The Beverton–Holt dynamic equation

The Cushing–Henson conjectures on time scales are presented and verified. The central part of these conjectures asserts that based on a model using the dynamic Beverton–Holt equation, a periodic environment is deleterious for the population. The proof technique is as follows. First, the Beverton–Holt equation is identified as a logistic dynamic equation. The usual substitution transforms this equation into a linear equation. Then the proof is completed using a recently established dynamic version of the generalized Jensen inequality.     

A Note on the Solution of a Stochastic Partial Differential Equation

Abstract

A physical model is described which justifies the appearance of a stochastic term in the two-dimensional Navier–Stokes equations. In this model, a linear oppositional control term accrues as well. The resulting stochastic partial differential equation is shown to have a unique stationary solution.     

Spectral discretization of Darcy equations coupled with Stokes equations by vorticity–velocity–pressure formulations

We propose to make the numerical analysis of a model coupling the Darcy equations in a porous medium with the Stokes equations in the cracks. The coupling is provided by a pressure continuity on the interface. We describe a discretization by spectral element methods. We derive a priori optimal error estimates and we present some numerical experiments which confirm the results of the analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1628–1651, 2017     

Positive solutions of a nonlocal singular elliptic equation by means of a non-standard bifurcation theory

We consider a nonlocal elliptic equation arising in a prey–predator model whose nonlocal term is singular. We use the Leray–Schauder degree to prove the existence of an unbounded continuum of positive solutions emanating from the trivial solution. As application, we study nonlocal and singular elliptic equations of the type logistic and Holling–Tanner.     

Meromorphic solutions in the FitzHugh–Nagumo model

The FitzHugh–Nagumo model is studied in the framework of analytic theory of differential equations. The Nevanlinna theory is used to find all meromorphic solutions of a second-order ordinary differential equation related to the FitzHugh–Nagumo model. As a consequence new exact solutions of the FitzHugh–Nagumo system are obtained in explicit form.     

General propagation lattice Boltzmann model for variable-coefficient non-isospectral KdV equation

In this paper, a general propagation lattice Boltzmann model for variable-coefficient non-isospectral Korteweg–de Vries (vc-nKdV) equation, which can describe the interfacial waves in a two layer liquid and Alfvén waves in a collisionless plasma, is proposed by selecting appropriate equilibrium distribution function and adding the compensate function. The Chapman–Enskog analysis shows that the vc-nKdV equation can be recovered correctly from the present model. Numerical simulation for the non-propagating one soliton of this equation in different situations is conducted as validation. It is found that the numerical results match well with the analytical solutions, which demonstrates that the current general propagation lattice Boltzmann model is a satisfactory and efficient method, and could be more stable and accurate than the standard lattice Bhatnagar–Gross–Krook model.     

Asymptotic analysis of Emden–Fowler type equation with an application to power flow models

Emden–Fowler type equations are nonlinear differential equations that appear in many fields such as mathematical physics, astrophysics and chemistry. In this paper, we perform an asymptotic analysis of a specific Emden–Fowler type equation that emerges in a queuing theory context as an approximation of voltages under a well-known power flow model. Thus, we place Emden–Fowler type equations in the context of electrical engineering. We derive properties of the continuous solution of this specific Emden–Fowler type equation and study the asymptotic behavior of its discrete analog. We conclude that the discrete analog has the same asymptotic behavior as the classical continuous Emden–Fowler type equation that we consider.     

Discrete potential Ablowitz–Kaup–Newell–Segur equation

We establish a discrete model for the potential Ablowitz–Kaup–Newell–Segur equation via a generalized Cauchy matrix approach. Soliton solutions and Jordan block solutions of this equation are presented by solving the determining equation set. By applying appropriate continuum limits, we obtain two semi-discrete potential Ablowitz–Kaup–Newell–Segur equations. The reductions to real and complex discrete and semi-discrete potential modified Korteweg-de Vries equations are also discussed.     

Option Pricing in Illiquid Markets with Jumps

ABSTRACT

The classical linear Black–Scholes model for pricing derivative securities is a popular model in the financial industry. It relies on several restrictive assumptions such as completeness, and frictionless of the market as well as the assumption on the underlying asset price dynamics following a geometric Brownian motion. The main purpose of this paper is to generalize the classical Black–Scholes model for pricing derivative securities by taking into account feedback effects due to an influence of a large trader on the underlying asset price dynamics exhibiting random jumps. The assumption that an investor can trade large amounts of assets without affecting the underlying asset price itself is usually not satisfied, especially in illiquid markets. We generalize the Frey–Stremme nonlinear option pricing model for the case the underlying asset follows a Lévy stochastic process with jumps. We derive and analyze a fully nonlinear parabolic partial-integro differential equation for the price of the option contract. We propose a semi-implicit numerical discretization scheme and perform various numerical experiments showing the influence of a large trader and intensity of jumps on the option price.     

Energy-dissipation in a coupled system of Allen–Cahn-type equation and Kobayashi–Warren–Carter-type model of grain boundary motion

In this paper, we consider a system of initial-boundary value problems for parabolic equations, as a generalized version of the “φ-η-θ model” of grain boundary motion, proposed by Kobayashi (2001). The system is a coupled system of an Allen–Cahn-type equation with a given temperature source and a phase-field model of grain boundary motion, known as “Kobayashi–Warren–Carter-type model.” The focus of the study is on a special kind of solution, called energy-dissipative solution, which is to reproduce the energy-dissipation of the governing energy in time. Under suitable assumptions, two Main Theorems, concerned with the existence of energy-dissipative solution and and the large-time behavior, will be demonstrated as the results of this paper.     

Amplitude equations for SPDEs with cubic nonlinearities

For a quite general class of stochastic partial differential equations with cubic nonlinearities, we derive rigorously amplitude equations describing the essential dynamics using the natural separation of timescales near a change of stability. Typical examples are the Swift–Hohenberg equation, the Ginzburg–Landau (or Allen–Cahn) equation and some model from surface growth. We discuss the impact of degenerate noise on the dominant behaviour, and see that additive noise has the potential to stabilize the dynamics of the dominant modes. Furthermore, we discuss higher order corrections to the amplitude equation.     

A mathematical model for phase separation: A generalized Cahn–Hilliard equation

In this paper we present a mathematical model to describe the phenomenon of phase separation, which is modelled as space regions where an order parameter changes smoothly. The model proposed, including thermal and mixing effects, is deduced for an incompressible fluid, so the resulting differential system couples a generalized Cahn–Hilliard equation with the Navier–Stokes equation. Its consistency with the second law of thermodynamics in the classical Clausius–Duhem form is finally proved. Copyright © 2011 John Wiley & Sons, Ltd.     

A nonconforming rectangular finite element pair for the Darcy–Stokes–Brinkman model

In this article, we consider the Darcy–Stokes–Brinkman model that can be sorted into three problems: the Darcy problems, the Stokes–Brinkman interface problems and the coupled Darcy–Stokes problems. We study finite element approximation of the model with Dirichlet boundary conditions and make a unified analysis of the three problems based on nonconforming element. Optimal error estimates for the fluid velocity and pressure are derived. Finally, we present some numerical examples verifying the theoretical predictions. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Optimal harvesting policies for a generalized Gordon–Schaefer model in randomly varying environment

A non‐linear stochastic model has been proposed and analized for fish harvesting. The model encompasses Gordon–Schaefer model and Pella–Tomlinson model that have been used in the study of extensive data on baleen whales, harp seals and Gulf of St. Lawrence cod. It has been established that the harvesting of those species is more profitable in terms of biomass for which data supports α>2, as compared to that for which α?2, where α is general index in the model: Copyright © 2003 John Wiley & Sons, Ltd.     

Exponential utility maximization for an insurer with time-inconsistent preferences

This paper studies the optimal consumption–investment–reinsurance problem for an insurer with a general discount function and exponential utility function in a non-Markovian model. The appreciation rate and volatility of the stock, the premium rate and volatility of the risk process of the insurer are assumed to be adapted stochastic processes, while the interest rate is assumed to be deterministic. The object is to maximize the utility of intertemporal consumption and terminal wealth. By the method of multi-person differential game, we show that the time-consistent equilibrium strategy and the corresponding equilibrium value function can be characterized by the unique solutions of a BSDE and an integral equation. Under appropriate conditions, we show that this integral equation admits a unique solution. Furthermore, we compare the time-consistent equilibrium strategies with the optimal strategy for exponential discount function, and with the strategies for naive insurers in two special cases.     

Effect of inhomogeneity in energy transfer through alpha helical proteins with interspine coupling

The dynamics of homogeneous and inhomogeneous alpha helical proteins with interspine coupling is under investigation in this paper by proposing a suitable model Hamiltonian. For specific choice of parameters, the dynamics of homogeneous alpha helical proteins is found to be governed by a set of completely integrable three coupled derivative nonlinear Schrödinger (NLS) equations (Chen–Lee–Liu equations). The effect of inhomogeneity is understood by performing a perturbation analysis on the resulting perturbed three coupled NLS equation. An equivalent set of integrable discrete three coupled derivative NLS equations is derived through an appropriate generalization of the Lax pair of the original Ablowitz–Ladik lattice and the nature of the energy transfer along the lattice is studied.     

A systems approach to the calibration of deterministic dynamic nonlinear simultaneous equation models with incomplete data

This paper develops an interactive three–stage systems approach for the calibration of the structural parameters and missing data within a deterministic, dynamic non–linear simultaneous equations model under arbitrary configurations of incomplete data. In Stage One, we minimize a quadratic loss function in the differences between the actual endogenous variables and the predicted solution values, relative to any feasible choice of the structural parameters. Missing exogenous variables and initial endogenous variables are treated as additional parameters to be calibrated; whereas missing current endogenous variables are treated by the missing data updating condition, in which the current solution values iteratively and sequentially replace those absent. Stage One may or may not lead to unique calibrations of the structural parameters—a fact that can be monitored a posteriori using singular value decompositions of the relevant Jacobian matrix. If not, there is an equivalence class of parameter values, all of which result in the same loss function value. If Stage Two is necessary, we attempt to exploit the non–linearity and simultaneity of the structural system to extract further information about the parameters from the same database, by minimizing the distance between the restricted and unrestricted reduced forms, while constraining the parameters also to lie within the Stage One equivalence class. This requires the use of higher–order numerical derivatives, and probably restricts its use in all but the simplest of cases to the next generation of supercomputers with massive numbers of parallel processors and much larger word–sizes. In Stage Three, various methods by which the original structural model can be simplified, given a non–unique Stage One calibration, are entertained.     

Coupling of mixed finite element and stabilized boundary element methods for a fluid–solid interaction problem in 3D

We introduce and analyze the coupling of a mixed finite element and a boundary element for a three‐dimensional time‐harmonic fluid–solid interaction problem. We consider a formulation in which the Cauchy stress tensor and the rotation are the main variables in the elastic structure and use the usual pressure formulation in the acoustic fluid. The mixed variational formulation in the solid is completed with boundary integral equations relating the Cauchy data of the acoustic problem on the coupling interface. A crucial point in our formulation is the stabilization technique introduced by Hiptmair and coworkers to avoid the well‐known instability issue appearing in the boundary element method treatment of the exterior Helmholtz problem. The main novelty of this formulation, with respect to a previous approach, consists in reducing the computational domain to the solid media and providing a more accurate treatment of the far field effect. We show that the continuous problem is well‐posed and propose a conforming Galerkin method based on the lowest‐order Arnold–Falk–Winther mixed finite element. Finally, we prove that the numerical scheme is convergent with optimal order.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1211–1233, 2014     

Dynamics of a spatiotemporal model on populations in a polluted river

In this paper, we investigate the dynamics for a reaction–diffusion–advection system which models populations in a polluted river. More precisely, we study the stability of steady states, which yields sufficient conditions that lead to population persistence or extinction. Furthermore, some dependence of the stability of the toxicant-only steady state and the population-toxicant coexistence steady state on the model parameters are given.