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Multi-scale Modelling, Synthesis, and Applications
Effektive Eigenschaften von Kompositen
Series: De Gruyter STEM
Beyond 2020
Advances on Convection Instabilities and Incompressible Fluid Flow

Abstract

In this paper, we introduce the mild solution for a new class of noninstantaneous and nonlocal impulsive Hilfer fractional stochastic integrodifferential equations with fractional Brownian motion and Poisson jumps. The existence of the mild solution is derived for the considered system by using fractional calculus, stochastic analysis and Sadovskii’s fixed point theorem. Finally, an example is also given to show the applicability of our obtained theory.

Abstract

This article is devoted to establish the existence of solution of $(α,β)$-order coupled implicit fractional differential equation with initial conditions, using Laplace transform method. The topological degree theory is used to obtain sufficient conditions for uniqueness and at least one solution of the considered system. Beside this, Ulam’s type stabilities are discussed for the proposed system. To support our main results, we present an example.

Abstract

This paper deals with the Constrained Regulation Problem (CRP) for linear continuous-times fractional-order systems. The aim is to find the existence conditions of linear feedback control law for CRP of fractional-order systems and to provide numerical solving method by means of positively invariant sets. Under two different types of the initial state constraints, the algebraic condition guaranteeing the existence of linear feedback control law for CRP is obtained. Necessary and sufficient conditions for the polyhedral set to be a positive invariant set of linear fractional-order systems are presented, an optimization model and corresponding algorithm for solving linear state feedback control law are proposed based on the positive invariance of polyhedral sets. The proposed model and algorithm transform the fractional-order CRP problem into a linear programming problem which can readily solved from the computational point of view. Numerical examples illustrate the proposed results and show the effectiveness of our approach.

Abstract

A general numerical framework is designed for the two-dimensional convection–diffusion–reaction (CDR) system. The compatibility of differential quadrature and finite difference methods (FDM) are utilized for the formulation. The idea is to switch one numerical scheme to another numerical scheme without changing the formulation. The only requirement is to input the weighting coefficients associated with the derivative discretizations to the general algorithm. Three numerical schemes comprising combinations of differential quadrature and FDMs are studied using the general algorithm. Properties of numerical schemes and the algorithm are analyzed by using the simulations of two-dimensional linear CDR system, Burgers’ equation, and Brusselator model.

Abstract

Consensus problem with faster convergence rate of consensus problem has been considered in this paper. Adding more edges such as that connecting each agent and its second-nearest neighbor or changing the consensus protocol such as mixing asymptotic terms and terms of finite-time has been proved to be possible ways in increasing the convergence rate of multi-agent system in this paper. Based on analysis of Laplacian matrix, increasing of the convergence rate has been proved using the second-smallest eigenvalue for the first method. Concerning the second method, advantages of asymptotic consensus protocol and finite-time consensus protocol have been mixed together with the help of homogeneity function and theory of Lyapunov. Simulation results using matlab are also presented to illustrate the newly designed consensus protocols in increasing the convergence rate.