How single node dynamics enhances synchronization in neural networks with electrical coupling

The stability of the completely synchronous state in neural networks with electrical coupling is analytically investigated applying both the Master Stability Function approach (MSF), developed by Pecora and Carroll (1998), and the Connection Graph Stability method (CGS) proposed by Belykh et al. (2004). The local dynamics is described by Morris–Lecar model for spiking neurons and by Hindmarsh–Rose model in spike, burst, irregular spike and irregular burst regimes. The combined application of both CGS and MSF methods provides an efficient estimate of the synchronization thresholds, namely bounds for the coupling strength ranges in which the synchronous state is stable. In all the considered cases, we observe that high values of coupling strength tend to synchronize the system. Furthermore, we observe a correlation between the single node attractor and the local stability properties given by MSF. The analytical results are compared with numerical simulations on a sample network, with excellent agreement.     

An elementary analysis of a procedure for sampling points in a convex body

In this paper we describe a new method for proving the polynomial-time convergence of an algorithm for sampling (almost) uniformly at random from a convex body in high dimension. Previous approaches have been based on estimating conductance via isoperimetric inequalities. We show that a more elementary coupling argument can be used to give a similar result. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12, 213–235, 1998     

Preconditioners for the p-version of the Galerkin method for a coupled finite element/boundary element system

We propose and analyze efficient preconditioners for solving systems of equations arising from the p-version for the finite element/boundary element coupling. The first preconditioner amounts to a block Jacobi method, whereas the second one is partly given by diagonal scaling. We use the generalized minimum residual method for the solution of the linear system. For our first preconditioner, the number of iterations of the GMRES necessary to obtain a given accuracy grows like log² p, where p is the polynomial degree of the ansatz functions. The second preconditioner, which is more easily implemented, leads to a number of iterations that behave like p log³ p. Computational results are presented to support this theory. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 47–61, 1998     

Stabilized spectral element approximation for the Navier–Stokes equations

The conforming spectral element methods are applied to solve the linearized Navier–Stokes equations by the help of stabilization techniques like those applied for finite elements. The stability and convergence analysis is carried out and essential numerical results are presented demonstrating the high accuracy of the method as well as its robustness. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 115–141, 1998     

A space decomposition method for parabolic equations

A convergence proof is given for an abstract parabolic equation using general space decomposition techniques. The space decomposition technique may be a domain decomposition method, a multilevel method, or a multigrid method. It is shown that if the Euler or Crank–Nicolson scheme is used for the parabolic equation, then by suitably choosing the space decomposition, only O(| log τ |) steps of iteration at each time level are needed, where τ is the time-step size. Applications to overlapping domain decomposition and to a two-level method are given for a second-order parabolic equation. The analysis shows that only a one-element overlap is needed. Discussions about iterative and noniterative methods for parabolic equations are presented. A method that combines the two approaches and utilizes some of the good properties of the two approaches is tested numerically. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 27–46, 1998     

Unconditional stability of a partitioned IMEX method for magnetohydrodynamic flows

Magnetohydrodynamic (MHD) flows are governed by Navier–Stokes equations coupled with Maxwell equations through coupling terms. We prove the unconditional stability of a partitioned method for the evolutionary full MHD equations, at high magnetic Reynolds number, written in the Elsässer variables. The method we analyze is a first-order one-step scheme, which consists of implicit discretization of the subproblem terms and explicit discretization of the coupling terms.     

A family of ELLAM schemes for advection-diffusion-reaction equations and their convergence analyses

A family of ELLAM (Eulerian–Lagrangian localized adjoint method) schemes is developed and analyzed for linear advection-diffusion-reaction transport partial differential equations with any combination of inflow and outflow Dirichlet, Neumann, or flux boundary conditions. The formulation uses space-time finite elements, with edges oriented along Lagrangian flow paths, in a time–stepping procedure, where space-time test functions are chosen to satisfy a local adjoint condition. This allows Eulerian–Lagrangian concepts to be applied in a systematic mass-conservative manner, yielding numerical schemes defined at each discrete time level. Optimal-order error estimates and superconvergence results are derived. Numerical experiments are performed to verify the theoretical estimates. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 739–780, 1998     

Applications of a space decomposition method to linear and nonlinear elliptic problems

This work presents some space decomposition algorithms for a convex minimization problem. The algorithms has linear rate of convergence and the rate of convergence depends only on four constants. The space decomposition could be a multigrid or domain decomposition method. We explain the detailed procedure to implement our algorithms for a two-level overlapping domain decomposition method and estimate the needed constants. Numerical tests are reported for linear as well as nonlinear elliptic problems. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 717–737, 1998     

Discretization of the stationary convection-diffusion-reaction equation

A finite volume method for the convection-diffusion-reaction equation is presented, which is a model equation in combustion theory. This method is combined with an exponential scheme for the computation of the fluxes. We prove that the numerical fluxes are second-order accurate, uniformly in the local Peclet numbers. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 607–625, 1998     

A second-order splitting combined with orthogonal cubic spline collocation method for the Rosenau equation

A second-order splitting method is applied to a KdV-like Rosenau equation in one space variable. Then an orthogonal cubic spline collocation procedure is employed to approximate the resulting system. This semidiscrete method yields a system of differential algebraic equations (DAEs) of index 1. Error estimates in L² and L^∞ norms have been obtained for the semidiscrete approximations. For the temporal discretization, the time integrator RADAU5 is used for the resulting system. Some numerical experiments have been conducted to validate the theoretical results and to confirm the qualitative behaviors of the Rosenau equation. Finally, orthogonal cubic spline collocation method is directly applied to BBM (Benjamin–Bona–Mahony) and BBMB (Benjamin–Bona–Mahony–Burgers) equations and the well-known decay estimates are demonstrated for the computed solution. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 695–716, 1998     

A numerical method for the first-order wave equation with discontinuous initial data

We introduce a method, constructed such that numerical solutions of the wave equation are well behaved when the solutions also contain discontinuities. The wave equation serves as a model problem for the Euler equations when the solution contains a contact discontinuity. Numerical computations of linear equations and the Euler equations in one and two dimensions are presented. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 353–365, 1998     

A first-second order splitting method for a third-order partial differential equation

A splitting of a third order partial differential equation into a first-order and a second-order one is proposed as the basis for a mixed finite element method to approximate its solution. A time-continuous numerical method is described and error estimates for its solution are demonstrated. Finally, a full discretization is described based on backward Euler finite differences in time, and error estimates for the resulting approximation are established. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 89–96, 1998     

Synchronization analysis through coupling mechanism in realistic neural models

Synchronization which relates to the system’s stability is important to many engineering and neural applications. In this paper, an attempt has been made to implement response synchronization using coupling mechanism for a class of nonlinear neural systems. We propose an OPCL (open-plus-closed-loop) coupling method to investigate the synchronization state of driver-response neural systems, and to understand how the behavior of these coupled systems depend on their inner dynamics. We have investigated a general method of coupling for generalized synchronization (GS) in 3D modified spiking and bursting Morris–Lecar (M-L) neural models. We have also presented the synchronized behavior of a network of four bursting Hindmarsh–Rose (H-R) neural oscillators using a bidirectional coupling mechanism. We can extend the coupling scheme to a network of N neural oscillators to reach the desired synchronous state. To make the investigations more promising, we consider another coupling method to a network of H-R oscillators using bidirectional ring type connections and present the effectiveness of the coupling scheme.     

He’s homotopy perturbation method for systems of integro-differential equations

In this article, the homotopy perturbation method [He JH. Homotopy perturbation technique. Comput Meth Appl Mech Eng 1999;178:257–62; He JH. A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int J Non-Linear Mech 2000;35(1):37–43; He JH. Comparison of homotopy perturbation method and homotopy analysis method. Appl Math Comput 2004;156:527–39; He JH. Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput 2003;135:73–79; He JH. The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl Math Comput 2004;151:287–92; He JH. Application of homotopy perturbation method to nonlinear wave equations Chaos, Solitons & Fractals 2005;26:695–700] is applied to solve linear and nonlinear systems of integro-differential equations. Some nonlinear examples are presented to illustrate the ability of the method for such system. Examples for linear system are so easy that has been ignored.     

The Method of Fundamental Solutions for Stokes flows with a free surface

We investigate the use of the Method of Fundamental Solutions (MFS) for solving Stokes flow problems with a free surface. We apply the method to the creeping planar Newtonian extrudate-swell problem and study the effect of the surface tension on the free surface. The results are in good agreement with existing finite element and boundary element solutions. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 667–678, 1998     

Dimensional splitting with front tracking and adaptive grid refinement

Front tracking in combination with dimensional splitting is analyzed as a numerical method for scalar conservation laws in two space dimensions. An analytic error bound is derived, and convergence rates based on numerical experiments are presented. Numerical experiments indicate that large CFL numbers can be used without loss of accuracy for a wide range of problems. A new method for grid refinement is introduced. The method easily allows for dynamical changes in the grid, using, for instance, the total variation in each grid cell as a criterion for introducing new or removing existing refinements. Several numerical examples are included, highlighting the features of the numerical method. A comparison with a high-resolution method confirms that dimensional splitting with front tracking is a highly viable numerical method for practical computations. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 627–648, 1998     

Guaranteed cost synchronization of complex networks with uncertainties and time‐Varying delays

This article focuses on the problem of Guaranteed cost synchronization of complex networks with uncertainties and time‐Varying delays. Sufficient conditions for the existence of the optimal guaranteed cost control laws are introduced in the light of linear matrix inequalities via the Lyapunov–Krasovskii stability theory. The time‐varying node delays and time‐varying coupling delays are simultaneously regarded in the complex network. The node uncertainties and coupling uncertainties are simultaneously considered as well. Numerical simulations are provided to account for the effectiveness and robustness of the proposed method. The results in this article generalize and improve the corresponding results of the recent works. © 2015 Wiley Periodicals, Inc. Complexity 21: 381–395, 2016     

Superconvergence in the projected-shear plate-bending finite element method

A projected-shear finite element method for periodic Reissner–Mindlin plate model are analyzed for rectangular meshes. A projection operator is applied to the shear stress term in the bilinear form. Optimal error estimates in the L²-norm, the H¹-norm, and the energy norm for both displacement and rotations are established and gradient superconvergence along the Gauss lines is justified in some weak senses. All the convergence and superconvergence results are uniform with respect to the thickness parameter t. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 367–386, 1998     

Numerical analysis of finite element methods for miscible displacements in porous media

Finite element methods are used to solve a coupled system of nonlinear partial differential equations, which models incompressible miscible displacement in porous media. Through a backward finite difference discretization in time, we define a sequentially implicit time-stepping algorithm that uncouples the system at each time-step. The Galerkin method is employed to approximate the pressure, and accurate velocity approximations are calculated via a post-processing technique involving the conservation of mass and Darcy's law. A stabilized finite element ( SUPG ) method is applied to the convection–diffusion equation delivering stable and accurate solutions. Error estimates with quasi-optimal rates of convergence are derived under suitable regularity hypotheses. Numerical results are presented confirming the predicted rates of convergence for the post-processing technique and illustrating the performance of the proposed methodology when applied to miscible displacements with adverse mobility ratios. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 519–548, 1998     

Moving finite element methods by use of space–time elements: I. Scalar problems

This article deals with moving finite element methods by use of the time-discontinuous Galerkin formulation in combination with oriented space–time meshes. A principle for mesh orientation in space–time based on minimization of the residual, related to adaptive error control via an a posteriori error estimate, is presented. The relation to Miller's moving finite element method is discussed. The article deals with scalar problems; systems will be treated in a companion article. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14:251–262, 1998