A multi‐stage ‘infection’ model of stock investors' reaction to new product announcement signal

Drawing on viral dynamics theory, this paper presents a differential equations model with time delay to investigate the stock investor behavior driven by new product announcement (NPA) signal. Visually, we look upon investors in stock market as cells in vivo and the NPA signals as free virus. The potential investors will be ‘infected’ by the dissociative NPA signal and then make investment decisions. In order to better understand the ‘infection’ process, we extract and establish a multi‐stage process during which NPA signal is delivered and ‘infects’ the potential investors. A time‐delay effect is employed to reflect the evaluation stage at which potential investors comprehensively evaluate and decide whether to invest or not. In addition, we introduce a set of external and internal factors into the model, including information sensitivity and investor sentiment, and so on, which are pivotal for examining investor behavior. Equilibrium analysis and numerical simulations are employed to check out the properties of the model and highlight the practical application values of the model. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical bifurcation analysis of a ‘car‐following’ model

We study a system of ordinary differential equations describing a car‐following model for the motion of N car around a circular highway. All cars behave in the same way. The acceleration of each car is determined as a function of the headway (optimal velocity function). This model is known to have a solution with constant velocities and headways which, in a certain parameter regime, is stable and, varying the density of the cars, the loss of stability is generally due to a super‐ or subcritical Hopf bifurcation. Guided by analytical results, we numerically investigate the global bifurcation diagram for periodic solutions and obtain a complete picture of the dynamics of the model. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A MICRO‐LEVEL ‘CONSUMER APPROACH’ TO SPECIES POPULATION DYNAMICS

ABSTRACT. In this paper we develop a micro ecosystem model whose basic entities are representative organisms which behave as if maximizing their net offspring under constraints. Net offspring is increasing in prey biomass intake, declining in the loss of own biomass to predators and Allee's law applies. The organism's constraint reflects its perception of how scarce its own biomass and the biomass of its prey is. In the short‐run periods prices (scarcity indicators) coordinate and determine all biomass transactions and net offspring which directly translates into population growth functions. We are able to explicitly determine these growth functions for a simple food web when specific parametric net offspring functions are chosen in the micro‐level ecosystem model. For the case of a single species our model is shown to yield the well‐known Verhulst‐Pearl logistic growth function. With two species in predator‐prey relationship, we derive differential equations whose dynamics are completely characterized and turn out to be similar to the predator‐prey model with Michaelis‐Menten type functional response. With two species competing for a single resource we find that coexistence is a knife‐edge feature confirming Tschirhart's [2002] result in a different but related model.     

Exchange rate uncertainty and employment: an algorithm describing ‘play’

The paper deals with the impacts of exchange rate uncertainty on the relationship between macroeconomic labour market variables. Under uncertainty, areas of weak reactions—so‐called ‘play’ areas—have to be considered at the macrolevel. The width of the play area is a positive function of the degree of uncertainty. When changes go beyond the play‐area suddenly strong reactions (‘spurts’) occur. These non‐linear dynamics are captured in a simplified linearized way. An algorithm describing linear play hysteresis is developed and implemented into a regression framework. As an empirical application, the exchange rate impacts on German employment are analysed considering play effects. Copyright © 2001 John Wiley & Sons, Ltd.     

A Lie symmetry analysis and explicit solutions of the two‐dimensional ∞‐Polylaplacian

In this work, we consider the Lie point symmetry analysis of a strongly nonlinear partial differential equation of third order, the ∞‐Polylaplacian, in two spatial dimensions. This equation is a higher order generalization of the ∞‐Laplacian, also known as Aronsson's equation, and arises as the analog of the Euler–Lagrange equations of a second‐order variational principle in L^∞. We obtain its full symmetry group, one‐dimensional Lie subalgebras and the corresponding symmetry reductions to ordinary differential equations. Finally, we use the Lie symmetries to construct new invariant ∞‐Polyharmonic functions.     

On martingale diffusions describing the ‘smile‐effect’ for implied volatilities

This paper discusses diffusion models describing the ‘smile‐effect’ of implied volatilities for option prices partly following the new approach of Bruno Dupire. If one restricts to the time homogeneous case, a careful study of this approach shows that the call option prices considered as a function of the price x of the underlying security, remaining time to maturity T–t and strike price K have necessarily to satisfy a certain functional equation, in order to fit into a coherent model. It is shown that for certain examples of empirically observed option prices which are reported in the literature, this functional equation does not hold. © 2000 John Wiley & Sons, Ltd.     

Spatial‐skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses

A spectral problem for the Laplace operator in a thick cascade junction with concentrated masses is considered. This cascade junction consists of the junction's body and a great number of ?‐alternating thin rods belonging to two classes. One class consists of rods of finite length, and the second one consists of rods of small length of order . The density of the junction is of order on the rods from the second class and outside of them. The asymptotic behavior of eigenvalues and eigenfunctions of this problem is studied as ? → 0. There exist five qualitatively different cases in the asymptotic behavior of eigenmagnitudes as ? → 0, namely the case of ‘light’ concentrated (α ∈ (0,1)), ‘middle’ concentrated (α = 1), and ‘heavy’ concentrated masses (α ∈ (1, + ∞ )) that we divide into ‘slightly heavy’ concentrated (α ∈ (1,2)), ‘intermediate heavy’ concentrated (α = 2), and ‘very heavy’ concentrated masses (α > 2). In the paper, we study in detail the influence of the concentrated masses on the asymptotic behavior if α ∈ (1,2). We construct the leading terms of asymptotic expansions both for the eigenvalues and eigenfunctions and prove the corresponding asymptotic estimates. Copyright © 2013 John Wiley & Sons, Ltd.     

Analysis of united boundary‐domain integro‐differential and integral equations for a mixed BVP with variable coefficient

The mixed (Dirichlet–Neumann) boundary‐value problem for the ‘Laplace’ linear differential equation with variable coefficient is reduced to boundary‐domain integro‐differential or integral equations (BDIDEs or BDIEs) based on a specially constructed parametrix. The BDIDEs/BDIEs contain integral operators defined on the domain under consideration as well as potential‐type operators defined on open sub‐manifolds of the boundary and acting on the trace and/or co‐normal derivative of the unknown solution or on an auxiliary function. Some of the considered BDIDEs are to be supplemented by the original boundary conditions, thus constituting boundary‐domain integro‐differential problems (BDIDPs). Solvability, solution uniqueness, and equivalence of the BDIEs/BDIDEs/BDIDPs to the original BVP, as well as invertibility of the associated operators are investigated in appropriate Sobolev spaces. Copyright © 2005 John Wiley & Sons, Ltd.     

New exact solutions for seventh‐order Sawada–Kotera–Ito,Lax and Kaup–Kupershmidt equations using Exp‐function method

In this work, Exp‐function method is used to solve three different seventh‐order nonlinear partial differential KdV equations. Sawada–Kotera–Ito, Lax and Kaup–Kupershmidt equations are well known and considered for solve. Exp‐function method can be used as an alternative to obtain analytic and approximate solutions of different types of differential equations applied in engineering mathematics. Ultimately this method is implemented to solve these equations and convenient and effective solutions are obtained. Copyright © 2009 John Wiley & Sons, Ltd.     

Computational methods for delay parabolic and time‐fractional partial differential equations

This article is concerned with ?‐methods for delay parabolic partial differential equations. The methodology is extended to time‐fractional‐order parabolic partial differential equations in the sense of Caputo. The fully implicit scheme preserves delay‐independent asymptotic stability and the solution continuously depends on the time‐fractional order. Several numerical examples of interest are included to demonstrate the effectiveness of the method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Integrable symplectic maps associated with discrete Korteweg‐de Vries‐type equations

In this paper, we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable partial difference equations which are the discrete counterparts of integrable partial differential equations of Korteweg‐de Vries‐type (KdV‐type). As a consequence it is demonstrated that several distinct Hamiltonian systems lead to one and the same difference equation by means of the Liouville integrability framework. Thus, these integrable symplectic maps may provide an efficient tool for characterizing, and determining the integrability of, partial difference equations.     

CFL condition,Boltzmann's H‐theorem,Onsager reciprocal relations and beyond

In this article, we show that the classical CFL condition from numerical analysis, Boltzmann's H‐theorem from kinetic theory and Onsager reciprocal relations from nonequilibrium thermodynamics are all related to some structural stability conditions for nonhomogeneous systems of first‐order partial differential equations. By doing this, we briefly review our previous works on first‐order partial differential equations. Copyright © 2013 John Wiley & Sons, Ltd.     

Semi‐invariant solutions of the Navier‐Stokes equations

We present new exact solutions and reduced differential systems of the Navier‐Stokes equations of incompressible viscous fluid flow. We apply the method of semi‐invariant manifolds, introduced earlier as a modification of the Lie invariance method. We show that many known solutions of the Navier‐Stokes equations are, in fact, semi‐invariant and that the reduced differential systems we derive using semi‐invariant manifolds generalize previously obtained results that used ad hoc methods. Many of our semi‐invariant solutions solve decoupled systems in triangular form that are effectively linear. We also obtain several new reductions of Navier‐Stokes to a single nonlinear partial differential equation. In some cases, we can solve reduced systems and generate new analytic solutions of the Navier‐Stokes equations or find their approximations, and physical interpretation.     

A new method for exact solutions of variant types of time‐fractional Korteweg‐de Vries equations in shallow water waves

The current article devoted on the new method for finding the exact solutions of some time‐fractional Korteweg–de Vries (KdV) type equations appearing in shallow water waves. We employ the new method here for time‐fractional equations viz. time‐fractional KdV‐Burgers and KdV‐mKdV equations for finding the exact solutions. We use here the fractional complex transform accompanied by properties of local fractional calculus for reduction of fractional partial differential equations to ordinary differential equations. The obtained results are demonstrated by graphs for the new solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence of global solutions for a three‐species predator‐prey model with cross‐diffusion

In this short note, we study a strongly coupled system of partial differential equations which models the dynamics of a two‐predator‐one‐prey ecosystem in which the prey exercises defense switching and the predators collaboratively take advantage of the prey's strategy. We prove the existence of global strong solutions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Regularity theory for fully nonlinear integro‐differential equations

We consider nonlinear integro‐differential equations like the ones that arise from stochastic control problems with purely jump Lévy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior C^{1, α} regularity for general fully nonlinear integro‐differential equations. Our estimates remain uniform as the degree of the equation approaches 2, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations. © 2008 Wiley Periodicals, Inc.     

The use of cubic B‐spline scaling functions for solving the one‐dimensional hyperbolic equation with a nonlocal conservation condition

Problems for parabolic partial differential equations with nonlocal boundary conditions have been studied in many articles, but boundary value problems for hyperbolic partial differential equations have so far remained nearly uninvestigated. In this article a numerical technique is presented for the solution of a nonclassical problem for the one‐dimensional wave equation. This method uses the cubic B‐spline scaling functions. Some numerical results are reported to support our study. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A relationship between λ‐symmetries and first integrals for ordinary differential equations

In this paper, we provide some geometric properties of λ‐symmetries of ordinary differential equations using vector fields and differential forms. According to the corresponding geometric representation of λ‐symmetries, we conclude that first integrals can also be derived if the equations do not possess enough symmetries. We also investigate the properties of λ‐symmetries in the sense of the deformed Lie derivative and differential operator. We show that λ‐symmetries have the exact analogous properties as standard symmetries if we take into consideration the deformed cases.     

Exact solutions of nonlinear time fractional partial differential equations by sub‐equation method

In this article, the sub‐equation method is presented for finding the exact solutions of a nonlinear fractional partial differential equations. For this, the fractional complex transformation method has been used to convert fractional‐order partial differential equation to ordinary differential equation. The fractional derivatives are described in Jumarie's the modified Riemann–Liouville sense. We apply to this method for the nonlinear time fractional differential equations. With the aid of symbolic computation, a variety of exact solutions for them are obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

The fully Sinc‐Galerkin method for time‐dependent boundary conditions

The fully Sinc‐Galerkin method is developed for a family of complex‐valued partial differential equations with time‐dependent boundary conditions. The Sinc‐Galerkin discrete system is formulated and represented by a Kronecker product form of those equations. The numerical solution is efficiently calculated and the method exhibits an exponential convergence rate. Several examples, some with a real‐valued solution and some with a complex‐valued solution, are used to demonstrate the performance of this method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004