Extended Fan’s sub-ODE technique and its application to a fractional nonlinear coupled network including multicomponents — LC blocks

Abstract

The present work deal with the improved Fan’s sub-ordinary differential equation (ODE) method and peruse abundant soliton solutions for a fractional nonlinear coupled network including multicomponents — LC blocks. The extended Fan’s sub-ODE technique is applied on a fractional nonlinear coupled electrical transmission network in addition — LC blocks and through some graphical representations we exhibit novelties. The integer version of the electrical circuit explored here is recently investigated but its fractional partner is weakly scrutinized. Then, we pay attention on the fractional nonlinear coupled electrical network including multicomponents — LC blocks and peruse peculiar voltage waves from the voltage wave equation governing their dynamics. The voltage waves dynamics equation is obtained by means of the Kirchhoff laws’ including the modified Riemann–Liouville derivatives. Through the fractional complex transform, we derive a nonlinear ordinary differential equation that is tackled by the suggested technique. The main gaps of the present investigation based not only on the use of the fractional version of the electrical model, but, also in the capacitor charge nonlinearity’s (the integer version used Qn,mt=C0Vn,m+b1Vn,m2, the fractional partner explored now Qn,mt=C0Vn,m+b2Vn,m3) and the proposed straightforward implemented. In addition, the technique performed here led us to discover new kind of voltage waves by using five auxiliary equation’s key parameters, more than two or three ones commonly involved into such straightforward method reported in the literature. We also prove the existence of our findings by examining the fixed points, phase portraits and related potentials according accurate electrical parameters ranges, coupling elements values wisely chosen. The current technique is reported in the present investigation in manner to be replicable to many other physical systems (electrical, mechanical, optics, acoustics to cite just a few) described by nonlinear evolution equations fractional or not.