Generating Layer-Adapted Meshes Using Mesh Partial Differential Equations

We present a new algorithm for generating layer-adapted meshes for the finite element solution of singularly perturbed problems based on mesh partial differential equations (MPDEs). The ultimate goal is to design meshes that are similar to the well-known Bakhvalov meshes, but can be used in more general settings: specifically two-dimensional problems for which the optimal mesh is not tensor-product in nature. Our focus is on the efficient implementation of these algorithms, and numerical verification of their properties in a variety of settings. The MPDE is a nonlinear problem, and the efficiency with which it can be solved depends adversely on the magnitude of the perturbation parameter and the number of mesh intervals. We resolve this by proposing a scheme based on $h$-refinement. We present fully working FEniCS codes [Alnaes et al., Arch. Numer. Softw., 3 (100) (2015)] that implement these methods, facilitating their extension to other problems and settings.     

Quantum and Integral (Co)Bordisms in Partial Differential Equations

Characterizations of quantum bordisms and integral bordisms in PDEs by means of subgroups of usual bordism groups are given. More precisely, it is proved that integral bordism groups can be expressed as extensions of quantum bordism groups and these last are extensions of subgroups of usual bordism groups. Furthermore, a complete cohomological characterization of integral bordism and quantum bordism is given. Applications to particular important classes of PDEs are considered. Finally, we give a complete characterization of integral and quantum singular bordisms by means of some suitable characteristic numbers. Some examples of interesting PDEs which arise in physics are also considered where existence of solutions with change of sectional topology (tunnel effect) is proved. As an application, we relate integral bordism to the spectral term that represents the space of conservation laws for PDEs. This also gives a general method to associate in a natural way a Hopf algebra to any PDE.     

Averaging of Backward Stochastic Differential Equations and Homogenization of Partial Differential Equations with Periodic Coefficients

Abstract

We study the limit of the solutions of systems of semi-linear partial differential equations (PDEs) of second order of parabolic type, with rapidly oscillating periodic coefficients, a singular drift, and singular coefficients of the zero and second order terms. Our basic tool is the approach given by Pardoux [14 Pardoux , E. 1999 . Homogenization of linear and semilinear second order parabolic PDEs with periodic coefficients: a probabilistic approach . J. Funct. Anal. 167 : 498 – 520 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]]. In particular, we use the weak convergence of an associated backward stochastic differential equation (BSDE).     

Dirichlet Series for Quasilinear Partial Differential Equations

We consider the construction of Dirichlet series for quasilinear partial differential equations. We obtain a remarkable result that for the class of equations under study, the only equations that admit such a series solution are transformable back onto the only known integrable equation within the class.     

Nash Equilibria for the Multiobjective Control of Linear Partial Differential Equations

This article is concerned with the numerical solution of multiobjective control problems associated with linear partial differential equations. More precisely, for such problems, we look for the Nash equilibrium, which is the solution to a noncooperative game. First, we study the continuous case. Then, to compute the solution of the problem, we combine finite-difference methods for the time discretization, finite-element methods for the space discretization, and conjugate-gradient algorithms for the iterative solution of the discrete control problems. Finally, we apply the above methodology to the solution of several tests problems.     

A Multipoint Problem with Complex Coefficients for Partial Differential Equations That Are Unsolvable for the Highest Derivative

Conditions of correct solvability in various functional spaces of a problem with multipoint conditions in the time variable and periodicity conditions in the space coordinate are established for partial differential equations that are unsolvable for the highest derivative with time. Metric theorems on lower estimates of small denominators that arise in the construction of solutions are proved.     

The Improved Modified Decomposition Method for the Treatment of Systems of Singular Nonlinear Partial Differential Equations with Initial Conditions

New developed inverse differential operators incorporated into the semi- analytical treatment of the modified decomposition method （MDM） are used to solve the systems of first and second-order singular nonlinear partial differential equations （PDEs） with initial conditions arising in physics. The new proposed method is called the improved modified decomposition method （IMDM）, and is used to the treatment of a few case study initial-value problems. The results obtained by the IMDM are in full agreement with the existing exact analytical solutions.     

Spatiotemporal dynamics and feedforward control of an infinite-dimensional,time-variant,thermal-coating process

In this contribution, a model for the spatiotemporal dynamics of a thermal-coating process is derived from first principles (the conservation of energy). The model is a dynamical system consisting of a system of hyperbolic, partial differential equations (PDEs), describing the evolution of the temperature distribution of the substrate. By studying the C ₀- semigroup generated by the system operator we find that in discrete time the infinite-dimensional plant may conveniently be represented by finite-dimensional operators defined on appropriately chosen Euclidean spaces. This representation provides the basis for numerical efficient solution of several optimal feedforward control problems associated with set point changes and launching of the process. Numerical and experimental studies highlight the value of this approach.     

非线性中立抛物型偏微分方程系统的振动性定理

研究一类非线性中立抛物型时滞偏微分方程系统解的振动性质,利用积分不等式和泛函微分方程的某些结果,获得了该类系统在第一类边值条件下所有解振动的若干充分条件.结论充分表明振动是由时滞量引起的.     

关于一类中立型偏微分方程

本文研究了一类一阶脉冲中立型偏泛函微分方程mild解的存在性和唯一性,改进并推广了文[11]的结果。     

Reduction of the Appell's System of Partial Differential Equations to the System of Total Differential Equations

In this paper it is shown that the Appell's system of partial differential equations, with two complex variables x and y, reduces to the system of total differential equations. Also, it is obtained the differential equation on the section y=const.     

关于某些非线性偏微分方程的准确解

以分层理论为基础，对偏微分方程给以一种全新的分类．     

Non-Commutative Polynomial Solutions to Partial Differential Equations

This paper extends the notion of a differential equation to the space of polynomials p in non-commutative variables x = (x ₁, . . . , x _g ). The directional derivative D[p, x _i , h] of p in x _i is a polynomial in x and a non-commuting variable h. When all variables commute, D[p, x _i , h] is equal to h?p/?x _i . This paper classifies all non-commutative polynomial solutions to a special class of partial differential equations, including the non-commutative extension of Laplace??s equation. Of interest also are non-commutative subharmonic polynomials. A non-commutative polynomial is subharmonic if its non-commutative Laplacian takes positive-semidefinite matrix values whenever matrices X ₁, . . . , X _g , H are substituted for the variables x ₁, . . . , x _g , h. This paper shows that a homogeneous subharmonic polynomial which is bounded below is a harmonic polynomial??that is, a non-commutative solution to Laplace??s equation??plus a sum of squares of harmonic polynomials.     

具非线性扩散系数的中立双曲型偏微分方程系统解的振动定理

考虑一类具非线性扩散系数的中立双曲型偏微分方程系统解的振动性.利用Green公式和Riccati变换,获得了该方程组在两类不同边值条件下振动的若干充分条件.     

具有连续时滞的中立型双曲偏微分方程系统解的振动性

研究了一类具有连续时滞变量的非线性中立型双曲型偏微分方程系统,解的振动性.获得了该方程组在Robin边值条件和Dirichlet边值条件下解振动的充分条件.     

一阶线性微分方程组的一种解法

利用矩阵知识给出了一阶线性微分方程组的一种用公式表达的解法,其优点在于一方面可以避免繁琐的复矩阵运算以及求复特征向量的运算,另一方面可以简化求解过程.     

Optimal Control of One-Dimensional Partial Differential Algebraic Equations with Applications

We present an approach to compute optimal control functions in dynamic models based on one-dimensional partial differential algebraic equations (PDAE). By using the method of lines, the PDAE is transformed into a large system of usually stiff ordinary differential algebraic equations and integrated by standard methods. The resulting nonlinear programming problem is solved by the sequential quadratic programming code NLPQL. Optimal control functions are approximated by piecewise constant, piecewise linear or bang-bang functions. Three different types of cost functions can be formulated. The underlying model structure is quite flexible. We allow break points for model changes, disjoint integration areas with respect to spatial variable, arbitrary boundary and transition conditions, coupled ordinary and algebraic differential equations, algebraic equations in time and space variables, and dynamic constraints for control and state variables. The PDAE is discretized by difference formulae, polynomial approximations with arbitrary degrees, and by special update formulae in case of hyperbolic equations. Two application problems are outlined in detail. We present a model for optimal control of transdermal diffusion of drugs, where the diffusion speed is controlled by an electric field, and a model for the optimal control of the input feed of an acetylene reactor given in form of a distributed parameter system.     

非线性偏微分方程的微局部分析

这是一篇介绍当前方程界十分重要的课题——非线性微局部分析——的综述文章。作为这一研究领域的开拓者,我们在不太长的篇幅里,从相当的理论高度简洁地介绍该领域近十年来一些最引人注目的工作。本文首先阐述了一般微局部分析的基本思想,然后介绍近十年来对非线性偏微分方程起重要推动作用的仿微分计算(如仿乘积,仿微分算子,仿复合等),以及有着更深刻内容的高次微局部的思想。同时,也大量介绍这些思想在非线性偏微分方程弱奇性分析中的应用,如奇性的传播,反射与绕射,余法型奇性的相互作用,非线性亚椭圆性,以及三个奇性波的相互作用等,这些均是当前方程界的热门课题。     

从超布朗运动到随机偏微分方程

随机偏微分方程(SPDE)是目前国内外广泛关注研究进展迅速的一个活跃的学术研究领域.该主题的研究涉及概率论(随机分析、随机场)、偏微分方程、调和分析等诸多分支学科方向.特别是随机偏微分方程其背景更多地源于现代物理学、化学、生物学、经济学等应用性学科,这使得该领域的研究显示出较强的意义和活力.本文从超布朗运动研究出发,发展性地提出有较强背景意义的典型类随机偏微分方程,并进而过渡到一般及更广泛类的随机偏微分方程的研究.同时我们系统地总结了关于高阶随机偏微分方程和随机波动方程的研究成果.     

Solutions of Discrete-Velocity Boltzmann Equations via Bateman and Riccati Equations

We propose several approaches for solving two discrete-velocity Boltzmann equations using the rescaling ansatz and the truncated Painlevé expansions. We use solutions of the two- and three-dimensional Bateman equations for the singularity manifold conditions to reduce the problem to Riccati equations. Both equations fail the Painlevé test.