Analogue Fractional-Order Generalized Memristive Devices

Memristor is a new electrical element which has been predicted and described in 1971 by Leon O. Chua and for the first time realized by HP laboratory in 2008. Chua proved that memristor behavior could not be duplicated by any circuit built using only the other three elements (resistor, capacitor, inductor), which is why the memristor is truly fundamental. Memristor is a contraction of memory resistor, because that is exactly its function: to remember its history. The memristor is a two-terminal device whose resistance depends on the magnitude and polarity of the voltage applied to it and the length of time that voltage has been applied. The missing element—the memristor, with memristance M—provides a functional relation between charge and flux, dφ = Mdq. In this paper, for the first time, the concept of (integer-order) memristive systems is generalized to non-integer order case using fractional calculus. We also show that the memory effect of such devices can be also used for an analogue implementation of the fractional-order operator, namely fractional-order integral and fractional-order derivatives. This kind of operators are useful for realization of the fractional-order controllers. We present theoretical description of such implementation and we proposed the practical realization and did some experiments as well.

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Fractional Lotka-Volterra-Type Cooperation Models: Impulsive Control on Their Stability Behavior

Entropy ◽

10.3390/e22090970 ◽

2020 ◽

Vol 22 (9) ◽

pp. 970 ◽

Cited By ~ 1

Author(s):

Rohisha Tuladhar ◽

Fidel Santamaria ◽

Ivanka Stamova

Keyword(s):

Fractional Order ◽

Fractional Derivatives ◽

Impulsive Control ◽

Control Strategies ◽

Volterra Type ◽

Stability Behavior ◽

Integer Order ◽

Order Case ◽

Cooperation Model ◽

Stability Results

We present a biological fractional n-species delayed cooperation model of Lotka-Volterra type. The considered fractional derivatives are in the Caputo sense. Impulsive control strategies are applied for several stability properties of the states, namely Mittag-Leffler stability, practical stability and stability with respect to sets. The proposed results extend the existing stability results for integer-order n−species delayed Lotka-Volterra cooperation models to the fractional-order case under impulsive control.

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Integer-and Fractional-Order Integral and Derivative Two-Port Summations: Practical Design Considerations

Applied Sciences ◽

10.3390/app10010054 ◽

2019 ◽

Vol 10 (1) ◽

pp. 54 ◽

Cited By ~ 3

Author(s):

Roman Sotner ◽

Ondrej Domansky ◽

Jan Jerabek ◽

Norbert Herencsar ◽

Jiri Petrzela ◽

...

Keyword(s):

Fractional Order ◽

Transfer Functions ◽

Constant Phase Element ◽

State Of The Art ◽

Active Components ◽

Integer Order ◽

Order Part ◽

Fractional Order Integral ◽

Practical Recommendations ◽

Good Agreement

This paper targets on the design and analysis of specific types of transfer functions obtained by the summing operation of integer-order and fractional-order two-port responses. Various operations provided by fractional-order, two-terminal devices have been studied recently. However, this topic needs to be further studied, and the topologies need to be analyzed in order to extend the state of the art. The studied topology utilizes the passive solution of a constant-phase element (with order equal to 0.5) implemented by parallel resistor–capacitor circuit (RC) sections operating as a fractional-order two-port. The integer-order part is implemented by operational amplifier-based lossless integrators and differentiators in branches with electronically adjustable gain, useful for time constant tuning. Four possible cases of the fractional-order and integer-order two-port interconnections are analyzed analytically, by PSpice simulations and also experimentally in the frequency range between 10 Hz and 1 MHz. Standard discrete active components are used in this design for laboratory verification. Practical recommendations for construction and also particular solutions overcoming possible issues with instability and DC offsets are also given. Experimental and simulated results are in good agreement with theory.

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Solvability of generalized fractional order integral equations via measures of noncompactness

Mathematical Sciences ◽

10.1007/s40096-020-00359-0 ◽

2021 ◽

Author(s):

Anupam Das ◽

Bipan Hazarika ◽

Vahid Parvaneh ◽

M. Mursaleen

Keyword(s):

Integral Equations ◽

Fractional Order ◽

Measures Of Noncompactness ◽

Fractional Order Integral

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Invariant Image Representation Using Novel Fractional-Order Polar Harmonic Fourier Moments

Sensors ◽

10.3390/s21041544 ◽

2021 ◽

Vol 21 (4) ◽

pp. 1544

Author(s):

Chunpeng Wang ◽

Hongling Gao ◽

Meihong Yang ◽

Jian Li ◽

Bin Ma ◽

...

Keyword(s):

Image Reconstruction ◽

Fractional Order ◽

Continuous Functions ◽

Kernel Functions ◽

Superior Performance ◽

Image Description ◽

Orthogonal Moments ◽

Integer Order ◽

Geometric Invariance ◽

Order Continuous

Continuous orthogonal moments, for which continuous functions are used as kernel functions, are invariant to rotation and scaling, and they have been greatly developed over the recent years. Among continuous orthogonal moments, polar harmonic Fourier moments (PHFMs) have superior performance and strong image description ability. In order to improve the performance of PHFMs in noise resistance and image reconstruction, PHFMs, which can only take integer numbers, are extended to fractional-order polar harmonic Fourier moments (FrPHFMs) in this paper. Firstly, the radial polynomials of integer-order PHFMs are modified to obtain fractional-order radial polynomials, and FrPHFMs are constructed based on the fractional-order radial polynomials; subsequently, the strong reconstruction ability, orthogonality, and geometric invariance of the proposed FrPHFMs are proven; and, finally, the performance of the proposed FrPHFMs is compared with that of integer-order PHFMs, fractional-order radial harmonic Fourier moments (FrRHFMs), fractional-order polar harmonic transforms (FrPHTs), and fractional-order Zernike moments (FrZMs). The experimental results show that the FrPHFMs constructed in this paper are superior to integer-order PHFMs and other fractional-order continuous orthogonal moments in terms of performance in image reconstruction and object recognition, as well as that the proposed FrPHFMs have strong image description ability and good stability.

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Fractional isospectral and non-isospectral AKNS hierarchies and their analytic methods for N-fractal solutions with Mittag-Leffler functions

Advances in Difference Equations ◽

10.1186/s13662-021-03374-0 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Bo Xu ◽

Yufeng Zhang ◽

Sheng Zhang

Keyword(s):

Spectral Problem ◽

Fractional Order ◽

Point Of View ◽

Adjoint Equations ◽

Scattering Data ◽

Bilinear Method ◽

Soliton Equation ◽

Scattering Transform ◽

First Time ◽

Hirota Bilinear

AbstractAblowitz–Kaup–Newell–Segur (AKNS) linear spectral problem gives birth to many important nonlinear mathematical physics equations including nonlocal ones. This paper derives two fractional order AKNS hierarchies which have not been reported in the literature by equipping the AKNS spectral problem and its adjoint equations with local fractional order partial derivative for the first time. One is the space-time fractional order isospectral AKNS (stfisAKNS) hierarchy, three reductions of which generate the fractional order local and nonlocal nonlinear Schrödinger (flnNLS) and modified Kortweg–de Vries (fmKdV) hierarchies as well as reverse-t NLS (frtNLS) hierarchy, and the other is the time-fractional order non-isospectral AKNS (tfnisAKNS) hierarchy. By transforming the stfisAKNS hierarchy into two fractional bilinear forms and reconstructing the potentials from fractional scattering data corresponding to the tfnisAKNS hierarchy, three pairs of uniform formulas of novel N-fractal solutions with Mittag-Leffler functions are obtained through the Hirota bilinear method (HBM) and the inverse scattering transform (IST). Restricted to the Cantor set, some obtained continuous everywhere but nondifferentiable one- and two-fractal solutions are shown by figures directly. More meaningfully, the problems worth exploring of constructing N-fractal solutions of soliton equation hierarchies by HBM and IST are solved, taking stfisAKNS and tfnisAKNS hierarchies as examples, from the point of view of local fractional order derivatives. Furthermore, this paper shows that HBM and IST can be used to construct some N-fractal solutions of other soliton equation hierarchies.

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Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2019-0030 ◽

2020 ◽

Vol 21 (6) ◽

pp. 571-587 ◽

Cited By ~ 2

Author(s):

Akbar Zada ◽

Sartaj Ali ◽

Tongxing Li

Keyword(s):

Differential Equations ◽

Fractional Order ◽

Fractional Differential Equations ◽

Sufficient Conditions ◽

Uniqueness Of Solutions ◽

Integer Order ◽

New Class ◽

Proposed Model ◽

Fractional Differential ◽

Sequential Fractional Differential Equations

AbstractIn this paper, we study an implicit sequential fractional order differential equation with non-instantaneous impulses and multi-point boundary conditions. The article comprehensively elaborate four different types of Ulam’s stability in the lights of generalized Diaz Margolis’s fixed point theorem. Moreover, some sufficient conditions are constructed to observe the existence and uniqueness of solutions for the proposed model. The proposed model contains both the integer order and fractional order derivatives. Thus, the exponential function appearers in the solution of the proposed model which will lead researchers to study fractional differential equations with well known methods of integer order differential equations. In the last, few examples are provided to show the applicability of our main results.

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Approximate analytical fractional view of convection–diffusion equations

Open Physics ◽

10.1515/phys-2020-0184 ◽

2020 ◽

Vol 18 (1) ◽

pp. 897-905

Author(s):

Hassan Khan ◽

Saima Mustafa ◽

Izaz Ali ◽

Poom Kumam ◽

Dumitru Baleanu ◽

...

Keyword(s):

Fractional Order ◽

Variational Iteration Method ◽

Absolute Error ◽

Form Solution ◽

Diffusion Equations ◽

Nonlinear Convection ◽

Convection Diffusion ◽

Suggested Technique ◽

First Time ◽

Modified Variational Iteration Method

Abstract In this article, a modified variational iteration method along with Laplace transformation is used for obtaining the solution of fractional-order nonlinear convection–diffusion equations (CDEs). The proposed technique is applied for the first time to solve fractional-order nonlinear CDEs and attain a series-form solution with the quick rate of convergence. Tabular and graphical representations are presented to confirm the reliability of the suggested technique. The solutions are calculated for fractional as well as for integer orders of the problems. The solution graphs of the solutions at various fractional derivatives are plotted. The accuracy is measured in terms of absolute error. The higher degree of accuracy is observed from the table and figures. It is further investigated that fractional solutions have the convergence behavior toward the solution at integer order. The applicability of the present technique is verified by illustrative examples. The simple and effective procedure of the current technique supports its implementation to solve other nonlinear fractional problems in different areas of applied science.

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