Existence of solutions of non-autonomous fractional differential equations with integral impulse condition

Kumar, Ashish; Chauhan, Harsh V. S.; Ravichandran, C.; Nisar, Kottakkaran Sooppy; Băleanu, Dumitru

doi:10.1186/s13662-020-02888-3

Cited by 55 publications

(24 citation statements)

References 38 publications

(27 reference statements)

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“…Among them, m − 1 < α < m. It can be seen from equation (5) that Caputo definition requires the first m-order derivative of the function to be integrable.…”

Section: Caputo Definition Of Fractional Calculus Caputo Fractional Calculus Is Defined Asmentioning

confidence: 99%

“…Fractional calculus is a mathematical problem to study integral and differential of arbitrary order. Fractional calculus has many applications, such as physical system modeling [1][2][3], control theory [4][5][6], and so on. In control theory, as fractional calculus is more and more applied to the design and analysis of controllers and filters, the research on approximation methods of fractional order systems or operator models has attracted more and more attention [7][8][9].…”

Section: Introductionmentioning

confidence: 99%

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Rational Implementation of Fractional Calculus Operator Based on Quadratic Programming

2021

Mathematical Problems in Engineering

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When fractional calculus operators and models are implemented rationally, there exist some problems such as low approximation accuracy of rational approximation function, inability to specify arbitrary approximation frequency band, or poor robustness. Based on the error criterion of the least squares method, a quadratic programming method based on the frequency-domain error is proposed. In this method, the frequency-domain error minimization between the fractional operator s ± r and its rational approximation function is transformed into a quadratic programming problem. The results show that the construction method of the optimal rational approximation function of fractional calculus operator is effective, and the optimal rational approximation function constructed can effectively approximate the fractional calculus operator and model for the specified approximation frequency band.

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“…Among them, m − 1 < α < m. It can be seen from equation (5) that Caputo definition requires the first m-order derivative of the function to be integrable.…”

Section: Caputo Definition Of Fractional Calculus Caputo Fractional Calculus Is Defined Asmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Rational Implementation of Fractional Calculus Operator Based on Quadratic Programming

2021

Mathematical Problems in Engineering

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“…Asymptotic behavior similar to Eq. (1.1) has been investigated in many documents during the last years (see, e.g., [2,3,[11][12][13][14][15] and the references therein).When ν = 0, Eq. (1.1) can be simplified to a usual reaction-diffusion equation, so the dynamical behavior of this equations has been investigated in many documents (see, e.g., [16][17][18][19] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Upper semicontinuity of attractors for nonclassical diffusion equations with arbitrary polynomial growth

Xie

Zhu

2021

Adv Differ Equ

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In this paper, we mainly investigate upper semicontinuity and regularity of attractors for nonclassical diffusion equations with perturbed parameters ν and the nonlinear term f satisfying the polynomial growth of arbitrary order $p-1$ p − 1 ($p \geq 2$ p ≥ 2 ). We extend the asymptotic a priori estimate method (see (Wang et al. in Appl. Math. Comput. 240:51–61, 2014)) to verify asymptotic compactness and upper semicontinuity of a family of semigroups for autonomous dynamical systems (see Theorems 2.2 and 2.3). By using the new operator decomposition method, we construct asymptotic contractive function and obtain the upper semicontinuity for our problem, which generalizes the results obtained in (Wang et al. in Appl. Math. Comput. 240:51–61, 2014). In particular, the regularity of global attractors is obtained, which extends and improves some results in (Xie et al. in J. Funct. Spaces 2016:5340489, 2016; Xie et al. in Nonlinear Anal. 31:23–37, 2016).

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“…Moreover, one can be noticed that noise or stochastic distress cannot be avoided in nature, including artificial systems. For this reason, stochastic differential systems have fascinated considerable attraction because of their large utilization in illustrating a lot of refined dynamical systems in biological, physical, and medical fields; one can verify [6,13,[26][27][28][29][32][33][34][35][36][37][38]. The differential equations of Sobolev-type come into sight commonly in the mathematical form of numerous physical phenomena similar to fluid flow over fissured rocks, thermodynamics, and so on, we can refer to [4,11,13,15,16,30,32,36,[38][39][40][41][42][43].…”

Section: Introductionmentioning

confidence: 99%

Results on approximate controllability results for second‐order Sobolev‐type impulsive neutral differential evolution inclusions with infinite delay

Vijayakumar

Panda²,

Nisar

et al. 2020

Numerical Methods Partial

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In this article, we determine a set of appropriate conditions for approximate controllability of second‐order Sobolev‐type impulsive neutral differential evolution inclusions with infinite delay. We verify the main results by applying ideas about the cosine function, sine functions, and fixed point approach. Next, we continue the discussion to the nonlocal second‐order Sobolev‐type differential system. In the end, we provide a theoretical application to assist in the effectiveness of our discussion.

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Existence of solutions of non-autonomous fractional differential equations with integral impulse condition

Abstract: In this paper, we investigate the existence of solution of non-autonomous fractional differential equations with integral impulse condition by the measure of non-compactness (MNC), fixed point theorems, and k-set contraction. The obtained results are verified via a supporting example.

Cited by 55 publications

References 38 publications

Rational Implementation of Fractional Calculus Operator Based on Quadratic Programming

Rational Implementation of Fractional Calculus Operator Based on Quadratic Programming

Upper semicontinuity of attractors for nonclassical diffusion equations with arbitrary polynomial growth

Results on approximate controllability results for second‐order Sobolev‐type impulsive neutral differential evolution inclusions with infinite delay

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