Extended crystal PDE’s stability,I: The general theory

This work, divided in two parts, follows some our previous works devoted to the algebraic topological characterization of PDE’s. In this first part, the stability of PDE’s is studied in some details in the framework of the geometric theory of PDE’s, and bordism groups theory of PDE’s. In particular we identify criteria to recognize PDE’s that are stable (in extended Ulam sense) and in their regular smooth solutions do not occur finite time unstabilities, (stable extended crystal PDE’s). Applications to some important PDE’s are carefully considered. (In the second part a stable extended crystal PDE, encoding anisotropic incompressible magnetohydrodynamics is obtained Ref. [A. Prástaro, Extended crystal PDE’s (submitted for publication)].)     

Extended crystal PDE’S stability,II: The extended crystal MHD-PDE’S

This paper is the second part of a work devoted to the algebraic topological characterization of PDE’s stability, and its relationship with an important class of PDE’s called extended crystals PDE’s in the sense introduced in [A. Prástaro, Extended crystal PDE’s (submitted for publication)]. In fact, their integral bordism groups can be considered as extensions of subgroups of crystallographic groups. This allows us to identify a characteristic class that measures the obstruction to the existence of global solutions. In part I [A. Prástaro, Extended crystal PDE’s stability, I: The general theory, Math. Comput. Modelling, 49 (9–10) (2009) 1759–1780] we identified criteria to recognize PDE’s that are stable (in extended Ulam sense) and in their regular smooth solutions, finite time instabilities do not occur (stable extended crystal PDE’s). Here, we study in some detail, a new PDE encoding anisotropic incompressible magnetohydrodynamics. Stable extended crystal MHD-PDE’s are obtained, where in their smooth solutions, instabilities do not occur in finite time. These results are considered first for systems without a body energy source, and later, by also introducing a contribution from an energy source, in order to take into account nuclear energy production. A condition in order that solutions satisfy the second principle of thermodynamics is given.     

A New Jacobi Elliptic Function Expansion Method for Solvinga Nonlinear PDE Describing Pulse Narrowing Nonlinear Transmission Lines

In this article, we apply the first elliptic function equation to find a new kind of solutions of nonlinear partial differential equations (PDEs) based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method. New exact solutions to the Jacobi elliptic functions of a nonlinear PDE describing pulse narrowing nonlinear transmission lines are given with the aid of computer program, e.g. Maple or Mathematica. Based on Kirchhoff's current law and Kirchhoff's voltage law, the given nonlinear PDE has been derived and can be reduced to a nonlinear ordinary differential equation (ODE) using a simple transformation. The given method in this article is straightforward and concise, and can be applied to other nonlinear PDEs in mathematical physics. Further results may be obtained.     

具有Size结构的捕食种群系统的最优收获策略

分析了一类捕食者种群带有Size结构的捕食-被捕食系统的最优收获问题. 利用不动点定理证明了状态系统及其共轭系统非负解的存在唯一性、解对控制变量的连续依赖性. 应用切锥法锥技巧导出了最优性条件, 借助Ekeland变分原理讨论了最优收获策略的存在唯一性, 推广了年龄结构种群模型中的相应结论.     

A PDE approach to risk measures of derivatives

This paper proposes a partial differential equation (PDE) approach to calculate coherent risk measures for portfolios of derivatives under the Black-Scholes economy. It enables us to define the risk measures in a dynamic way and to deal with American options in a relatively effective way. Our risk measure is based on the representation form of coherent risk measures. Through the use of some earlier results the PDE satisfied by the risk measures are derived. The PDE resembles the standard Black-Scholes type PDE which can be solved using standard techniques from the mathematical finance literature. Indeed, these results reveal that the PDE approach can provide practitioners with a more applicable and flexible way to implement coherent risk measures for derivatives in the context of the Black-Scholes model.     

Quantum exotic PDE’s

Following the previous works on the Prástaro’s formulation of algebraic topology of quantum (super) PDE’s, it is proved that a canonical Heyting algebra (integral Heyting algebra) can be associated to any quantum PDE. This is directly related to the structure of its global solutions. This allows us to recognize a new inside in the concept of quantum logic for microworlds. Furthermore, the Prástaro’s geometric theory of quantum PDE’s is applied to the new category of quantum hypercomplex manifolds, related to the well-known Cayley–Dickson construction for algebras. Theorems of existence for local and global solutions are obtained for (singular) PDE’s in this new category of noncommutative manifolds. Finally, the extension of the concept of exotic PDE’s, recently introduced by Prástaro, has been extended to quantum PDE’s. Then a smooth quantum version of the quantum (generalized) Poincaré conjecture is given too. These results extend ones for quantum (generalized) Poincaré conjecture, previously given by Prástaro.     

Existence of weak solutions to a system of nonlinear partial differential equations modelling ice streams

This paper deals with the mathematical analysis of a nonlinear system of three differential equations of mixed type. It describes the generation of fast ice streams in ice sheets flowing along soft and deformable beds. The system involves a nonlinear parabolic PDE with a multivalued term in order to deal properly with a free boundary which is naturally associated to the problem of determining the basal water flux in a drainage system. The other two equations in the system are an ODE with a nonlocal (integral) term for the ice thickness, which accounts for mass conservation and a first order PDE describing the ice velocity of the system. We first consider an iterative decoupling procedure to the system equations to obtain the existence and uniqueness of solutions for the uncoupled problems. Then we prove the convergence of the iterative decoupling scheme to a bounded weak solution for the original system.     

Rank–Crank-type PDEs and generalized Lambert series identities

We show how Rank–Crank-type PDEs for higher order Appell functions due to Zwegers may be obtained from a generalized Lambert series identity due to the first author. Special cases are the Rank–Crank PDE due to Atkin and the third author and a PDE for a level 5 Appell function also found by the third author. These two special PDEs are related to generalized Lambert series identities due to Watson, and Jackson, respectively. The first author’s Lambert series identity is a common generalization. We also show how Atkin and Swinnerton-Dyer’s proof using elliptic functions can be extended to prove these generalized Lambert series identities.     

Families of traveling impulses and fronts in some models with cross-diffusion

An analysis of traveling wave solutions of partial differential equation (PDE) systems with cross-diffusion is presented. The systems under study fall in a general class of the classical Keller–Segel models to describe chemotaxis. The analysis is conducted using the theory of the phase plane analysis of the corresponding wave systems without a priory restrictions on the boundary conditions of the initial PDE. Special attention is paid to families of traveling wave solutions. Conditions for existence of front–impulse, impulse–front, and front–front traveling wave solutions are formulated. In particular, the simplest mathematical model is presented that has an impulse–impulse solution; we also show that a non-isolated singular point in the ordinary differential equation (ODE) wave system implies existence of free-boundary fronts. The results can be used for construction and analysis of different mathematical models describing systems with chemotaxis.     

A two species competition model under the simultaneous effect of toxicant and disease

A mathematical model is proposed to study the simultaneous effects of toxicants and infectious diseases on a competing species system. It is assumed that the competing populations are adversely affected by the toxicant and one of them is vulnerable to an infectious disease. In this paper, two models are studied separately. The first model is developed to study the effect of only infectious diseases on the existence of a two competing species system in the absence of a toxicant, whereas in the second model the presence of a toxicant is also taken into account. In both the models, conditions for the existence of interior equilibria are derived. The models are analyzed using stability theory, and conditions for the nonlinear stability of the interior equilibria are obtained using Lyapunov’s direct method. Further, the models are studied numerically by taking two sets of numerical values for each model and the results are compared.     

Semi-analytical approximations for a class of multi-parameter eigenvalue problems related to tensile buckling

This work complements the first author’s recent asymptotic investigations on a class of boundary-eigenvalue problems for tensile buckling of thin elastic plates. In particular, it is shown here that the approximations for the critical buckling load can be improved by using a modified energy method that relies directly on the asymptotic results derived previously. We also explore a number of additional mathematical features that have an intrinsic interest in the context of multi-parameter eigenvalue problems.     

A class of multi-period semi-variance portfolio selection with a four-factor futures price model

Considering the stochastic exchange rate, a four-factor futures model with the underling asset, convenience yield, instantaneous risk free interest rate and exchange rate, is established. These processes follow jump-diffusion processes (Wiener process and Poisson process). The corresponding partial differential equation (PDE) of the futures price is derived. The general solution with parameters of the PDE is drawn. The weight least squares approach is applied to obtain the parameters of above PDE. Variance is substituted by semi-variance in Markovitz’s portfolio selection model. Therefore, a class of multi-period semi-variance model is formulated originally. A hybrid genetic algorithm (GA) with particle swarm optimizer (PSO) is proposed to solve the multi-period semi-variance model. Finally, an example, which are fuel futures in Shanghai exchange market, is selected to demonstrate the effectiveness of above models and methods.     

Modelling and control of a shell structure based on a finite dimensional variational formulation

A mathematical model of a controlled shell structure based on Hamilton’s principle and the generalized Ritz–Galerkin method is proposed in this paper. The problem of minimizing the stress energy is solved explicitly for a static version of this model. For the dynamical system under consideration, a procedure for estimating external disturbances and the state vector is derived. We also propose an observer design scheme and solve the stabilization problem for an arbitrary dimension of the linearized model. This approach allows us to perform control design for double-curved shells of complex geometry by combining analytical computation of the controller parameters with numerical data that represent the reference configuration and modal displacements of the shell. As an example, the parameters of our model are validated by results of a finite element analysis for the Stuttgart SmartShell structure.     

Automatic Differentiation for Solving Nonlinear Partial Differential Equations: An Efficient Operator Overloading Approach

By resorting to Automatic Differentiation (AD) users of nonlinear PDE solvers can be relieved from the extra work of linearising a nonlinear PDE system and at the same time improve on the computational efficiency. This paper describes the main AD techniques and discusses how the operator overloading approach of AD can be extended to eliminate the overhead generally incurred with operator overloading. A recent AD system FastDer++, specially designed for this purpose, is integrated into a Least Squares solver. The necessary modifications to the general FEM algorithms. Code fragments and timing results demonstrate that (1) integrating AD with nonlinear PDE solvers leads to highly flexible code with a close resemblance to the mathematical expression of the problem, (2) coding and debugging efforts are greatly reduced, and (3) the computational efficiency is improved.     

Boundary stabilization of non-classical micro-scale beams

In this paper, the problem of boundary stabilization of a vibrating non-classical micro-scale Euler–Bernoulli beam is considered. In non-classical micro-beams, the governing Partial Differential Equation (PDE) of motion is obtained based on the non-classical continuum mechanics which introduces material length scale parameters. In this research, linear boundary control laws are constructed to stabilize the free vibration of non-classical micro-beams which its governing PDE is derived based on the modified strain gradient theory as one of the most inclusive non-classical continuum theories. Well-posedness and asymptotic stabilization of the closed loop system are investigated for both cases of complete and incomplete boundary control inputs. To illustrate the performance of the designed controllers, the closed loop PDE model of the system is simulated via Finite Element Method (FEM). To this end, new strain gradient beam element stiffness and mass matrices are derived in this work.     

Time-consistent investment-reinsurance strategies towards joint interests of the insurer and the reinsurer under CEV models

The present paper studies time-consistent solutions to an investment-reinsurance problem under a mean-variance framework.The paper is distinguished from other literature by taking into account the interests of both an insurer and a reinsurer jointly.The claim process of the insurer is governed by a Brownian motion with a drift.A proportional reinsurance treaty is considered and the premium is calculated according to the expected value principle.Both the insurer and the reinsurer are assumed to invest in a risky asset,which is distinct for each other and driven by a constant elasticity of variance model.The optimal decision is formulated on a weighted sum of the insurer’s and the reinsurer’s surplus processes.Upon a verification theorem,which is established with a formal proof for a more general problem,explicit solutions are obtained for the proposed investment-reinsurance model.Moreover,numerous mathematical analysis and numerical examples are provided to demonstrate those derived results as well as the economic implications behind.     

基于改进型Bessel-Legendre不等式的电力系统时滞相关鲁棒稳定性研究

基于改进型Bessel-Legendre不等式方法,探讨了电力系统时滞相关鲁棒稳定性问题.首先构建了含有时滞环节的电力系统数学模型表达式,并推广出其含有不确定参数的模型表达式,通过应用时滞系统理论分析方法,有效地处理了泛函导数中的积分项,从而推导出了一个具有更小保守性的电力系统时滞相关鲁棒稳定新判据.通过三个数值实例对仿真结果进行了对比与验证,结果表明推导出的新判据具有有效性与优越性.     

Stability Analysis of Decoupled Solution Strategies for Coupled Multi-field Problems – A General Framework

A coupled partial differential equation (PDE) system, stemming from the mathematical modelling of a coupled phenomenon, is usually solved numerically following a monolithic or a decoupled solution method. In spite of the potential unconditional stability offered by monolithic solvers, their usage for solving complex problems sometimes proves cumbersome. This has motivated the development of various partitioned and staggered solution strategies, generally known as decoupled solution schemes. To this end, the problem is broken down into several isolated yet communicating sub-problems that are independently advanced in time, possibly by different integrators. Nevertheless, using a decoupled solver introduces additional errors to the system and, therefore, may jeopardise the stability of the solution [1]. Consequently, to scrutinise the stability of the solution scheme becomes a pertinent step in proposing decoupled solution strategies. Here, we endeavour to present a practical stability analysis algorithm, which can readily be used to reveal the stability condition of numerical solvers. To illustrate its capabilities, the algorithm is then utilised for the stability analysis of solution schemes applied to multi variate coupled PDE systems resulting from the mathematical modelling of surface- and volume-coupled multi-field problems. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hopf bifurcation in doubly fed induction generator under vector control

This paper first presents the Hopf bifurcation phenomena of a vector-controlled doubly fed induction generator (DFIG) which is a competitive choice in wind power industry. Using three-phase back-to-back pulse-width-modulated (PWM) converters, DFIG can keep stator frequency constant under variable rotor speed and provide independent control of active and reactive power output. Main results are illustrated by “exact” cycle-by-cycle simulations. The detailed mathematical model of the closed-loop system is derived and used to analyze the observed bifurcation phenomena. The loci of the Jacobian’s eigenvalues are computed and the analysis shows that the system loses stability via a Hopf bifurcation. Moreover, the boundaries of Hopf bifurcation are also given to facilitate the selection of practical parameters for guaranteeing stable operation.