In this paper,we propose a new analytical modelling of the well-known fractional generalized Kuramoto-Sivashinky equation(FGKSE)using fractional operator with non-singular kernel and the homotopy anal-ysis transform method via J-transform method.Also,using fixed-point theorem,we prove the existence and uniqueness of our proposed solution to the fractional Kuramoto-Sivashinky equation.To further val-idate the efficiency of the suggested technique,we proved the convergence analysis of the method and provide the error estimate.The obtained solutions of the FGKSE,describing turbulence processes in the field of ocean engineering are analytically and numerically compared to show the behaviors of many parameters of the present model.
This paper presents new synchronization conditions for second-order phase-coupled Kuramoto oscillators in terms of edge dynamics.Two types of network-underlying graphs are studied,the positively weighted and signed graphs,respectively.We apply an edge Laplacian matrix for a positively weighted network to represent the edge connections.The properties of the edge Laplacian matrix are analyzed and incorporated into the proposed conditions.These conditions take account of the dynamics of edge-connected oscillators instead of all oscillator pairs in conventional studies.For a network with positive and negative weights,we represent the network by its spanning tree dynamics,and derive conditions to evaluate the synchronization state of this network.These conditions show that if all edge weights in the spanning tree are positive,and the tree-induced dynamics are in a dominant position over the negative edge dynamics,then this network achieves synchronization.The theoretical findings are validated by numerical examples.
