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This paper deals with the application of fractional derivatives in the modeling of electrical circuits RC, RL, RLC, power electronic devices and nonlinear loads, the equations are obtained by replacing the time derivative by fractional derivatives of type Riemann–Liouville, Grünwald–Letnikov, Liouville–Caputo and the fractional definition recently introduced by Caputo and Fabrizio. The fractional equations in the time domain considers derivatives in the range of α ∈ (0; 1], analytical and numerical results are presented considering different source terms introduced in the fractional equation. The resulting solutions modified the capacitance, inductance, also, the resistance exhibits fluctuations or fractality of time in different scales. Furthermore, the results showed the existence of heterogeneities in the electrical components causing irreversible dissipative effects. The classical models are recovered when the order of the fractional derivatives are equal to 1. [ABSTRACT FROM AUTHOR]

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 σ x and σ t 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. [ABSTRACT FROM AUTHOR]