In this paper we present an alternative representation of the diffusion equation and the diffusion–advection equation using the fractional calculus approach, the spatial-time derivatives are ...approximated using the fractional definition recently introduced by Caputo and Fabrizio in the range β,γ∈(0;2 for the space and time domain respectively. In this representation two auxiliary parameters σx and σt are introduced, these parameters related to equation results in a fractal space–time geometry provide an entire new family of solutions for the diffusion processes. The numerical results showed different behaviors when compared with classical model solutions. In the range β,γ∈(0;1), the concentration exhibits the non-Markovian Lévy flights and the subdiffusion phenomena; when β=γ=1 the classical case is recovered; when β,γ∈(1;2 the concentration exhibits the Markovian Lévy flights and the superdiffusion phenomena; finally when β=γ=2 the concentration is anomalous dispersive and we found ballistic diffusion.
•Fractional calculus is applied to the diffusion and the diffusion–advection equation.•The Caputo–Fabrizio fractional derivative is applied.•The generalization of the equations in space–time exhibits anomalous behavior.•To keep the dimensionality an auxiliary parameter σ is introduced.•The numerical solutions are obtained using the numerical Laplace transform algorithm.
•A new fractional order learning algorithm is proposed in this paper.•The proposed fractional order neural network (neural network trained by the proposed fractional order learning algorithm) leads ...to accurate and simple system identification models.•On the three different systems that were identified, the proposed fractional order neural network reaches the best accuracy with less number of parameters.
Neural networks and fractional order calculus have shown to be powerful tools for system identification. In this paper we combine both approaches to propose a fractional order neural network (FONN) for system identification. The learning algorithm was generalized considering the Grünwald-Letnikov fractional derivative. This new black box modeling approach is validated by the identification of three different systems (two benchmark systems and a real system). Comparisons vs others approaches showed that the proposed FONN model reached better accuracy with less number of parameters.
•This paper presents the space-time fractional diffusion equation.•The diffusion is related with the electromagnetic transient transmission lines.•The generalization of the equations in space-time ...exhibit anomalous behavior.•An analysis of the fractional time constant is presented.•The solutions are given in terms of the Mittag-Leffler function.•To keep the dimensionality an auxiliary parameter σ is introduced.
In this paper, the space-time fractional diffusion equation related to the electromagnetic transient phenomena in transmission lines is studied, three cases are presented; the diffusion equation with fractional spatial derivative, with fractional temporal derivative and the case with fractional space-time derivatives. For the study cases, the order of the spatial and temporal fractional derivatives are 0 < β, γ ≤ 2, respectively. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional diffusion equation. The general solutions of the proposed equations are expressed in terms of the multivariate Mittag-Leffler functions; these functions depend only on the parameters β and γ and preserve the appropriated physical units for any value of the fractional derivative exponent. Furthermore, an analysis of the fractional time constant was made in order to indicate the change of the medium properties and the presence of dissipation mechanisms. The proposed mathematical representation can be useful to understand electrochemical phenomena, propagation of energy in dissipative systems, irreversible thermodynamics, quantum optics or turbulent diffusion, thermal stresses, models of porous electrodes, the description of gel solvents and anomalous complex processes.
In this research, the corrosion type evaluation on the 6061-T6 aluminum alloy exposed to three different solutions using the Electrochemical Noise (EN) technique evaluated by the Synchrosqueezing ...Transform (SST), and the Shannon energy (SSE) methods is presented. The solutions used for the tests were Sulfuric Acid (15% H2SO4), Sodium Chloride (3.5% NaCl), and demineralized Water (diH2O), these solutions were chosen to evaluate the type of corrosion because of each one of them produces a different corrosion type on the Aluminum alloy. To carry out the evaluation, firstly, the EN signals (Electrochemical Potential Noise (EPN) and Electrochemical Current Noise (ECN)) are obtained from probes. After, these signals are evaluated by the SST-SSE method without removing the DC drift. Finally, the EPN and ECN are evaluated using only the SST method. To show the effectiveness of the SST-SSE and SST methods, a comparison with the Wavelet Transform and the Localization Index is carried out.
•Corrosion type identification using the Synchrosqueezing Trans-form•Corrosion type identification using the Synchrosqueezing and Shannon Energy•Comparison between the proposed methodology and the Wavelet Transform technique
The aim of this work is to study the non-local dynamic behavior of triple pendulum-type systems. We use the Euler-Lagrange and the Hamiltonian formalisms to obtain the dynamic models, based on the ...Riemann-Liouville, Liouville-Caputo, Caputo-Fabrizio and Atangana-Baleanu-Caputo fractional derivative definitions. In these representations, an auxiliary parameter σ is introduced, to define the equations in a fractal temporal geometry, which provides an entire new family of solutions for the dynamic behavior of the pendulum-type systems. The phase diagrams allow to visualize the effect of considering the fractional order approach, the classical behavior is recovered when the order of the fractional derivative is 1.
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In this paper, we approximate the solution of fractional differential equations with delay using a new approach based on artificial neural networks. We consider fractional differential equations of ...variable order with the Mittag-Leffler kernel in the Liouville-Caputo sense. With this new neural network approach, an approximate solution of the fractional delay differential equation is obtained. Synaptic weights are optimized using the Levenberg-Marquardt algorithm. The neural network effectiveness and applicability were validated by solving different types of fractional delay differential equations, linear systems with delay, nonlinear systems with delay and a system of differential equations, for instance, the Newton-Leipnik oscillator. The solution of the neural network was compared with the analytical solutions and the numerical simulations obtained through the Adams-Bashforth-Moulton method. To show the effectiveness of the proposed neural network, different performance indices were calculated.
In this paper, a Fractional-Order Extended Kalman Filter (FOEKF) to estimate the State-of-Charge (SOC) of a LiFePO4 battery is proposed. Today there is no sensor to measure the state of charge, due ...to this, it is necessary to use various methods to calculate an estimated value of this parameter. For designing the FOEKF a battery model based on an equivalent electric circuit and the fractional derivative of Atangana-Baleanu type were used. Different numerical simulations using Matlab® were carried out in order to compare it with experimental results from a laboratory prototype and show the performance and accuracy of the FOEKF for estimating the SOC.
Fractional calculus is a powerful tool for describing diffusion phenomena, anomalous behaviors, and in general, systems with highly complex dynamics. However, the application of fractional operators ...for modeling purposes, produces a dimensional problem. In this paper, the fractional models of the RC, RL, RLC electrical circuits, a supercapacitor, a bank of supercapacitors, a LiFePO4 battery and a direct current motor are presented. A correction parameter is included in their formulation in order to preserve dimensionality in the physical equations. The optimal value of this parameter was determined via particle swarm optimization algorithm using numerical simulations and experimental data. Thus, a direct and effective approach for the construction of dimensionally corrected fractional models with power, exponential-decay and constant proportional Caputo hybrid derivative is presented. To show the effectiveness of the procedure, the time-response of the models is compared with experimental data and the modeling error is computed. The numerical solutions of the models were obtained using a numerical method based on the Adams methods.
•Fractional models of electrical circuits, supercapacitor, and LiFePO4 battery are presented.•A correction parameter is included in order to preserve dimensionality.•The optimal value of this parameter was determined via particle swarm optimization algorithm.•The effectiveness of the procedure is compared with experimental.•Numerical solutions were obtained using a numerical method based on the Adams methods.
In this work, an experimental fault tolerant control (FTC) implementation is presented. The FTC is based on a multi-input multi-output (MIMO) model predictive control (MPC). The aim of the FTC is to ...keep on operating a double-pipe counter-current heat exchanger even if the main actuator of the heat exchanger is stuck open. To develop the FTC, an adaptive observer was implemented in order to design a fault detection and isolation (FDI) system. In the FDI system, the cold and hot water flow rate estimations by the adaptive observer are compared to the control signals provided by the MPC. The results of the implementation of the FTC using a MIMO model predictive control were compared to the results obtained in a previous work which was developed using model-following control.
A subclass of dynamical systems with a time rate of change of acceleration are called Newtonian jerky dynamics. Some mechanical and acoustic systems can be interpreted as jerky dynamics. In this ...paper we show that the jerk dynamics are naturally obtained for electrical circuits using the fractional calculus approach with order γ . We consider fractional LC and RL electrical circuits with 1 ⩽ γ < 2 for different source terms. The LC circuit has a frequency ω dependent on the order of the fractional differential equation γ , since it is defined as ω ( γ ) = ω 0 γ γ 1 - γ , where ω 0 is the fundamental frequency. For γ = 3 / 2 , the system is described by a third-order differential equation with frequency ω ~ ω 0 3 / 2 , and assuming γ = 2 the dynamics are described by a fourth differential equation for jerk dynamics with frequency ω ~ ω 0 2 .
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