A fully Sinc-Galerkin method for Euler–Bernoulli beam models

A fully Sinc-Galerkin method in both space and time is presented for fourth-order time-dependent partial differential equations with fixed and cantilever boundary conditions. The sine discretizations for the second-order temporal problem and the fourth-order spatial problems are presented. Alternate formulations for variable parameter fourth-order problems are given, which prove to be especially useful when applying the forward techniques of this article to parameter recovery problems. The discrete system that corresponds to the time-dependent partial differential equations of interest are then formulated. Computational issues are discussed and an accurate and efficient algorithm for solving the resulting matrix system is outlined. Numerical results that highlight the method are given for problems with both analytic and singular solutions as well as fixed and cantilever boundary conditions.     

Dynamic complexities in predator–prey ecosystem models with age-structure for predator

Natural populations, whose generations are non-overlapping, can be modelled by difference equations that describe how the populations evolve in discrete time-steps. In the 1970s ecological research detected chaos and other forms of complex dynamics in simple population dynamics models, initiating a new research tradition in ecology. However, in former studies most of the investigations of complex population dynamics were mainly concentrated on single populations instead of higher dimensional ecological systems. This paper reports a recent study on the complicated dynamics occurring in a class of discrete-time models of predator–prey interaction based on age-structure of predator. The complexities include (a) non-unique dynamics, meaning that several attractors coexist; (b) antimonotonicity; (c) basins of attraction (defined as the set of the initial conditions leading to a certain type of an attractor) with fractal properties, consisting of pattern of self-similarity and fractal basin boundaries; (d) intermittency; (e) supertransients; and (f) chaotic attractors.     

Continuous-time mean–variance portfolio selection with liability and regime switching

A continuous-time mean–variance model for individual investors with stochastic liability in a Markovian regime switching financial market, is investigated as a generalization of the model of Zhou and Yin [Zhou, X.Y., Yin, G., 2003. Markowitz’s mean–variance portfolio selection with regime switching: A continuous-time model, SIAM J. Control Optim. 42 (4), 1466–1482]. We assume that the risky stock’s price is governed by a Markovian regime-switching geometric Brownian motion, and the liability follows a Markovian regime-switching Brownian motion with drift, respectively. The evolution of appreciation rates, volatility rates and the interest rates are modulated by the Markov chain, and the Markov switching diffusion is assumed to be independent of the underlying Brownian motion. The correlation between the risky asset and the liability is considered. The objective is to minimize the risk (measured by variance) of the terminal wealth subject to a given expected terminal wealth level. Using the Lagrange multiplier technique and the linear-quadratic control technique, we get the expressions of the optimal portfolio and the mean–variance efficient frontier in closed forms. Further, the results of our special case without liability is consistent with those results of Zhou and Yin [Zhou, X.Y., Yin, G., 2003. Markowitz’s mean–variance portfolio selection with regime switching: A continuous-time model, SIAM J. Control Optim. 42 (4), 1466–1482].     

An approach to understanding avalanche statistics in mean‐field driven threshold systems

Driven threshold models that produce complex histories of avalanches are used to simulate the dynamics of many complex interacting systems, such as earthquake generating faults and neural networks. A mean‐field model may be formulated in a way that makes avalanches Abelian, so the final size of the avalanche depends only on the initial conditions, not the algorithm. If the initial stress distribution is statistically stationary, the avalanche size distribution is generated by the first intersection of a random process with a curvilinear boundary. Solutions show that such mean‐field models are never truly critical, but always exhibit dissipation or finite‐size effects. © 2005 Wiley Periodicals, Inc. Complexity 10:68–72, 2005     

Terminal perturbation method for the backward approach to continuous time mean–variance portfolio selection

A terminal perturbation method is introduced to study the backward approach to continuous time mean–variance portfolio selection with bankruptcy prohibition in a complete market model. Using Ekeland’s variational principle, we obtain a necessary condition, i.e. the stochastic maximum principle, which the optimal terminal wealth satisfies. This method can deal with nonlinear wealth equation with bankruptcy prohibition and several examples are given to show applications of our results.     

New exact solutions for the classical Drinfel’d–Sokolov–Wilson equation

In this paper, we investigate the classical Drinfel’d–Sokolov–Wilson equation (DSWE)

where p, q, r, s are some nonzero parameters. Some explicit expressions of solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain solitary wave solutions, blow-up solutions, periodic solutions, periodic blow-up solutions and kink-shaped solutions. Some previous results are extended.     

Conservation laws for Lotka–Volterra models

We derive necessary and sufficient conditions on a Lotka–Volterra model to admit a conservation law of Volterra's type. The result and the proof for the corresponding linear algebra problem are given in graph‐theoretical terms; they refer to the directed graph which is defined by the coefficients of the differential equation system. Copyright © 2003 John Wiley & Sons, Ltd.     

Self‐attracting Fermi–Dirac particles in canonical and microcanonical setting

The existence of stationary solutions for the system describing the interaction of gravitationally attracting particles that obey the Fermi–Dirac statistics is studied depending on the relations between mass, the energy and the temperature of the particles. Copyright © 2005 John Wiley & Sons, Ltd.     

Turing instability and Hopf bifurcation in a diffusive Leslie–Gower predator–prey model

In this paper, a reaction‐diffusion predator–prey system that incorporates the Holling‐type II and a modified Leslie‐Gower functional responses is considered. For ODE, the local stability of the positive equilibrium is investigated and the specific conditions are obtained. For partial differential equation, we consider the dissipation and persistence of solutions, the Turing instability of the equilibrium solutions, and the Hopf bifurcation. By calculating the normal form, we derive the formulae, which can determine the direction and the stability of Hopf bifurcation according to the original parameters of the system. We also use some numerical simulations to illustrate our theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Asymptotic formulae of Mehler–Heine-type for certain classical polyorthogonal polynomials

The purpose of this article is to give some asymptotic formulae of polyorthogonal polynomials with respect to some classical measures. The formulae are analogous to the Mehler–Heine formulae for Jacobi and Laguerre polynomials.     

A general converse theorem for mean‐value theorems in linear elasticity

We prove a rather general mean‐value formula in the theory of elasticity, which expresses the value of the displacement at the centre of a sphere in terms of certain combinations of integral averages over the sphere itself of the traction and the displacement. We also establish the corresponding converse to this mean‐value formula under minimal smoothness assumptions on the displacement. Copyright © 2006 John Wiley & Sons, Ltd.     

System Identification with piecewise linear models in the frequency domain – application to an servopneumatic actuator

By means of a real world application a system identification method was investigated for nonlinear systems from input‐output measurements. This approach is based on a blended multiple model structure, which describes the global behaviour of the system over the whole operating range. Depending on the operating point twenty local linear blackbox models were identified in the frequency domain from a finite number of measurements of the inputs and outputs. A comparative study was made of a model, which have been derived using physical laws [4] and measurements of several process states to estimate unknown parameters.     

On the Eigenvalue Problem for Self–Adjoint Operators with Singular Perturbations

We investigate the eigenvalue problem for self–adjoint operators with singular perturbations. The general results presented here include weakly as well as strongly singular cases. We illustrate these results on two models which correspond to so–called additive strongly singular perturbations.     

Robust test for structural instability in dynamic factor models

Annals of the Institute of Statistical Mathematics - In this paper, we consider a robust test for structural breaks in dynamic factor models. The proposed framework considers structural changes...     

‐differential equations for ‐classical polynomials and ‐Jacobi–Stirling numbers

We introduce, characterise and provide a combinatorial interpretation for the so‐called q‐Jacobi–Stirling numbers. This study is motivated by their key role in the (reciprocal) expansion of any power of a second order q‐differential operator having the q‐classical polynomials as eigenfunctions in terms of other even order operators, which we explicitly construct in this work. The results here obtained can be viewed as the q‐version of those given by Everitt et al. and by the first author, whilst the combinatorics of this new set of numbers is a q‐version of the Jacobi–Stirling numbers given by Gelineau and the second author.     

Stability and instability of periodic standing wave solutions for some Klein–Gordon equations

In the present paper we show some results concerning the orbital stability of dnoidal standing wave solutions and orbital instability of cnoidal standing wave solutions to the following Klein–Gordon equation:

u_tt−u_xx+u−|u|²u=0.