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Two occurrences of fractional actions in nonlinear dynamics

  • Rami Ahmad El-Nabulsi EMAIL logo
Published/Copyright: August 12, 2021
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International Journal of Nonlinear Sciences and Numerical Simulation
From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 24 Issue 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 .

□

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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Received: 2020-12-15
Revised: 2021-06-09
Accepted: 2021-07-20
Published Online: 2021-08-12

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  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
Keywords for this article
action with two-occurrences of integrals; dissipation; nonlinear dynamics

Articles in the same Issue

  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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