The pulse narrowing nonlinear transmission lines model within the local fractional calculus on the Cantor sets

作者：Kang‐Jia Wang, Guo‐Dong Wang, Feng Shi · 发表于：COMPEL The International Journal for Computation and Mathematics in Electrical and Electronic Engineering · 年份：2023 · DOI：10.1108/compel-11-2022-0390 · 被引用次数：35 · 研究领域：Fractional Differential Equations Solutions、Nonlinear Waves and Solitons、Advanced Differential Equations and Dynamical Systems

Purpose The fractal and fractional calculus have obtained considerable attention in the electrical and electronic engineering since they can model many complex phenomena that the traditional integer-order calculus cannot. The purpose of this paper is to develop a new fractional pulse narrowing nonlinear transmission lines model within the local fractional calculus for the first time and derive a novel method, namely, the direct mapping method, to seek for the nondifferentiable (ND) exact solutions. Design/methodology/approach By defining some special functions via the Mittag–Leffler function on the Cantor sets, a novel approach, namely, the direct mapping method is derived via constructing a group of the nonlinear local fractional ordinary differential equations. With the aid of the direct mapping method, four groups of the ND exact solutions are obtained in just one step. The dynamic behaviors of the ND exact solutions on the Cantor sets are also described through the 3D graphical illustration. Findings It is found that the proposed method is simple but effective and can construct four sets of the ND exact solutions in just one step. In addition, one of the ND exact solutions becomes the exact solution of the classic pulse narrowing nonlinear transmission lines model for the special case 9 = 1, which strongly proves the correctness and effectiveness of the method. The ideas in the paper can be used to study the other fractal partial differential equations (PDEs) within the l...