This work presents the homotopy perturbation transform method for nonlinear fractional partial differential equations of the Caputo-Fabrizio fractional operator. Perturbative expansion polynomials are considered to obtain an infinite series solution. The effectiveness of this method is demonstrated by finding the exact solutions of the fractional equations proposed, for the special case when the limit of the integral order of the time derivative is considered.
The Cattaneo-Vernotte equation is a generalization of the heat and particle diffusion equations; this mathematical model combines waves and diffusion with a finite velocity of propagation. In disordered systems the diffusion can be anomalous. In these kinds of systems, the mean-square displacement is proportional to a fractional power of time not equal to one. The anomalous diffusion concept is naturally obtained from diffusion equations using the fractional calculus approach. In this paper we present an alternative representation of the Cattaneo-Vernotte equation using the fractional calculus approach; the spatial-time derivatives of fractional order are approximated using the Caputo-type derivative in the range(0,2]. 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 Cattaneo-Vernotte equation. Finally, consider the Dirichlet conditions, the Fourier method was used to find the full solution of the fractional Cattaneo-Vernotte equation in analytic way, and Caputo and Riesz fractional derivatives are considered. The advantage of our representation appears according to the comparison between our model and models presented in the literature, which are not acceptable physically due to the dimensional incompatibility of the solutions. The classical cases are recovered when the fractional derivative exponents are equal to1.
The current study presents a new fractional-order three-echelon supply chain model. The chaotic behaviour of the proposed model is demonstrated, and after its synchronization is studied. To this end, a new control technique is offered for the proposed fractional-order system. In the design of the controller, it is assumed that all parameters of the model are unknown. Hence, the proposed sliding mode controller is equipped with an adaptive mechanism, which estimates all unknown parameters of the controller. On the basis of the Lyapunov stability theorem and Barbalat’s lemma, the stability of the system is proven. In order to enhance the performance of the proposed controller and prevent the probable chattering phenomenon which could occur due to the discontinuous function in the sliding mode controller, a fuzzy controller is proposed along with the main controller. Not only is the offered controller robust against uncertainties, but also it is able to estimate parameters of the systems and avoids chattering in the system. Finally, the proposed controller’s excellent performance for synchronization of the fractional-order three-echelon supply chain system is demonstrated through numerical results.
This paper aims to analyze the quality of insulation in high voltage underground cables XLPE using a prototype which classifies the following usual types of patterns of partial discharge (PD): (1) internal PD, (2) superficial PD, (3) corona discharge in air, and (4) corona discharge in oil, in addition to considering two new PD patterns: (1) false contact and (2) floating ground. The tests and measurements to obtain the patterns and study cases of partial discharges were performed at the Testing Laboratory Equipment and Materials (LEPEM) of the Federal Electricity Commission of Mexico (CFE) using a measuring equipment LDIC and norm IEC60270. To classify the six patterns of partial discharges mentioned above a Probabilistic Neural Network Bayesian Modified (PNNBM) method having the feature of using a large amount of data will be used and it is not saturated. In addition, PNN converges, always finding a solution in a short period of time with low computational cost. The insulation of two high voltage cables with different characteristics was analyzed. The test results allow us to conclude which wire has better insulation.
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