Solutions of fractional order electrical circuits via Laplace transform and nonstandard finite difference method

Integral representations of Mittag-Leffler function on the positive real axis

“…The classical Laplace transform is one of the most widely tools used in the literature for solving integral equations and ordinary or partial differential equations, involving integer or fractional order derivatives [1,8,31,33]. It is also used in many others applications such as electrical circuit and signal processing [15,21,35]. In general, the Laplace inversion is done numerically due to the impossibility of the exact inversion by means of an integration on the complex plane [7,32].…”

Fractional operator without singular kernel: Applications to linear electrical circuits

“…The first operator is the derivative of the convolution of a given function and a power-law kernel, and the second one is the convolution of the local derivative of a given function with power-law function. In other studies, [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42] authors developed electrical circuits models using FC; they used the Riemann-Liouville or Liouville-Caputo fractional-order derivative operators. The Liouville-Caputo operator is more suitable for modeling real-world problems since it allows using initial conditions.…”

“…[19][20][21][22][23][24][25] Due to FC is one of the most powerful mathematical tools used in the recent decades to model real-world problems, several existing electrical circuits models have been generalized, and fractional derivatives models have been developed to represent the behavior of fractional linear electrical systems, for the designing of analog and digital filters, as well as to describe the magnetically coupled coils or the behavior of circuits and systems with memristors, meminductors, or memcapacitors. In other studies, [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42] authors developed electrical circuits models using FC; they used the Riemann-Liouville or Liouville-Caputo fractional-order derivative operators.…”

Three-parameter Mittag-Leffler function with an integral representation on the positive real axis

“…The classical Laplace transform is one of the most widely tools used in the literature for solving integral equations and ordinary or partial differential equations, involving integer or fractional order derivatives [1,5,21,24]. It is also used in many others applications such as electrical circuit solving and signal processing [12,23]. In general, the Laplace inversion is done numerically due to the impossibility of the exact inversion by means of an integration on the complex plane [4].…”