Some Hermite–Jensen–Mercer type inequalities for k-Caputo-fractional derivatives and related results

Butt, Saad Ihsan; Nazeer, Waqas; Nasir, Jamshed; Umar, Muhammad; Liu, Ya

doi:10.1186/s13662-020-02693-y

Cited by 16 publications

(11 citation statements)

References 5 publications

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“…e concentration of analytical solution is obtained by the iterative Laplace transform technique, and the concentration is plotted for different input parameters. For more modern fractionalorder mathematical model developments, the reader can refer to [9][10][11][12][13][14][15][16]. e transport equation due to Doulati Ardejani et al [17] for the absorption process is given as…”

Section: Mathematical Modelling Using the Caputo-fabrizio Fractional Derivative Without The Singular Kernelmentioning

confidence: 99%

Application of Green Synthesized Metal Nanoparticles in the Photocatalytic Degradation of Dyes and Its Mathematical Modelling Using the Caputo–Fabrizio Fractional Derivative without the Singular Kernel

Dave

Khan

Purohıt

et al. 2021

Journal of Mathematics

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Textile dyes are untreated discharge into the environment which results in a significant increase in water pollution levels worldwide. Due to the continuous addition of toxic organic dyes, a necessary strategic model is required for the complete degradation of dyes in textile effluent. This paper considers the possibility of biological synthesis of silver and iron nanoparticles and their use in photocatalytic degradation. The immediate change of silver nitrate solution occurring from colorless to brown is observed after the addition of the aqueous leaf extract, indicating the successive reduction of Ag+ ions to the Ag nanoparticles. These formed Ag nanoparticles were subjected to examine the photocatalytic activity under the solar radiation for the degradation of methyl orange. Green synthesized Ag nanoparticles were found to successfully degrade methyl orange up to 95% between 70 hours than the initial exposure time. The absorbance of methyl orange was measured at 465 nm. The present paper is focused on fractional mathematical modelling of dye degradation in textile effluents using the Caputo–Fabrizio fractional derivative without the singular kernel. The iterative Laplace transform method is employed to obtain an analytic solution for the absorption transport equation. The obtained experimental results showing significant removal of dyes from textile wastewater are compared using modelling results. The innovative approach is in outstanding agreement with the findings of the experiment. The mathematical modelling for the dye removal process helps to design suitable environmental management studies to reduce the adverse effect caused by toxic wastewater. Model validation has been shown by comparing analytical simulated solutions with experimental results for photocatalytic degradation using silver and iron nanoparticles as eco-friendly and low-cost agents.

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Section: Mathematical Modelling Using the Caputo-fabrizio Fractional Derivative Without The Singular Kernelmentioning

confidence: 99%

Application of Green Synthesized Metal Nanoparticles in the Photocatalytic Degradation of Dyes and Its Mathematical Modelling Using the Caputo–Fabrizio Fractional Derivative without the Singular Kernel

Dave

Khan

Purohıt

et al. 2021

Journal of Mathematics

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“…A number of mathematicians in the field of applied and pure mathematics have dedicated their efforts to extend, generalize, counterpart, and refine Hermite-Hadamard's inequality (H − H) for different classes of convex functions. For more recent results obtained on inequality (H − H), we refer the reader to references [7][8][9][10].…”

Section: Definition 1 a Function λmentioning

confidence: 99%

Fractional Hermite–Jensen–Mercer Integral Inequalities with respect to Another Function and Application

Butt

Umar²,

Khan

et al. 2021

Complexity

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In this paper, authors prove new variants of Hermite–Jensen–Mercer type inequalities using ψ –Riemann–Liouville fractional integrals with respect to another function via convexity. We establish generalized identities involving ψ –Riemann–Liouville fractional integral pertaining first and twice differentiable convex function λ , and these will be used to derive novel estimates for some fractional Hermite–Jensen–Mercer type inequalities. Some known results are recaptured from our results as special cases. Finally, an application from our results using the modified Bessel function of the first kind is established as well.

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“…[23] established the Caputo fractional derivatives for exponential s-convex functions. Some new k-Caputo fractional derivative inequalities were established in [24] by using Hermite-Hadamard-Mercer type inequalities for differentiable mapping. Ref.…”

Section: Introductionmentioning

confidence: 99%

Switched Fractional Order Multiagent Systems Containment Control with Event-Triggered Mechanism and Input Quantization

Yuan

Chen

2022

Fractal Fract

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This paper studies the containment control problem for a class of fractional order nonlinear multiagent systems in the presence of arbitrary switchings, unmeasured states, and quantized input signals by a hysteresis quantizer. Under the framework of the Lyapunov function theory, this paper proposes an event-triggered adaptive neural network dynamic surface quantized controller, in which dynamic surface control technology can avoid “explosion of complexity” and obtain fractional derivatives for virtual control functions continuously. Radial basis function neural networks (RBFNNs) are used to approximate the unknown nonlinear functions, and an observer is designed to obtain the unmeasured states. The proposed distributed protocol can ensure all the signals remain semi-global uniformly ultimately bounded in the closed-loop system, and all followers can converge to the convex hull spanned by the leaders’ trajectory. Utilizing the combination of an event-triggered scheme and quantized control technology, the controller is updated aperiodically only at the event-sampled instants such that transmitting and computational costs are greatly reduced. Simulations compare the event-triggered scheme without quantization control technology with the control method proposed in this paper, and the results show that the event-triggered scheme combined with the quantization mechanism reduces the number of control inputs by 7% to 20%.

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Some Hermite–Jensen–Mercer type inequalities for k-Caputo-fractional derivatives and related results

Cited by 16 publications

References 5 publications

Application of Green Synthesized Metal Nanoparticles in the Photocatalytic Degradation of Dyes and Its Mathematical Modelling Using the Caputo–Fabrizio Fractional Derivative without the Singular Kernel

Application of Green Synthesized Metal Nanoparticles in the Photocatalytic Degradation of Dyes and Its Mathematical Modelling Using the Caputo–Fabrizio Fractional Derivative without the Singular Kernel

Fractional Hermite–Jensen–Mercer Integral Inequalities with respect to Another Function and Application

Switched Fractional Order Multiagent Systems Containment Control with Event-Triggered Mechanism and Input Quantization

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