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Abstract:In this work a fractional differential equation for the electrical RLC circuit is studied. The order of the derivative being considered is 0 < γ ≤ 1. To keep the dimensionality of the physical quantities R L and C an auxiliary parameter σ is introduced. This parameter characterizes the existence of fractional components in the system. It is shown that there is a relation between γ and σ through the physical parameters RLC of the circuit. Due to this relation, the analytical solution is given in terms of the Mittag-Leffler function depending on the order γ of the fractional differential equation.PACS (2008)

Making use of supersymmetry breaking selection rules under local n = 2 conformal supersymmetry we are able to obtain a normalizable wavefunction (at zero energy) of the universe for the FRW cosmological model (k = 0). For this purpose it is necessary to `weight' the inner product with the scalar factor R. The appropriate Hermitian operators are constructed and the expectation value of R is calculated giving us the size of the universe.

Abstract:This paper provides an analysis in the time and frequency domain of an RC electrical circuit described by a fractional differential equation of the order 0 < α ≤ 1. We use the Laplace transform of the fractional derivative in the Caputo sense. In the time domain we emphasize on the delay, rise and settling times, while in the frequency domain the interest is in the cutoff frequency, the bandwidth and the asymptotes in low and high frequencies. All these quantities depend on the order of differential equation. PACS

Summary In the last 3 years, the fractional conformable derivative and its properties have been introduced. Unlike other definitions, this new fractional derivative is based on the basic limit definition of the derivative and satisfies the same formulas of derivation, such as product and quotient of 2 functions and the chain rule. Using this new derivative, we obtain a new class of linear ordinary differential equations with noninteger power variable coefficients for the Resistance Capacitance (RC), Inductance Capacitance (LC), and Resistance, Inductance Capacitance (RLC) electric circuits. The numerical solutions are solved through the Matlab software. Solutions depend on time and on the fractional order parameter 0 < γ ≤ 1. The computing using this new derivative is much easier than using other definitions of fractional derivative. It has been shown that in the particular case γ = 1, these solutions become the ordinary ones. Also, a comparison has been made with the Caputo fractional derivative for the case of the RC circuit with constant source.

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The superfield action for Bianchi type models is formulated on the basis of local n = 2 supersymmetry. It is shown that the supersymmetric action for these models has the form of the localized version of n = 2 supersymmetric quantum mechanics. This local symmetry procedure provides a well defined scheme for including matter.

Using the fractional calculus approach, we present the Laplace analysis of an equivalent electrical circuit for a multilayered system, which includes distributed elements of the Cole model type. The Bode graphs are obtained from the numerical simulation of the corresponding transfer functions using arbitrary electrical parameters in order to illustrate the methodology. A numerical Laplace transform is used with respect to the simulation of the fractional differential equations. From the results shown in the analysis, we obtain the formula for the equivalent electrical circuit of a simple spectrum, such as that generated by a real sample of blood tissue, and the corresponding Nyquist diagrams. In addition to maintaining consistency in adjusted electrical parameters, the advantage of using fractional differential equations in the study of the impedance spectra is made clear in the analysis used to determine a compact formula for the equivalent electrical circuit, which includes the Cole model and a simple RC model as special cases.