MIP-based instantaneous control of mixed-integer PDE-constrained gas transport problems

We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics described by the system of isothermal Euler equations, which are partial differential equations in time and 1-dimensional space. This combination leads to mixed-integer optimization problems subject to nonlinear hyperbolic partial differential equations on a graph. We propose an instantaneous control approach in which suitable Euler discretizations yield systems of ordinary differential equations on a graph. This networked system of ordinary differential equations is shown to be well-posed and affine-linear solutions of these systems are derived analytically. As a consequence, finite-dimensional mixed-integer linear optimization problems are obtained for every time step that can be solved to global optimality using general-purpose solvers. We illustrate our approach in practice by presenting numerical results on a realistic gas transport network.     

Similarity solutions to nonlinear heat conduction and Burgers/Korteweg–deVries fractional equations

We analyze self-similar solutions to a nonlinear fractional diffusion equation and fractional Burgers/Korteweg–deVries equation in one spatial variable. By using Lie-group scaling transformation, we determined the similarity solutions. After the introduction of the similarity variables, both problems are reduced to ordinary nonlinear fractional differential equations. In two special cases exact solutions to the ordinary fractional differential equation, which is derived from the diffusion equation, are presented. In several other cases the ordinary fractional differential equations are solved numerically, for several values of governing parameters. In formulating the numerical procedure, we use special representation of a fractional derivative that is recently obtained.     

A new neural network for solving linear programming problems

We propose and analyse a new class of neural network models for solving linear programming (LP) problems in real time. We introduce a novel energy function that transforms linear programming into a system of nonlinear differential equations. This system of differential equations can be solved on-line by a simplified low-cost analog neural network containing only one single artificial neuron with adaptive synaptic weights. The network architecture is suitable for currently available CMOS VLSI implementations. An important feature of the proposed neural network architecture is its flexibility and universality. The correctness and performance of the proposed neural network is illustrated by extensive computer simulation experiments.     

Stability analysis of stochastic functional differential equations with infinite delay and its application to recurrent neural networks

In this paper, we investigate the stochastic functional differential equations with infinite delay. Some sufficient conditions are derived to ensure the pth moment exponential stability and pth moment global asymptotic stability of stochastic functional differential equations with infinite delay by using Razumikhin method and Lyapunov functions. Based on the obtained results, we further study the pth moment exponential stability of stochastic recurrent neural networks with unbounded distributed delays. The result extends and improves the earlier publications. Two examples are given to illustrate the applicability of the obtained results.     

Asymptotic description of stochastic neural networks. I. Existence of a large deviation principle

We study the asymptotic law of a network of interacting neurons when the number of neurons becomes infinite. The dynamics of the neurons is described by a set of stochastic differential equations in discrete time. The neurons interact through the synaptic weights that are Gaussian correlated random variables. We describe the asymptotic law of the network when the number of neurons goes to infinity. Unlike previous works which made the biologically unrealistic assumption that the weights were i.i.d. random variables, we assume that they are correlated. We introduce the process-level empirical measure of the trajectories of the solutions into the equations of the finite network of neurons and the averaged law (with respect to the synaptic weights) of the trajectories of the solutions into the equations of the network of neurons. The result ( Theorem 3.1 below) is that the image law through the empirical measure satisfies a large deviation principle with a good rate function. We provide an analytical expression of this rate function in terms of the spectral representation of certain Gaussian processes.     

Solvable models of layered neural networks based on their differential structure

We introduce the notion of solvable models of artificial neural networks, based on the theory of ordinary differential equations. It is shown that a solvable, three layer, neural network can be realized as a solution of an ordinary differential equation. Several neural networks in standard use are shown to be solvable. This leads to a new, two-step, non-recursive learning paradigm: estimate the differential equation which the target function satisfies approximately, and then approximate the target function in the solution space of that differential equation. It is shown experimentally that the proposed algorithm is useful for analyzing the generalization problem in artificial neural networks. Connections with wavelet analysis are also pointed out.     

Composition constants for raising the orders of unconventional schemes for ordinary differential equations

Many models of physical and chemical processes give rise to ordinary differential equations with special structural properties that go unexploited by general-purpose software designed to solve numerically a wide range of differential equations. If those properties are to be exploited fully for the sake of better numerical stability, accuracy and/or speed, the differential equations may have to be solved by unconventional methods. This short paper is to publish composition constants obtained by the authors to increase efficiency of a family of mostly unconventional methods, called reflexive.

     

Lasalle method and general decay stability of stochastic neural networks with mixed delays

This paper investigates the general decay pathwise stability conditions on a class of stochastic neural networks with mixed delays by applying Lasalle method. The mixed time delays comprise both time-varying delays and infinite distributed delays. The contributions are as follows: (1)?we extend the Lasalle-type theorem to cover stochastic differential equations with mixed delays; (2)?based on the stochastic Lasalle theorem and the M-matrix theory, new criteria of general decay stability, which includes the almost surely exponential stability and the almost surely polynomial stability and the partial stability, for neural networks with mixed delays are established. As an application of our results, this paper also considers a two-dimensional delayed stochastic neural networks model.     

Boundary equations in the finite transfer method for solving differential equation systems

The finite transfer method is going to be used to solve a p system of linear ordinary differential equations. The complete problem is extended by adding the p boundary equations involved. It is chosen a fourth order scheme to obtain finite transfer expressions. A recurrence strategy is used in these equations and permits one to relate different points in the domain where boundary equations are defined. Finally a 2p algebraic system of equations is noted and solved. To show the efficiency and accuracy, the method is applied to determine the structural behavior of a bending beam with different supports and to solve a differential equation of second degree with different boundary conditions.     

A generalization of Vinograd's theorem for dynamical systems

The theory of dynamical systems has been expanded by the introduction of local dynamical systems [10, 4, 9] and local semidynamical systems [1]. Using integral curves of autonomous ordinary differential equations to illustrate these generalizations, we find that, roughly, the integral curves form a local dynamical system if solutions exist and are unique without requiring existence for all time, and the integral curves form a local semidynamical system if solutions exist and are unique in the positive sense but need not exist for all positive time. In addition to autonomous ordinary differential equations, the enlarged theory of dynamical systems has applications to nonautonomous ordinary differential equations, certain partial differential equations, functional differential equations, and Volterra Integral equations [9, 1, 2, 8], respectively. All of these have metric phase spaces. Since many dynamic considerations are invariant to reparameterizations, it is of interest to known when a local dynamical (or semidynamical) system can be reparameterized to yield a “global” dynamical (or semidynamical) system. For autonomous ordinary differential equations, Vinograd [7] has shown that the local dynamical system on an open subset ofRⁿ formed by integral curves is isomorphic (in the sense of Nemytskii and Stepanov) to a global dynamical system. In an extensive study of isomorphisms, Ura [12] has expanded the Gottschalk-Hedlund notion of an isomorphism and restated Vinograd's result in terms of a reparameterization. In this paper we study the problem of finding a global dynamical (or semidynamical) system which is isomorphic to a given local system. A necessary and sufficient condition is found which is then used to show that the Vinograd result holds on metric spaces.     

Strang‐type preconditioners applied to ordinary and neutral differential‐algebraic equations

This paper deals with boundary‐value methods (BVMs) for ordinary and neutral differential‐algebraic equations. Different from what has been done in Lei and Jin (Lecture Notes in Computer Science, vol. 1988. Springer: Berlin, 2001; 505–512), here, we directly use BVMs to discretize the equations. The discretization will lead to a nonsymmetric large‐sparse linear system, which can be solved by the GMRES method. In order to accelerate the convergence rate of GMRES method, two Strang‐type block‐circulant preconditioners are suggested: one is for ordinary differential‐algebraic equations (ODAEs), and the other is for neutral differential‐algebraic equations (NDAEs). Under some suitable conditions, it is shown that the preconditioners are invertible, the spectra of the preconditioned systems are clustered, and the solution of iteration converges very rapidly. The numerical experiments further illustrate the effectiveness of the methods. Copyright © 2011 John Wiley & Sons, Ltd.     

Landesman-Lazer type conditions for a system of p-Laplacian like operators

We study the existence of periodic solutions for a nonlinear second order system of ordinary differential equations of p-Laplacian type. Assuming suitable Nagumo and Landesman-Lazer type conditions we prove the existence of at least one solution applying topological degree methods. We extend a celebrated result by Nirenberg for resonant systems.     

Group method analysis of combined heat and mass transfer by MHD non-Darcy non-Newtonian natural convection adjacent to horizontal cylinder in a saturated porous medium

The group theoretic method is applied for solving problem of combined magneto-hydrodynamic heat and mass transfer of non-Darcy natural convection about an impermeable horizontal cylinder in a non-Newtonian power law fluid embedded in porous medium under coupled thermal and mass diffusion, inertia resistance, magnetic field, thermal radiation effects. The application of one-parameter groups reduces the number of independent variables by one and consequently, the system of governing partial differential equations with the boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The ordinary differential equations are solved numerically for the velocity using shooting method. The effects of magnetic parameter M, Ergun number Er, power law (viscosity) index n, buoyancy ratio N, radiation parameter R_d, Prandtl number Pr and Lewis number Le on the velocity, temperature fields within the boundary layer, heat and mass transfer are presented graphically and discussed.     

Analysis of flow and heat transfer of viscous fluid between contracting rotating disks

The present work investigates the effects of disks contracting, rotation and heat transfer on the viscous fluid between heated contracting rotating disks. By introducing the Von Kármán type similarity transformations through which we reduced the highly nonlinear partial differential equation to a system of ordinary differential equations. This system of differential equations with appropriate boundary conditions is responsible for the flow behavior between large but finite coaxial rotating and heated disks. It is important to note that the lower disk is rotating with angular velocity Ω while the upper one with SΩ, the disks are also contracting and the temperatures of the upper and lower disks are T₁ and T₀, respectively. The agents which driven the flow are the contraction and also the rotation of the disks. On the other hand the velocity components and especially radial component of velocity strongly influence the temperature distribution inside the flow regime. The basic equations which govern the flow are the Navier Stokes equations with well known continuity equation for incompressible flow. The final system of ordinary differential equations is then solved numerically with given boundary conditions. In addition, the effect of physical parameters, the Reynolds number (Re), the wall contraction ratio (γ) and the rotation ratio (S) on the velocity and pressure gradient, as well as, the effect of Prandtl number (Pr) on temperature distribution are also observed.     

Blow-up the symmetry analysis for the Hirota-Satsuma equations

The paper investigates the invariance and integrability properties of Hirota-Satsuma equations. Painleve analysis for the general similarity reduced ordinary differential equation is performed. Using Rung-Kutta-Merson method in shooting and matching technique, the nonlinear ordinary differential equations are solved that were numerically converted from similarity reduction.     

Lie symmetry group analysis of magnetic field effects on free convective flow of a nanofluid over a semi-infinite stretching sheet

We investigate the steady two-dimensional flow of an incompressible water based nanofluid over a linearly semi-infinite stretching sheet in the presence of magnetic field numerically. The basic boundary layer equations for momentum and heat transfer are non-linear partial differential equations. Lie symmetry group transformations are used to convert the boundary layer equations into non-linear ordinary differential equations. The dimensionless governing equations for this investigation are solved numerically using Nachtsheim–Swigert shooting iteration technique together with fourth order Runge–Kutta integration scheme. Effects of the nanoparticle volume fraction ϕ, magnetic parameter M, Prandtl number Pr on the velocity and the temperature profiles are presented graphically and examined for different metallic and non-metallic nanoparticles. The skin friction coefficient and the local Nusselt number are also discussed for different nanoparticles.     

Non-linear waves in a ring of neurons

^** Corresponding author. Email: shangjguo{at}etang.com In this paper, we study the effect of synaptic delay of signaltransmission on the pattern formation and some properties ofnon-linear waves in a ring of identical neurons. First, linearstability of the model is investigated by analyzing the associatedcharacteristic transcendental equation. Regarding the delayas a bifurcation parameter, we obtained the spontaneous bifurcationof multiple branches of periodic solutions and their spatio-temporalpatterns. Second, global continuation conditions for Hopf bifurcatingperiodic orbits are derived by using the equivariant degreetheory developed by Geba et al. and independently by Ize &Vignoli. Third, we show that the coincidence of these periodicsolutions is completely determined either by a scalar delaydifferential equation if the number of neurons is odd, or bya system of two coupled delay differential equations if thenumber of neurons is even. Fourth, we summarize some importantresults about the properties of Hopf bifurcating periodic orbits,including the direction of Hopf bifurcation, stability of theHopf bifurcating periodic orbits, and so on. Fifth, in an excitatoryring network, solutions of most initial conditions tend to stableequilibria, the boundary separating the basin of attractionof these stable equilibria contains all of periodic orbits andhomoclinic orbits. Finally, we discuss a trineuron network toillustrate the theoretical results obtained in this paper andconclude that these theoretical results are important to complementthe experimental and numerical observations made in living neuronssystems and artificial neural networks, in order to understandthe mechanisms underlying the system dynamics better.     

Estimates for eigenvalues of quasilinear elliptic systems

In this paper we introduce the generalized eigenvalues of a quasilinear elliptic system of resonant type. We prove the existence of infinitely many continuous eigencurves, which are obtained by variational methods. For the one-dimensional problem, we obtain an hyperbolic type function defining a region which contains all the generalized eigenvalues (variational or not), and the proof is based on a suitable generalization of Lyapunov's inequality for systems of ordinary differential equations. We also obtain a family of curves bounding by above the kth variational eigencurve.     

Stagnation-point flow and heat transfer over an exponentially shrinking sheet

An analysis is carried out to investigate the stagnation-point flow and heat transfer over an exponentially shrinking sheet. Using the boundary layer approximation and a similarity transformation in exponential form, the governing mathematical equations are transformed into coupled, nonlinear ordinary differential equations which are then solved numerically by a shooting method with fourth order Runge-Kutta integration scheme. The analysis reveals that a solution exists only when the velocity ratio parameter satisfies the inequality −1.487068 ? c/a. Also, the numerical calculations exhibit the existence of dual solutions for the velocity and the temperature fields; and it is observed that their boundary layers are thinner for the first solution (in comparison with the second). Moreover, the heat transfer from the sheet increases with an increase in c/a for the first solution, while the heat transfer decreases with increasing c/a for the second solution, and ultimately heat absorption occurs.     

一类带有周期输入的人工神经网络的渐近性质

本文利用常微分方程定性理论研究一类带有周期输入的人工神经网络，得到了该网络周期吸引子的存在性。