L1/LDG Method for the Generalized Time-Fractional Burgers Equation in Two Spatial Dimensions

This paper aims to numerically study the generalized time-fractional Burgers equation in two spatial dimensions using the L1/LDG method. Here the L1 scheme is used to approximate the time-fractional derivative, i.e., Caputo derivative, while the local discontinuous Galerkin (LDG) method is used to discretize the spatial derivative. If the solution has strong temporal regularity, i.e., its second derivative with respect to time being right continuous, then the L1 scheme on uniform meshes (uniform L1 scheme) is utilized. If the solution has weak temporal regularity, i.e., its first and/or second derivatives with respect to time blowing up at the starting time albeit the function itself being right continuous at the beginning time, then the L1 scheme on non-uniform meshes (non-uniform L1 scheme) is applied. Then both uniform L1/LDG and non-uniform L1/LDG schemes are constructed. They are both numerically stable and the \(L^2\) optimal error estimate for the velocity is obtained. Numerical examples support the theoretical analysis.

Modeling and Computing of Fractional Convection Equation

In this paper,we derive the fractional convection(or advection)equations(FCEs)(or FAEs)to model anomalous convection processes.Through using a continuous time random walk(CTRW)with power-law jump length distributions,we formulate the FCEs depicted by Riesz derivatives with order in(0,1).The numerical methods for fractional convection operators characterized by Riesz derivatives with order lying in(0,1)are constructed too.Then the numerical approximations to FCEs are studied in detail.By adopting the implicit Crank-Nicolson method and the explicit Lax-WendrofT method in time,and the secondorder numerical method to the Riesz derivative in space,we,respectively,obtain the unconditionally stable numerical scheme and the conditionally stable numerical one for the FCE with second-order convergence both in time and in space.The accuracy and efficiency of the derived methods are verified by numerical tests.The transport performance characterized by the derived fractional convection equation is also displayed through numerical simulations.

Pinning Synchronization Between Two General Fractional Complex Dynamical Networks With External Disturbances

In this paper, the pinning synchronization between two fractional complex dynamical networks with nonlinear coupling, time delays and external disturbances is investigated. A Lyapunov-like theorem for the fractional system with time delays is obtained. A class of novel controllers is designed for the pinning synchronization of fractional complex networks with disturbances. By using this technique, fractional calculus theory and linear matrix inequalities, all nodes of the fractional complex networks reach complete synchronization. In the above framework, the coupling-configuration matrix and the innercoupling matrix are not necessarily symmetric. All involved numerical simulations verify the effectiveness of the proposed scheme. © 2017 Chinese Association of Automation.

Preface to the Focused Issue on Fractional Derivatives and General Nonlocal Models

Fractional calculus,which has two prominent characteristics-singularity and nonlocality,comprises the integration and differentiation of any positive real(and even complex)order.It has a more than three-hundred-year history and can be traced back to a letter from Leib-niz to UHopital,dated 30 September 1695,in which the meaning of the one-half order derivative was first discussed and some remarks about its feasibility were made.Abel was probably the first who rendered an application of fractional calculus.He used the derivatives of arbitrary order to solve the tautochrone(isochrone)problem in 1823.Fractional calculus underwent two periods:from its beginning to the 1970s,and after 1970s.During the first period,fractional calculus was studied mainly by mathematicians as an abstract field containing only pure mathematical manipulations of little applications,except for sporadic applications in rheology.During the second period,the paradigm shifted from pure mathematical research to applications in various realms,including anomalous diffusion,anomalous convection,power laws,allometric scaling laws,history dependence,long?range interactions,and so on.

STABILITY OF N-DIMENSIONAL LINEAR SYSTEMS WITH MULTIPLE DELAYS AND APPLICATION TO SYNCHRONIZATION

This paper further investigates the stability of the n-dimensional linear systems with multiple delays. Using Laplace transform, we introduce a definition of characteristic equation for the n-dimensional linear systems with multiple delays. Moreover, one sufficient condition is attained for the Lyapunov globally asymptotical stability of the general multi-delay linear systems. In particular, our result shows that some uncommensurate linear delays systems have the similar stability criterion as that of the commensurate linear delays systems. This result also generalizes that of Chen and Moore （2002）. Finally, this theorem is applied to chaos synchronization of the multi-delay coupled Chua＇s systems.

L1/Local Discontinuous Galerkin Method for the Time-Fractional Stokes Equation

In this paper,L1/local discontinuous Galerkin method seeking the numerical solution to the time-fractional Stokes equation is displayed,where the timefractional derivative is in the sense of Caputo with derivative orderα∈(0,1).Although the time-fractional derivative is used,its solution may be smooth since such examples can be easily constructed.In this case,we use the uniform L1 scheme to approach the temporal derivative and use the local discontinuous Galerkin(LDG)method to approximate the spatial derivative.If the solution has a certain weak regularity at the initial time,we use the non-uniform L1 scheme to discretize the time derivative and still apply LDG method to discretizing the spatial derivative.The numerical stability and error analysis for both situations are studied.Numerical experiments are also presented which support the theoretical analysis.

Fractional Buffer Layers:Absorbing Boundary Conditions for Wave Propagation

We develop fractional buffer layers(FBLs)to absorb propagating waves without reflection in bounded domains.Our formulation is based on variable-order spatial fractional derivatives.We select a proper variable-order function so that dissipation is induced to absorb the coming waves in the buffer layers attached to the domain.In particular,we first design proper FBLs for the one-dimensional one-way and two-waywave propagation.Then,we extend our formulation to two-dimensional problems,wherewe introduce a consistent variable-order fractionalwave equation.In each case,we obtain the fully discretized equations by employing a spectral collocation method in space and Crank-Nicolson or Adams-Bashforth method in time.We compare our results with a finely tuned perfectly matched layer(PML)method and show that the proposed FBL is able to suppress reflected waves including corner reflections in a two-dimensional rectangular domain.We also demonstrate that our formulation is more robust and uses less number of equations.