On high order fractional integro-differential equations including the Caputo–Fabrizio derivative

Aydoğan, Melike; Băleanu, Dumitru; Mousalou, Asef; Rezapour, Shahram

doi:10.1186/s13661-018-1008-9

Cited by 160 publications

(104 citation statements)

References 12 publications

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“…x(t) as = A cos(Ωt + φ). (27) From the system transfer function H 10 , the expressions of the steady-state response amplitude and phase shift of the system can be obtained:…”

Section: Steady-state Response Of the First Moment And Output Amplitudementioning

confidence: 99%

“…For most of the real systems, such as viscoelastic processes, however, the damping term no longer depends only on the current speed but also the historical speed, i.e., one has to take into account the memory effects. Meanwhile, the fractional derivative, naturally has a good ability to describe such memory property [20][21][22][23][24][25][26][27]. Therefore, we proposed a power-law kernel function to model such processes which are associated with the Caputo fractional derivative.…”

Section: Introductionmentioning

confidence: 99%

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Stochastic resonance of fractional-order Langevin equation driven by periodic modulated noise with mass fluctuation

Yang

Deng²,

Ren

2020

Adv Differ Equ

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The stochastic resonance (SR) of a second-order harmonic oscillator subject to mass fluctuation and periodic modulated noise in viscous media is studied. The mass fluctuation noise is modeled as dichotomous noise and the memory of viscous media is characterized by fractional power kernel function. By using the Shapiro-Loginov formula and Laplace transform, we got the analytical expression of the first moment of the steady-state response and studied the relationship between the system response and the system parameters in the long-time limit. The simulation results show the non-monotonic dependence between the response amplitude and the input signal frequency, noise parameters of the system, etc, which indicates that the bona fide resonance and the generalized SR phenomena appear. Furthermore, the mass fluctuation noise, modulation noise, and the fractional order work together, producing more complex dynamic phenomena than the integral-order system. For example, there is a transition from bimodal resonance to unimodal resonance between the amplitude and the driving frequency under different fractional orders.

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“…x(t) as = A cos(Ωt + φ). (27) From the system transfer function H 10 , the expressions of the steady-state response amplitude and phase shift of the system can be obtained:…”

Section: Steady-state Response Of the First Moment And Output Amplitudementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stochastic resonance of fractional-order Langevin equation driven by periodic modulated noise with mass fluctuation

Yang

Deng²,

Ren

2020

Adv Differ Equ

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“…Quadratic perturbation in fractional differential equations (FDEs) has become a significant tool to model many physical phenomena in physics, biology, engineering, and control theory. Some recent developments can be found in a series of papers . It is observed that a class of DEs representing certain physical phenomenon may not be easily solvable or analyzed.…”

Section: Introductionmentioning

confidence: 99%

“…Some recent developments can be found in a series of papers. [1][2][3][4][5][6][7][8][9][10] It is observed that a class of DEs representing certain physical phenomenon may not be easily solvable or analyzed. However, employing perturbations in such problems makes it possible to study with available methods.…”

Section: Introductionmentioning

confidence: 99%

Dhage iterative principle for quadratic perturbation of fractional boundary value problems with finite delay

Gupta

Bora

Nieto

2019

Math Methods in App Sciences

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In this article, by employing Dhage iterative method embodied in current hybrid fixed point theorem (HFPT) of Dhage, we derive an algorithm for the numerical solutions via construction of a sequence of successive approximations for a fractional order boundary value problem (FBVP) with finite delay. By using this technique, we obtain existence as well as approximation of solutions under weaker partial Lipschitz and partial compactness type conditions in a partially ordered Banach space. Additionally, we prove an existence and uniqueness theorem under a weaker partial nonlinear Lipschitz condition. The assumptions and main outcomes are also illustrated by two examples.

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“…In the last two years, many authors used Caputo-Fabrizio fractional derivative operator in many interesting models and with many known methods, see [11][12][13][14][15][16][17][18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

An Alternative Approach for Nonlinear Optimization Problem with Caputo - Fabrizio Derivative

Evirgen

Yavuz

2018

ITM Web Conf.

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In this study, a fractional mathematical model with steepest descent direction is proposed to find optimal solutions for a class of nonlinear programming problem. In this sense, Caputo-Fabrizio derivative is adapted to the mathematical model. To demonstrate the solution trajectory of the mathematical model, we use the multistage variational iteration method (MVIM). Numerical simulations and comparisons on some test problems show that the mathematical model generated using Caputo-Fabrizio fractional derivative is both feasible and efficient to find optimal solutions for a certain class of equality constrained optimization problems.

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On high order fractional integro-differential equations including the Caputo–Fabrizio derivative

Abstract: By using the fractional Caputo-Fabrizio derivative, we introduce two types new high order derivations called CFD and DCF. Also, we study the existence of solutions for two such type high order fractional integro-differential equations. We illustrate our results by providing two examples.

Cited by 160 publications

References 12 publications

Stochastic resonance of fractional-order Langevin equation driven by periodic modulated noise with mass fluctuation

Stochastic resonance of fractional-order Langevin equation driven by periodic modulated noise with mass fluctuation

Dhage iterative principle for quadratic perturbation of fractional boundary value problems with finite delay

An Alternative Approach for Nonlinear Optimization Problem with Caputo - Fabrizio Derivative

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