ∂ L 1 ∂ D τ α a x ε ∂ L 2 ∂ x ε + τ D b β ∂ L 2 ∂ a D τ β x ε + ( − 1 ) γ + 1 ( b − t ) γ − β − 1 Γ ( 1 − β ) ( t − τ ) α − 1 ∂ L 2 ∂ a D τ β x ε = − ( t − τ ) α − β ∂ S ∂ S 2 ∂ S ∂ S 1 .

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Appendix C: Proof of Theorem 3

Let again x ɛ (τ) = x(τ) + ɛY(τ), ɛ ≪ 1 where Y ( τ ) ∈ C 1 ( [ a , b ] ; R ) satisfying the boundary conditions Y(a) = Y(b) = 0 and that its left and right fractional derivatives are continuous. From S = [S α , S β ] we have

d S d ε = ∂ S ∂ ε ︸ = 0 s i n c e S d o e s n o t d e p e n d e x p l i c i t l y o n ε + ∂ S ∂ S 1 d S 1 d ε + ∂ S ∂ S 2 d S 2 d ε ,

where

d S 1 d ε = 1 Γ ( α ) ∫ a b Y ( τ ) ( t − τ ) β − 1 ∂ L 1 ∂ x ε + a D τ α Y ( τ ) ( t − τ ) β − 1 ∂ L 1 ∂ D τ α a x ε + τ D b χ Y ( τ ) ( t − τ ) β − 1 ∂ L 1 ∂ τ D b χ q ε d τ ,

d S 2 d ε = 1 Γ ( γ ) ∫ a b Y ( τ ) ( t − τ ) γ − 1 ∂ L 2 ∂ x ε + a D τ η Y ( τ ) ( t − τ ) γ − 1 ∂ L 2 ∂ a D τ η x ε + τ D b ζ Y ( τ ) ( t − τ ) γ − 1 ∂ L 2 ∂ τ D b ζ q ε d τ .

Using the integration by parts rule for fractional derivatives:

∫ a b D τ α a Y ( τ ) ( t − τ ) β − 1 ∂ L 1 ∂ D τ α a x ε d τ = Y ( τ ) I b 1 − α τ ( t − τ ) β − 1 ∂ L 1 ∂ D τ α a x ε a b ︸ = 0 Y ( a ) = Y ( b ) = 0 + ∫ a b Y ( τ ) D b α τ ( t − τ ) β − 1 ∂ L 1 ∂ D τ α a x ε d τ ,

∫ a b D b ζ τ Y ( τ ) ( t − τ ) γ − 1 ∂ L 2 ∂ a D τ ζ x ε d τ = Y ( τ ) I τ 1 − ζ a ( t − τ ) γ − 1 ∂ L 2 ∂ a D τ ζ x ε a b ︸ = 0 Y ( a ) = Y ( b ) = 0 + ∫ a b Y ( τ ) τ D b ζ ( t − τ ) γ − 1 ∂ L 2 ∂ a D τ ζ x ε d τ ,

we can write:

d S 1 d ε = 1 Γ ( α ) ∫ a b Y ( τ ) ( t − τ ) β − 1 ∂ L 1 ∂ x ε + Y ( τ ) D b α τ ( t − τ ) β − 1 ∂ L 1 ∂ D τ α a x ε + Y ( τ ) τ D b χ ( t − τ ) β − 1 ∂ L 2 ∂ a D τ χ x ε d τ ,

d S 2 d ε = 1 Γ ( β ) ∫ a b Y ( τ ) ( t − τ ) γ − 1 ∂ L 2 ∂ x ε + Y ( τ ) τ D b ζ ( t − τ ) γ − 1 ∂ L 1 ∂ a D τ ζ x ε + Y ( τ ) τ D b ζ ( t − τ ) γ − 1 ∂ L 2 ∂ a D τ ζ x ε d τ .

Substituting into d S d ε , we get:

d S d ε = ∂ S ∂ S 1 ∫ a b Y ( τ ) ( t − τ ) β − 1 ∂ L 1 ∂ x ε + τ D b α ( t − τ ) β − 1 ∂ L 1 ∂ D τ α a x ε + τ D b χ ( t − τ ) β − 1 ∂ L 2 ∂ a D τ χ x ε d τ + ∂ S ∂ S 2 ∫ a b Y ( τ ) ( t − τ ) γ − 1 ∂ L 2 ∂ x ε + D b η τ ( t − τ ) γ − 1 ∂ L 2 ∂ a D τ η x ε + τ D b ζ ( t − τ ) γ − 1 ∂ L 2 ∂ a D τ ζ x ε d τ .

Since we have

D b α τ ( t − τ ) β − 1 ∂ L 1 ∂ D τ α a x ε = ( t − τ ) β − 1 D b α τ ∂ L 1 ∂ D τ α a x ε + τ D b α ( t − τ ) β − 1 ∂ L 1 ∂ D τ α a x ε ,

D b χ τ ( t − τ ) β − 1 ∂ L 2 ∂ a D τ χ x ε = ( t − τ ) β − 1 D b χ τ ∂ L 2 ∂ a D τ χ x ε + τ D b χ ( t − τ ) β − 1 ∂ L 2 ∂ a D τ χ x ε ,

D b η τ ( t − τ ) γ − 1 ∂ L 2 ∂ a D τ η x ε = ( t − τ ) γ − 1 D b η τ ∂ L 2 ∂ a D τ η x ε + D b η τ ( t − τ ) γ − 1 ∂ L 2 ∂ a D τ η x ε ,

D b ζ τ ( t − τ ) γ − 1 ∂ L 2 ∂ a D τ ζ x ε = ( t − τ ) γ − 1 D b ζ τ ∂ L 2 ∂ a D τ ζ x ε + τ D b ζ ( t − τ ) γ − 1 ∂ L 2 ∂ a D τ ζ x ε ,

with

D b α τ ( t − τ ) β − 1 = ( − 1 ) β + 1 ( b − t ) β − α − 1 Γ ( 1 − α ) , β > α ,

D b χ τ ( t − τ ) β − 1 = ( − 1 ) β + 1 ( b − t ) β − χ − 1 Γ ( 1 − χ ) , β > χ ,

D b η τ ( t − τ ) γ − 1 = ( − 1 ) γ + 1 ( b − t ) γ − η − 1 Γ ( 1 − η ) , γ > η ,

D b ζ τ ( t − τ ) γ − 1 = ( − 1 ) γ + 1 ( b − t ) γ − ζ − 1 Γ ( 1 − ζ ) , γ > ζ ,

we find:

d S d ε = ∂ S ∂ S 1 ∫ a b Y ( τ ) ( t − τ ) β − 1 ∂ L 1 ∂ x ε + ( t − τ ) β − 1 D b α τ ∂ L 1 ∂ D τ α a x ε + ( − 1 ) β + 1 ( b − t ) β − α − 1 Γ ( 1 − α ) ∂ L 1 ∂ D τ α a x ε + ( t − τ ) β − 1 D b χ τ ∂ L 1 ∂ a D τ χ x ε + ( − 1 ) β + 1 ( b − t ) β − χ − 1 Γ ( 1 − χ ) ∂ L 1 ∂ a D τ β x ε d τ + ∂ S ∂ S 2 ∫ a b Y ( τ ) ( t − τ ) γ − 1 ∂ L 2 ∂ x ε + ( t − τ ) γ − 1 D b η τ ∂ L 2 ∂ a D τ η x ε + D b η τ ( t − τ ) γ − 1 ∂ L 2 ∂ a D τ η x ε + ( t − τ ) γ − 1 D b ζ τ ∂ L 2 ∂ a D τ ζ x ε + ( − 1 ) γ + 1 ( b − t ) γ − ζ − 1 Γ ( 1 − ζ ) ∂ L 2 ∂ a D τ ζ x ε d τ .

Since d S d ε = 0 is the necessary condition for S = [S 1, S 2] to have an extremum, then for any increment Y we get

∂ S ∂ S 1 ( t − τ ) β − 1 ∂ L 1 ∂ x ε + ( t − τ ) β − 1 D b α τ ∂ L 1 ∂ D τ α a x ε + ( − 1 ) β + 1 ( b − t ) β − α − 1 Γ ( 1 − α ) ∂ L 1 ∂ D τ α a x ε

+ ( t − τ ) β − 1 D b χ τ ∂ L 1 ∂ a D τ χ x ε + ( − 1 ) β + 1 ( b − t ) β − χ − 1 Γ ( 1 − χ ) ∂ L 1 ∂ a D τ β x ε d τ

+ ∂ S ∂ S 2 ( t − τ ) γ − 1 ∂ L 2 ∂ x ε + ( t − τ ) γ − 1 D b η τ ∂ L 2 ∂ a D τ η x ε + ( − 1 ) γ + 1 ( b − t ) γ − η − 1 Γ ( 1 − η ) ∂ L 2 ∂ a D τ η x ε

+ ( t − τ ) γ − 1 D b ζ τ ∂ L 2 ∂ a D τ ζ x ε + ( − 1 ) γ + 1 ( b − t ) γ − ζ − 1 Γ ( 1 − ζ ) ∂ L 2 ∂ a D τ ζ x ε d τ ,

or

∂ L 1 ∂ x ε + τ D b α ∂ L 1 ∂ D τ α a x ε + D b χ τ ∂ L 1 ∂ a D τ χ x ε + ( − 1 ) β + 1 ( b − t ) β − α − 1 Γ ( 1 − α ) ( t − τ ) β − 1 ∂ L 1 ∂ D τ α a x ε + Γ ( 1 − α ) ( b − t ) α − χ Γ ( 1 − χ ) ∂ L 1 ∂ a D τ χ x ε ∂ L 2 ∂ x ε + D b η τ ∂ L 2 ∂ a D τ η x ε + D b ζ τ ∂ L 2 ∂ a D τ ζ x ε + ( − 1 ) γ + 1 ( b − t ) γ − η − 1 Γ ( 1 − η ) ( t − τ ) γ − 1 ∂ L 2 ∂ a D τ η x ε + Γ ( 1 − η ) ( b − t ) η − ζ Γ ( 1 − ζ ) ∂ L 2 ∂ a D τ ζ x ε = − ( t − τ ) γ − β ∂ S ∂ S 2 ∂ S ∂ S 1

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References

[1] K. S. Miller and B. Ross, Eds. An Introduction to the Fractional Calculus and Fractional Differential Equations, 1st ed. New York, John Wiley & Sons, 1993.Suche in Google Scholar

[2] K. B. Oldham and J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order (Mathematics in Science and Engineering, V), Hardcover Publisher, California, Academic Press, 1974.Suche in Google Scholar

[3] I. Podlubny, Fractional Differential Equations, San Diego, CA, Academic Press, 1999.Suche in Google Scholar

[4] M. Suzuki, “Unified variational theory of reversible and irreversible dynamics-Discovery of dissipative Lagrangians weighted in time,” Proc. Jpn. Acad. B, vol. 95, pp. 419–429, 2019. https://doi.org/10.2183/pjab.95.029.Suche in Google Scholar PubMed PubMed Central

[5] B. N. Lundstrom, M. H. Higgs, W. J. Spain, and A. L. Fairhall, “Fractional differentiation by neocortical pyramidal neurons,” Nat. Neurosci., vol. 11, pp. 1335–1342, 2008. https://doi.org/10.1038/nn.2212.Suche in Google Scholar PubMed PubMed Central

[6] R. L. Magin, “Fractional calculus models of complex dynamics in biological tissues,” Comput. Math. Appl., vol. 59, pp. 1586–1593, 2010. https://doi.org/10.1016/j.camwa.2009.08.039.Suche in Google Scholar

[7] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, London, Imperial College Press, 2010.10.1142/p614Suche in Google Scholar

[8] R. Najafi, F. Bahrami, and M. S. Hashemi, “Classical and nonclassical Lie symmetry analysis to a class of nonlinear time-fractional differential equations,” Nonlinear Dynam., vol. 87, pp. 1785–1796, 2017. https://doi.org/10.1007/s11071-016-3152-z.Suche in Google Scholar

[9] O. Agrawal, “Formulation of Euler–Lagrange equations for fractional variational problems,” J. Math. Anal. Appl., vol. 272, pp. 368–379, 2002. https://doi.org/10.1016/s0022-247x(02)00180-4.Suche in Google Scholar

[10] R. Almeida, A. B. Malinowksa, and D. F. M. Torres, “A fractional calculus of variations for multiple integrals with application to vibrating string,” J. Math. Phys., vol. 51, no. 3, pp. 035503–035515, 2010. https://doi.org/10.1063/1.3319559.Suche in Google Scholar

[11] T. Bakkyaraj and R. Sahadevan, “Invariant analysis of nonlinear fractional ordinary differential equations with Riemann–Liouville fractional derivative,” Nonlinear Dynam., vol. 80, pp. 447–455, 2015. https://doi.org/10.1007/s11071-014-1881-4.Suche in Google Scholar

[12] R. A. El-Nabulsi, “A fractional approach to non-conservative Lagrangian dynamical systems,” Fiz. Atmos., vol. 14, pp. 289–298, 2005.Suche in Google Scholar

[13] R. A. El-Nabulsi and D. F. M. Torres, “Fractional action-like variational problems,” J. Math. Phys., vol. 49, 2008, Art no. 052521. https://doi.org/10.1063/1.2929662.Suche in Google Scholar

[14] R. A. El-Nabulsi, “Fractional Dirac operators and deformed field theory on Clifford algebra,” Chaos, Solit. Fractals, vol. 42, pp. 2614–2622, 2009. https://doi.org/10.1016/j.chaos.2009.04.002.Suche in Google Scholar

[15] R. A. El-Nabulsi, “Fractional dynamics, fractional weak bosons masses and physics beyond the standard model,” Chaos, Solit. Fractals, vol. 4, pp. 2262–2270, 2009.10.1016/j.chaos.2008.08.033Suche in Google Scholar

[16] R. A. El-Nabulsi, “Complexified quantum field theory and “mass without mass” from multidimensional fractional action like variational approach with dynamical fractional exponents,” Chaos, Solit. Fractals, vol. 42, pp. 2384–2398, 2009. https://doi.org/10.1016/j.chaos.2009.03.115.Suche in Google Scholar

[17] R. A. El-Nabulsi and G.-c. Wu, “Fractional complexified field theory from Saxena–Kumbhat fractional integral, fractional derivative of order (α, β) and dynamical fractional integral exponent,” African Disp. J. Math., vol. 13, pp. 45–61, 2012.Suche in Google Scholar

[18] R. A. El-Nabulsi, “Fractional functional with two occurrences of integrals and asymptotic optimal change of drift in the Black–Scholes model,” Acta Math. Viet., vol. 40, p. 689, 2015. https://doi.org/10.1007/s40306-014-0079-7.Suche in Google Scholar

[19] F. Riewe, “Nonconservative Lagrangian and Hamiltonian mechanics,” Phys. Rev. E, vol. 53, no. 2, pp. 1890–1899, 1996. https://doi.org/10.1103/physreve.53.1890.Suche in Google Scholar PubMed

[20] D. Baleanu and S. I. Muslih, “Lagrangian formulation of classical fields within Riemann–Liouville fractional derivatives,” Phys. Scripta, vol. 72, pp. 119–121, 2005. https://doi.org/10.1238/physica.regular.072a00119.Suche in Google Scholar

[21] D. Baleanu, “Fractional variational principles in action,” Phys. Scripta, vol. T136, 2009, Art no. 014006. https://doi.org/10.1088/0031-8949/2009/t136/014006.Suche in Google Scholar

[22] D. Ferst, “Pricing Asian options by importance sampling,” DIPLOMARBEIT, Wien, Im Mai, Ausgefuhrt am Institut fur Wirtschaftsmathematik der Technischen Universitat Wien, 2012.Suche in Google Scholar

[23] Z. E. Musielak, “General conditions for the existence of non-standard Lagrangians for dissipative dynamical systems,” Chaos, Solit. Fractals, vol. 42, no. 15, pp. 2645–2652, 2009. https://doi.org/10.1016/j.chaos.2009.03.171.Suche in Google Scholar

[24] A. B. Malinowska, M. R. S. Ammi, and D. F. M. Torres, “Composition functionals in fractional calculus of variations,” Comm. Frac. Calc., vol. 1, pp. 32–40, 2010.Suche in Google Scholar

[25] N.-e. Tatar, “A wave equation with fractional damping,” J. Math. Anal. Appl., vol. 22, pp. 609–617, 2003. https://doi.org/10.4171/zaa/1165.Suche in Google Scholar

[26] A. Tofighi, “The intrinsic damping of the fractional oscillator,” Phys. A: Stat. Mech. Appl., vol. 329, pp. 29–34, 2003. https://doi.org/10.1016/s0378-4371(03)00598-3.Suche in Google Scholar

[27] A. Al-rabtah, V. S. Ertürk, and S. Momani, “Solutions of a fractional oscillator by using differential transform method,” Comput. Math. Appl., vol. 59, pp. 1356–1362, 2010. https://doi.org/10.1016/j.camwa.2009.06.036.Suche in Google Scholar

[28] W. S. Chung, and M. Jung, “Fractional damped oscillators and fractional forced oscillators,” J. Kor. Phys. Soc., vol. 64, pp. 186–191, 2014. https://doi.org/10.3938/jkps.64.186.Suche in Google Scholar

[29] T. Odzijewicz, A. B. Malinowska, and D. F. M. Torres, “Fractional calculus of variations in terms of a generalized fractional integral with applications to physics,” Abstr. Appl. Anal., vol. 2012, 2012, Art no. 871912. https://doi.org/10.1155/2012/871912.Suche in Google Scholar

[30] M. R. Alaimia and N.-e. Tatar, “Blow up for the wave equation with a fractional damping,” J. Appl. Anal., vol. 11, pp. 133–144, 2005. https://doi.org/10.1515/jaa.2005.133.Suche in Google Scholar

[31] V. Georgiev and G. Todorova, “Existence of a solution of the wave equation with nonlinear damping and source terms,” J. Differ. Equ., vol. 109, pp. 295–308, 1994. https://doi.org/10.1006/jdeq.1994.1051.Suche in Google Scholar

[32] D. Ingman and J. Suzdalnitsky, “Control of damping oscillations by fractional differential operator with time-dependent order,” Comput. Methods Appl. Mech. Eng., vol. 193, pp. 5585–5595, 2004. https://doi.org/10.1016/j.cma.2004.06.029.Suche in Google Scholar

[33] M. Kirane and N.-e. Tatar, “Exponential growth for a fractionally damped wave equation,” Z. Anal. Anw., vol. 22, pp. 167–177, 2003. https://doi.org/10.4171/zaa/1137.Suche in Google Scholar

[34] B. N. N. Achar, J. W. Hanneken, T. Enck, and T. Clarke, “Dynamics of the fractional oscillator,” Phys. A: Stat. Mech. Appl., vol. 297, pp. 361–367, 2001. https://doi.org/10.1016/s0378-4371(01)00200-x.Suche in Google Scholar

[35] V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Beijing, Berlin, Springer, HEP, 2010.10.1007/978-3-642-14003-7_11Suche in Google Scholar
Received: 2020-12-15
Revised: 2021-06-09
Accepted: 2021-07-20
Published Online: 2021-08-12

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Original Research Articles
  3. Frequency responses for induced neural transmembrane potential by electromagnetic waves (1 kHz to 1 GHz)
  4. Investigating existence results for fractional evolution inclusions with order r ∈ (1, 2) in Banach space
  5. Optimal control of non-instantaneous impulsive second-order stochastic McKean–Vlasov evolution system with Clarke subdifferential
  6. Generalized forms of fractional Euler and Runge–Kutta methods using non-uniform grid
  7. Controllability of coupled fractional integrodifferential equations
  8. A new generalized approach to study the existence of solutions of nonlinear fractional boundary value problems
  9. Rational soliton solutions in the nonlocal coupled complex modified Korteweg–de Vries equations
  10. Buoyancy driven flow characteristics inside a cavity equiped with diamond elliptic array
  11. Analysis and numerical effects of time-delayed rabies epidemic model with diffusion
  12. Two occurrences of fractional actions in nonlinear dynamics
  13. Multiwave interaction solutions for a (3 + 1)-dimensional B-type Kadomtsev–Petviashvili equation in fluid dynamics
  14. Effects of mixed time delays and D operators on fixed-time synchronization of discontinuous neutral-type neural networks
  15. Solvability and stability of nonlinear hybrid ∆-difference equations of fractional-order
  16. Weak and strong boundedness for p-adic fractional Hausdorff operator and its commutator
  17. Simulation and modeling of different cell shapes for closed-cell LM-13 alloy foam for compressive behavior
  18. Pandemic management by a spatio–temporal mathematical model
  19. Adaptive ADI difference solution of quenching problems based on the 3D convection–reaction–diffusion equation
  20. Null controllability of Hilfer fractional stochastic integrodifferential equations with noninstantaneous impulsive and Poisson jump
  21. Application of modified Mickens iteration procedure to a pendulum and the motion of a mass attached to a stretched elastic wire
  22. Modeling the spatiotemporal intracellular calcium dynamics in nerve cell with strong memory effects
  23. Switched coupled system of nonlinear impulsive Langevin equations involving Hilfer fractional-order derivatives
  24. Improving the dynamic behavior of the plate under supersonic air flow by using nonlinear energy sink
https://doi.org/10.1515/ijnsns-2020-0282
Schlagwörter für diesen Artikel
action with two-occurrences of integrals; dissipation; nonlinear dynamics

Artikel in diesem Heft

  1. Frontmatter
  2. Original Research Articles
  3. Frequency responses for induced neural transmembrane potential by electromagnetic waves (1 kHz to 1 GHz)
  4. Investigating existence results for fractional evolution inclusions with order r ∈ (1, 2) in Banach space
  5. Optimal control of non-instantaneous impulsive second-order stochastic McKean–Vlasov evolution system with Clarke subdifferential
  6. Generalized forms of fractional Euler and Runge–Kutta methods using non-uniform grid
  7. Controllability of coupled fractional integrodifferential equations
  8. A new generalized approach to study the existence of solutions of nonlinear fractional boundary value problems
  9. Rational soliton solutions in the nonlocal coupled complex modified Korteweg–de Vries equations
  10. Buoyancy driven flow characteristics inside a cavity equiped with diamond elliptic array
  11. Analysis and numerical effects of time-delayed rabies epidemic model with diffusion
  12. Two occurrences of fractional actions in nonlinear dynamics
  13. Multiwave interaction solutions for a (3 + 1)-dimensional B-type Kadomtsev–Petviashvili equation in fluid dynamics
  14. Effects of mixed time delays and D operators on fixed-time synchronization of discontinuous neutral-type neural networks
  15. Solvability and stability of nonlinear hybrid ∆-difference equations of fractional-order
  16. Weak and strong boundedness for p-adic fractional Hausdorff operator and its commutator
  17. Simulation and modeling of different cell shapes for closed-cell LM-13 alloy foam for compressive behavior
  18. Pandemic management by a spatio–temporal mathematical model
  19. Adaptive ADI difference solution of quenching problems based on the 3D convection–reaction–diffusion equation
  20. Null controllability of Hilfer fractional stochastic integrodifferential equations with noninstantaneous impulsive and Poisson jump
  21. Application of modified Mickens iteration procedure to a pendulum and the motion of a mass attached to a stretched elastic wire
  22. Modeling the spatiotemporal intracellular calcium dynamics in nerve cell with strong memory effects
  23. Switched coupled system of nonlinear impulsive Langevin equations involving Hilfer fractional-order derivatives
  24. Improving the dynamic behavior of the plate under supersonic air flow by using nonlinear energy sink
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