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The present work aims to approximate the solution of linear time fractional PDE with Caputo Fabrizio derivative. For the said purpose Laplace transform with local radial basis functions is used. The Laplace transform is applied to obtain the corresponding time independent equation in Laplace space and then the local RBFs are employed for spatial discretization. The solution is then represented as a contour integral in the complex space, which is approximated by trapezoidal rule with high accuracy. The application of Laplace transform avoids the time stepping procedure which commonly encounters the time instability issues. The convergence of the method is discussed also we have derived the bounds for the stability constant of the differentiation matrix of our proposed numerical scheme. The efficiency of the method is demonstrated with the help of numerical examples. For our numerical experiments we have selected three different domains, in the first test case the square domain is selected, for the second test the circular domain is considered, while for third case the L-shape domain is selected.

Accurate and fast characterization of voltage variations helps to evaluate their severity on equipment and activate protections. In this paper, a methodology for tracking and characterization of voltage variations, sample to sample, is presented. It consists of a Hilbert transform to estimate the voltage of the signal’s envelope, a fuzzy logic system to track down the type of voltage variation, and a rule-based method for the final identification and decision making according to IEEE Std 1159-2009. Unlike some techniques presented in the literature for tracking voltage variations such as the Kalman filter and adaptive linear network techniques, the proposed methodology requires neither a harmonic model nor an algorithm to adjust the model parameters, which in many cases increases the computational burden and time tracking. It is worth mentioning that the proposed classification stage does not need a training stage; therefore, its development is easier and its efficiency does not depend on a data training set. The performance of the proposed methodology is validated and tested using synthetic signals as well as real measurements of voltage variations. In addition, an implementation of our methodology into an field-programmable gate array based system is performed in an effort to offer a low-cost and portable system-on-a-chip solution for online and real-time monitoring of voltage variations. [ABSTRACT FROM PUBLISHER]