Analytical and numerical solutions of a two‐dimensional multi‐term time‐fractional Oldroyd‐B model

In this paper, we consider a two‐dimensional multi‐term time‐fractional Oldroyd‐B equation on a rectangular domain. Its analytical solution is obtained by the method of separation of variables. We employ the finite difference method with a discretization of the Caputo time‐fractional derivative to obtain an implicit difference approximation for the equation. Stability and convergence of the approximation scheme are established in the L_∞ ‐norm. Two examples are given to illustrate the theoretical analysis and analytical solution. The results indicate that the present numerical method is effective for this general two‐dimensional multi‐term time‐fractional Oldroyd‐B model.     

On measure‐valued solutions to a two‐dimensional gravity‐driven avalanche flow model

This paper concerns measure‐valued solutions for the two‐dimensional granular avalanche flow model introduced by Savage and Hutter. The system is similar to the isentropic compressible Euler equations, except for a Coulomb–Mohr friction law in the source term. We will partially follow the study of measure‐valued solutions given by DiPerna and Majda. However, due to the multi‐valued nature of the friction law, new more sensitive measures must be introduced. The main idea is to consider the class of x‐dependent maximal monotone graphs of non‐single‐valued operators and their relation with 1‐Lipschitz, Carathéodory functions. This relation allows to introduce generalized Young measures for x‐dependent maximal monotone graph. Copyright © 2005 John Wiley & Sons, Ltd.     

A quasi‐static boundary value problem in multi‐surface elastoplasticity: Part 1—Analysis

The quasi‐static evolution of an elastoplastic body with a multi‐surface constitutive law of linear kinematic hardening type allows the modelling of curved stress–strain relations. It generalizes classical small‐strain elastoplasticity from one to various plastic phases. This paper presents the mathematical models and proves existence and uniqueness of the solution of the corresponding initial‐boundary value problem. The analysis involves an explicit estimate for the effective ellipticity constant. Copyright © 2004 John Wiley & Sons, Ltd.     

ESS,NIS and GIS for multi‐player matrix game in single population

In this paper, we distinguish the concept of global invader strategy (GIS) from that of neighborhood invader strategy (NIS), and discuss the concepts and the properties of ESS, NIS and GIS and relationship among them in the scenario of multi‐player matrix game. We show that a GIS is always an ESS and GIS is unique for any multi‐player. We also show that NIS is equivalent to ESS for pairwise game and there are some results in multi‐player game different from those in pairwise game. Copyright © 2009 John Wiley & Sons, Ltd.     

A multi‐regional epidemic model for controlling the spread of Ebola: awareness,treatment, and travel‐blocking optimal control approaches

Ebola virus disease (EVD) can rapidly cause death to animals and people, for less than 1month. In addition, EVD can emerge in one region and spread to its neighbors in unprecedented durations. Such cases were reported in Guinea, Sierra Leone, and Liberia. Thus, by blocking free travelers, traders, and transporters, EVD has had also impacts on economies of those countries. In order to find effective strategies that aim to increase public knowledge about EVD and access to possible treatment while restricting movements of people coming from regions at high risk of infection, we analyze three different optimal control approaches associated with awareness campaigns, treatment, and travel‐blocking operations that health policy‐makers could follow in the war on EVD. Our study is based on the application of Pontryagin's maximum principle, in a multi‐regional epidemic model we devise here for controlling the spread of EVD. The model is in the form of multi‐differential systems that describe dynamics of susceptible, infected, and removed populations belonging to p different geographical domains with three control functions incorporated. The forward–backward sweep method with integrated progressive‐regressive Runge–Kutta fourth‐order schemes is followed for resolving the multi‐points boundary value problems obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

Topological sensitivity analysis of a multi‐scale constitutive model considering a cracked microstructure

This paper deals with the sensitivity analysis of the macroscopic elasticity tensor to topological microstructural changes of the underlying material. In particular, the microstucture is topologicaly perturbed by the nucleation of a small circular inclusion. The derivation of the proposed sensitivity relies on the concept of topological derivative, applied within a variational multi‐scale constitutive framework where the macroscopic strain and stress at each point of the macroscopic continuum are defined as volume averages of their microscopic counterparts over a representative volume element (RVE) of material associated with that point. We consider that the RVE can contain a number of voids, inclusions and/or cracks. It is assumed that non‐penetration conditions are imposed at the crack faces, which do not allow the opposite crack faces to penetrate each other. The derived sensitivity leads to a symmetric fourth‐order tensor field over the unperturbed RVE domain, which measures how the macroscopic elasticity parameters estimated within the multi‐scale framework changes when a small circular inclusion is introduced at the micro‐scale level. Copyright © 2009 John Wiley & Sons, Ltd.     

Single and multi‐interval Legendre spectral methods in time for parabolic equations

In this article, we take the parabolic equation with Dirichlet boundary conditions as a model to present the Legendre spectral methods both in spatial and in time. Error analysis for the single/multi‐interval schemes in time is given. For the single interval spectral method in time, we obtain a better error estimate in L²‐norm. For the multi‐interval spectral method in time, we obtain the L²‐optimal error estimate in spatial. By choosing approximate trial and test functions, the methods result in algebraic systems with sparse forms. A parallel algorithm is constructed for the multi‐interval scheme in time. Numerical results show the efficiency of the methods. The methods are also applied to parabolic equations with Neumann boundary conditions, Robin boundary conditions and some nonlinear PDEs. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

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.     

New Ore‐Type Conditions for H‐Linked Graphs

For a fixed (multi)graph H, a graph G is H‐linked if any injection f: V(H)→V(G) can be extended to an H‐subdivision in G. The notion of an H ‐linked graph encompasses several familiar graph classes, including k‐linked, k‐ordered and k‐connected graphs. In this article, we give two sharp Ore‐type degree sum conditions that assure a graph G is H ‐linked for arbitrary H. These results extend and refine several previous results on H ‐linked, k‐linked, and k‐ordered graphs. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:69–77, 2012     

Phase‐field systems for multi‐dimensional Prandtl–Ishlinskii operators with non‐polyhedral characteristics

Hysteresis operators have recently proved to be a powerful tool in modelling phase transition phenomena which are accompanied by the occurrence of hysteresis effects. In a series of papers, the present authors have proposed phase‐field models in which hysteresis non‐linearities occur at several places. A very important class of hysteresis operators studied in this connection is formed by the so‐called Prandtl–Ishlinskii operators. For these operators, the corresponding phase‐field systems are in the multi‐dimensional case only known to admit unique solutions if the characteristic convex sets defining the operators are polyhedrons. In this paper, we use approximation techniques to extend the known results to multi‐dimensional Prandtl–Ishlinskii operators having non‐polyhedral convex characteristicsets. Copyright © 2002 John Wiley & Sons, Ltd.     

Shrinkage drift parameter estimation for multi‐factor Ornstein–Uhlenbeck processes

We consider some inference problems concerning the drift parameters of multi‐factors Vasicek model (or multivariate Ornstein–Uhlebeck process). For example, in modeling for interest rates, the Vasicek model asserts that the term structure of interest rate is not just a single process, but rather a superposition of several analogous processes. This motivates us to develop an improved estimation theory for the drift parameters when homogeneity of several parameters may hold. However, the information regarding the equality of these parameters may be imprecise. In this context, we consider Stein‐rule (or shrinkage) estimators that allow us to improve on the performance of the classical maximum likelihood estimator (MLE). Under an asymptotic distributional quadratic risk criterion, their relative dominance is explored and assessed. We illustrate the suggested methods by analyzing interbank interest rates of three European countries. Further, a simulation study illustrates the behavior of the suggested method for observation periods of small and moderate lengths of time. Our analytical and simulation results demonstrate that shrinkage estimators (SEs) provide excellent estimation accuracy and outperform the MLE uniformly. An over‐ridding theme of this paper is that the SEs provide powerful extensions of their classical counterparts. Copyright © 2009 John Wiley & Sons, Ltd.     

Multiscale agent‐based consumer market modeling

Consumer markets have been studied in great depth, and many techniques have been used to represent them. These have included regression‐based models, logit models, and theoretical market‐level models, such as the NBD‐Dirichlet approach. Although many important contributions and insights have resulted from studies that relied on these models, there is still a need for a model that could more holistically represent the interdependencies of the decisions made by consumers, retailers, and manufacturers. When the need is for a model that could be used repeatedly over time to support decisions in an industrial setting, it is particularly critical. Although some existing methods can, in principle, represent such complex interdependencies, their capabilities might be outstripped if they had to be used for industrial applications, because of the details this type of modeling requires. However, a complementary method—agent‐based modeling—shows promise for addressing these issues. Agent‐based models use business‐driven rules for individuals (e.g., individual consumer rules for buying items, individual retailer rules for stocking items, or individual firm rules for advertizing items) to determine holistic, system‐level outcomes (e.g., to determine if brand X's market share is increasing). We applied agent‐based modeling to develop a multi‐scale consumer market model. We then conducted calibration, verification, and validation tests of this model. The model was successfully applied by Procter & Gamble to several challenging business problems. In these situations, it directly influenced managerial decision making and produced substantial cost savings. © 2010 Wiley Periodicals, Inc. Complexity, 2010     

Bankruptcy prediction by generalized additive models

We compare several accounting‐based models for bankruptcy prediction. The models are developed and tested on large data sets containing annual financial statements for Norwegian limited liability firms. Out‐of‐sample and out‐of‐time validation shows that generalized additive models significantly outperform popular models like linear discriminant analysis, generalized linear models and neural networks at all levels of risk. Further, important issues like default horizon and performance depreciation are examined. We clearly see a performance depreciation as the default horizon is increased and as time goes by. Finally a multi‐year model, developed on all available data from three consecutive years, is compared with a one‐year model, developed on data from the most recent year only. The multi‐year model exhibits a desirable robustness to yearly fluctuations that is not present in the one‐year model. Copyright © 2006 John Wiley & Sons, Ltd.     

Multi‐switching combination–combination synchronization of non‐identical fractional‐order chaotic systems

In this paper, multi‐switching combination–combination synchronization scheme has been investigated between a class of four non‐identical fractional‐order chaotic systems. The fractional‐order Lorenz and Chen's systems are taken as drive systems. The combination–combination of multi drive systems is then synchronized with the combination of fractional‐order Lü and Rössler chaotic systems. In multi‐switching combination–combination synchronization, the state variables of two drive systems synchronize with different state variables of two response systems simultaneously. Based on the stability of fractional‐order chaotic systems, the multi‐switching combination–combination synchronization of four fractional‐order non‐identical systems has been investigated. For the synchronization of four non‐identical fractional‐order chaotic systems, suitable controllers have been designed. Theoretical analysis and numerical results are presented to demonstrate the validity and feasibility of the applied method. Copyright © 2017 John Wiley & Sons, Ltd.     

Application of the exp‐function method to nonlinear lattice differential equations for multi‐wave and rational solutions

In this paper, we extend the basic Exp‐function method to nonlinear lattice differential equations for constructing multi‐wave and rational solutions for the first time. We consider a differential‐difference analogue of the Korteweg–de Vries equation to elucidate the solution procedure. Our approach is direct and unifying in the sense that the bilinear formalism of the equation studied becomes redundant. Copyright © 2011 John Wiley & Sons, Ltd.     

Evaluating an hepatic enzyme induction mechanism through coarse‐ and fine‐grained measurements of an in silico liver

Results from simulation experiments falsified the hypothesis that a uniform distribution of simulated drug passing through an in silico liver (ISL) will produce a uniform extent of enzyme induction (EI). Wet‐lab EI experiments, as formulated, are infeasible. The simulated EI is intended to have a hepatic counterpart. The ISL is synthetic, physiologically based, fine‐grained, and multi‐agent. It has been validated against in situ drug disposition data. We discuss methodological considerations regarding the phenomenal manifold, multi‐level observation, and manipulation of synthetic models and their referents. Interestingly, a lower probability of metabolism caused higher EI and, counter‐intuitively, more extraction. © 2008 Wiley Periodicals, Inc. Complexity, 2009     

Global stability and periodic oscillations for an SIV infection model with immune response and intracellular delays

In this paper, we consider the combined effects of cytotoxic T lymphocyte (CTL) responses on the competition dynamics of two Simian immunodeficiency virus (SIV) strains model. One of strains concerns a relatively slowly replicating and mildly cytopathic virus in the early infection (SIVMneCL8), the other is faster replicating and more cytopathic virus at later stages of the infection (SIVMne170). It is shown that the global dynamics of the ordinary differential equations can be determined by several threshold parameters, and we prove the global stability of the equilibria by rigorous mathematical analysis. To account for a series of infection mechanism leading to viral production, we incorporate time delays in the infection term. Using the methods of constructing suitable Lyapunov functionals and LaSalle’s invariance principle, we obtain the sufficient conditions for the global attractiveness of infection-free equilibrium with both virus strains going extinct, single-infection equilibrium with one of two virus strains out-competing the other one and the two strains coexisting infection equilibrium. We establish that the intracellular delays can destabilize the single-infection equilibrium leading to Hopf bifurcation and periodic oscillations. We show that introduction of immune responses is responsible for the coexistence of two virus strains and the intracellular delays may alter the two-strain competition results. Numerical simulations are presented to illustrate the theoretical conclusions.     

Positive solutions for singular systems of multi‐point boundary value problems

We study the existence of positive solutions for systems of singular nonlinear second‐order ordinary differential equations subject to multi‐point boundary conditions. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical analysis nonlinear multi‐term time fractional differential equation with collocation method via fractional B‐spline

In this work, we present numerical analysis for nonlinear multi‐term time fractional differential equation which involve Caputo‐type fractional derivatives for . The proposed method is based on utilization of fractional B‐spline basics in collocation method. The scheme can be readily obtained efficient and quite accurate with less computational work numerical result. The proposal approach transform nonlinear multi‐term time fractional differential equation into a suitable linear system of algebraic equations which can be solved by a suitable numerical method. The numerical experiments will be verify to demonstrate the effectiveness of our method for solving one‐ and two‐dimensional multi‐term time fractional differential equation.