Fractional calculus and its applications

Li, Changpin; Chen, YangQuan; Kurths, Jürgen

doi:10.1098/rsta.2013.0037

Cited by 39 publications

(22 citation statements)

References 17 publications

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“…An imperfection is introduced in the previously studied problem. That is, the same geometry, see Figure 1, and the same equations, see (28), with the same parameters are considered. However, now a zero thickness imperfection of a prescribed length is introduced in the middle of the ply just between the inner boundary and the outer boundary and centered at the measuring point.…”

Section: Multi-ply Composite Cylinder: Response Verification and Impementioning

confidence: 99%

“…Finally, it is important to note that in spite of the large amount of scientific contributions using a frequency domain description in solid dynamics, this approach is not standard, to the author's knowledge, for thermal models subjected to dynamical forced thermal loads. Certainly, time domain approximations of thermal models are both efficient and robust, and moreover, model order reduction is successfully applied in this setting [19][20][21][22][23][24][25][26], whereas frequency domain approaches for thermal studies are scarce [27][28][29][30][31].…”

Section: Introductionmentioning

confidence: 99%

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Real‐time monitoring of thermal processes by reduced‐order modeling

Aguado

Huerta

Chinesta

et al. 2014

Numerical Meth Engineering

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SUMMARYThis work presents a simple technique for real-time monitoring of thermal processes. Real-time simulationbased control of thermal processes is a big challenge because high-fidelity numerical simulations are costly and cannot be used, in general, for real-time decision making. Very often, processes are monitored or controlled with a few measurements at some specific points. Thus, the strategy presented here is centered on fast evaluation of the response only where it is needed. To accomplish this, classical harmonic analysis is combined with recent model reduction techniques. This leads to an advanced harmonic methodology, which solves in real-time the transient heat equation at the monitored point. In order to apply the reciprocity principle, harmonic analysis is used in the space-frequency domain. Then, Proper Generalized Decomposition, a reduced order approach, pre-computes a transfer function able to produce the output response for a given excitation. This transfer function is computed offline and only once. The response at the monitoring point can be recovered performing a computationally inexpensive post-processing step. This last step can be performed online for real-time monitoring of the thermal process. Examples show the applicability of this approach for a wide range of problems ranging from fast temperature evaluation to inverse problems.

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Section: Multi-ply Composite Cylinder: Response Verification and Impementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Real‐time monitoring of thermal processes by reduced‐order modeling

Aguado

Huerta

Chinesta

et al. 2014

Numerical Meth Engineering

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“…This dependency varies subject to t being included in the denominator. If one wants to apply the polynomial (2) …”

Section: Fractional Time Derivative Approximationmentioning

confidence: 99%

“…The recently growing interest in the application of fractional derivatives [1,2] naturally motivates the design of analytical and numerical methods. The author's interest lies mostly in the aspect of fractional derivative models in electric circuits (i.e.…”

Section: Introductionmentioning

confidence: 99%

A subinterval-based method for circuits with fractional order elements

Sowa¹

2014

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. The paper deals with the solution of problems that concern fractional time derivatives. Specifically the author's interest lies in solving circuit problems with so called fractional capacitors and fractional inductors. A numerical method is proposed that involves polynomial interpolation and the division of the entire time interval (for which computations are performed) into subintervals. Analytical formulae are derived for the integro-differentiation according to the Caputo fractional derivative. The rules that concern the subinterval dynamics throughout the computation are also presented in the paper. For exemplary linear circuit problems (AC and transient) involving fractional order elements the solutions have been obtained. These solutions are compared with ones obtained by means of traditional methods.

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“…[14][15][16] In the last decades, fractional calculus have gained considerable research attention for its more advantages than classical integer-order ones in describing memory and hereditary properties of many materials and processes. [17][18][19][20][21][22][23] It was recognized that many fractional-order (FO) nonlinear differential systems, including the FO Duffing oscillator, 24 the FO Chua circuit, 25 or the so-called FO R€ ossler, 26 FO Chen,27 FO Lorenz, 28 FO L€ u, 29 and FO Liu systems, 30 exhibit complex bifurcations and chaotic phenomena. Moreover, the research on the multi-directional MSCA generated from FO linear autonomous systems by adding different functions has also been reported.…”

Section: Introductionmentioning

confidence: 99%

Design and implementation of grid multi-scroll fractional-order chaotic attractors

Chen

Pan

et al. 2016

Chaos: An Interdisciplinary Journal of Nonlinear Science

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This paper proposes a novel approach for generating multi-scroll chaotic attractors in multidirections for fractional-order (FO) systems. The stair nonlinear function series and the saturated nonlinear function are combined to extend equilibrium points with index 2 in a new FO linear system. With the help of stability theory of FO systems, stability of its equilibrium points is analyzed, and the chaotic behaviors are validated through phase portraits, Lyapunov exponents, and Poincar e section. Choosing the order 0.96 as an example, a circuit for generating 2-D grid multiscroll chaotic attractors is designed, and 2-D 9 Â 9 grid FO attractors are observed at most. Numerical simulations and circuit experimental results show that the method is feasible and the designed circuit is correct.With the development of fractional-order (FO) calculus, it is well verified that many nonlinear FO differential systems, such as the FO Duffing oscillator, the FO Chua circuit, or the so-called FO R€ ossler, FO Chen, FO Lorenz, and FO L€ u systems, among others, exhibit complex bifurcations and chaotic phenomena. These systems can generate single-scroll or double-scroll chaotic attractors. Multi-scroll chaotic attractors (MSCA) admit much more complex dynamic behaviors, more adjustability, and more encryption parameters. Therefore, MSCA have more potential application to communications, cryptography, and many other fields. Naturally, the design and implementation of fractional MSCA is an interesting and challenging topic. In this paper, a new MSCA generation method will be introduced, which is different from the existing ones. In addition, circuit experimental results of the generation of fractional MSCA are presented.

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Fractional calculus and its applications

Abstract: One contribution of 14 to a Theme Issue 'Fractional calculus and its applications' .

Cited by 39 publications

References 17 publications

Real‐time monitoring of thermal processes by reduced‐order modeling

Real‐time monitoring of thermal processes by reduced‐order modeling

A subinterval-based method for circuits with fractional order elements

Design and implementation of grid multi-scroll fractional-order chaotic attractors

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