The Modified ABC Operator in Fractional Calculus: Applications to Linear Circuits and Nonlinear Logistic Model

This study employs the modified Atangana–Baleanu Caputo (MABC) fractional derivative to investigate fractional linear electrical systems. The MABC operator is defined through a convolution representation that extends the admissible function space from H10,1 to the broader class L10,1; on the smooth source terms and nonlinearity considered here, this representation coincides exactly with the classical Atangana–Baleanu derivative in the Caputo sense (ABC) operator, since it employs the same nonsingular Mittag–Leffler kernel and normalization function. The extended domain of applicability, rather than any modification of the memory kernel, is therefore the operator's principal formal distinguishing feature relative to ABC is a distinction not exercised numerically in the present study, which treats only smooth data. The Laplace transform method is used to develop analytical solutions for fractional models, offering a reliable comparison between fractional and classical electrical systems. The same MABC formulation is further used to solve nonlinear fractional logistic differential equations, illustrating its applicability beyond linear circuit models. The analytical results are supported by independent numerical simulations, which further highlight the useful benefits of fractional modeling in capturing complicated system behaviors. The proposed formulation is compared quantitatively with the Caputo operator, which employs a structurally different, singular power-law kernel, and convergence conditions for the analytical series solutions are established, while preserving the classical integer-order limit. This study demonstrates how fractional calculus and the MABC formulation's extended domain of applicability in particular, can improve our knowledge of and ability to describe electrical circuits and nonlinear systems.