实Clifford分析中,双正则函数向量的非线性边值问题

研究实Clifford分析中关于多个未知函数的双正则函数向量的一个非线性边值问题     

实Clifford分析中超正则函数的Plemelj公式

给出了超正则函数的定义,讨论了实Clifford分析中超正则函数的Plemelj公式,它在边值问题中有广泛的应用.     

Clifford分析中双曲调和函数的一个带位移的非线性边值问题

讨论了Clifford分析中双曲调和函数的一个带位移的非线性边值问题,先讨论了解析函数的一个边值问题的解的存在性,然后利用Clifford分析中双曲调和函数与解析函数的关系讨论了此边值问题的解,并给出了解的积分表达式.     

双hypergenic函数向量的边值问题

讨论实Clifford分析中双hypergenic函数向量的边值问题.首先利用积分方程和压缩映射原理证明了其线性边值问题HL解的存在唯一性,并给出其积分表达式,再利用积分方程和Schauder不动点定理证明了其边值问题HB解的存在性.     

双hypergenic函数向量的带Haseman位移带共轭的边值问题

讨论实Clifford分析中双hypergenic函数向量的带Haseman位移带共轭的边值问题.首先利用积分方程和Schauder不动点定理证明了其边值问题解的存在性,再运用压缩映射原理证明了其线性边值问题解的存在唯一性,并给出解的积分表达式.     

实Clifford分析中,广义双正则函数向量的带位移带共轭的非线性边值问题

首先得到了广义正则函数向量的 Plemelj公式 ,然后利用积分方程的方法和 Arzela- Ascoli定理 ,讨论了实 Clifford分析中广义双正则函数向量的带位移带共轭的非线性边值问题解的存在性 .     

双正则函数向量的带Haseman位移带共轭值的线性及非线性边值问题

首先利用积分方程的方法和Arzela-Ascoli定理讨论了实Clifford分析中双正则函数向量的带Haseman位移带共轭值的非线性边值问题解的存在性及其积分表达式,其次利用压缩映射原理解决了其线性边值问题解的存在唯一性及其积分表达式.     

广义双正则函数向量的带位移带共轭的边值问题

本讨论实Clifford分析中广义双正则函数向量的带位移带共轭的边值问题,首先得到其Plemelj公式,然后用积分方程的方法和压缩映射原理讨论了这个边值问题的存在唯一性。     

广义双正则函数向量的非线性边值问题

利用Schauder不动点定理和积分方程的方法，讨论了实Clifford分析中广义双正则函数向量的非线性边值问题解的存在性及其解的积分表达式。     

实Clifford分析中双超正则函数的Cauchy积分公式

第1部分给出了实Clifford分析中双超正则函数的定义,并运用拟置换的思想得到了双超正则函数的等价条件,第2部分讨论了实Clifford分析中双超正则函数的柯西积分公式.     

On a singular two dimensional nonlinear evolution equation with nonlocal conditions

This paper deals with a nonclassical initial boundary value problem for a two dimensional parabolic equation with Bessel operator. We prove the existence and uniqueness of the weak solution of the given nonlinear problem. We start by solving the associated linear problem. After writing this latter in its operator form, we establish an a priori bound from which we deduce the uniqueness of the strong solution. For the solvability of the associated linear problem, we prove that the range of the operator generated by the considered problem is dense. On the basis of the obtained results of the linear problem, we apply an iterative process to establish the existence and uniqueness of the nonlinear problem.     

An approach for solving nonlinear programming problems

In this paper we use measure theory to solve a wide range of the nonlinear programming problems. First, we transform a nonlinear programming problem to a classical optimal control problem with no restriction on states and controls. The new problem is modified into one consisting of the minimization of a special linear functional over a set of Radon measures; then we obtain an optimal measure corresponding to functional problem which is then approximated by a finite combination of atomic measures and the problem converted approximately to a finite-dimensional linear programming. Then by the solution of the linear programming problem we obtain the approximate optimal control and then, by the solution of the latter problem we obtain an approximate solution for the original problem. Furthermore, we obtain the path from the initial point to the admissible solution.     

Multiphysics mixed finite element method with Nitsche's technique for Stokes-poroelasticity problem

In this article, we propose a multiphysics mixed finite element method with Nitsche's technique for Stokes-poroelasticity problem. Firstly, we reformulate the poroelasticity part of the original problem by introducing two pseudo-pressures to into a “fluid–fluid” coupled problem so that we can use the classical stable finite element pairs to deal with this problem conveniently. Then, we prove the existence and uniqueness of weak solution of the reformulated problem. And we use Nitsche's technique to approximate the coupling condition at the interface to propose a loosely-coupled time-stepping method to solve three subproblems at each time step–a Stokes problem, a generalized Stokes problem and a mixed diffusion problem. And the proposed method does not require any restriction on the choice of the discrete approximation spaces on each side of the interface provided that appropriate quadrature methods are adopted. Also, we give the stability analysis and error estimates of the loosely-coupled time-stepping method. Finally, we give the numerical tests to show that the proposed numerical method has a good stability and no “locking” phenomenon.     

Optimal control of a two-dimensional contact problem

We consider a frictionless contact problem with unilateral constraints for a 2D bar. We describe the problem, then we derive its weak formulation, which is in the form of an elliptic variational inequality of the first kind. Next, we establish the existence of a unique weak solution to the problem and prove its continuous dependence with respect to the applied tractions and constraints. We proceed with the study of an associated control problem for which we prove the existence of an optimal pair. Finally, we consider a perturbed optimal control problem for which we prove a convergence result.     

A study of mixed discrete bilevel programs using semidefinite and semi-infinite programming

In this paper, we study the bilevel programming problem with discrete polynomial lower level problem. We start by transforming the problem into a bilevel problem comprising a semidefinite program (SDP for short) in the lower level problem. Then, we are able to deduce some conditions of existence of solutions for the original problem. After that, we again change the bilevel problem with SDP in the lower level problem into a semi-infinite program. With the aid of the exchange technique, for simple bilevel programs, an algorithm for computing a global optimal solution is suggested, the convergence is shown, and a numerical example is given.     

Periodic Solutions for Second Order Differential Inclusions with Nonconvex and Unbounded Multifunction

In this paper we consider a second order multivalued periodic boundary value problem with a nonconvex and unbounded orientor field (set-valued vector field). Using a directionally continuous selector, through its Filippov regularization we produce a convex-valued, bounded multifunction and with this as orientor field we introduce a new multivalued periodic problem. Using the Leray-Schauder principle, we solve the convex problem and then we show that its solutions also solve the original nonconvex problem.     

An unconstrained minimization method for solving low-rank SDP relaxations of the maxcut problem

In this paper we consider low-rank semidefinite programming (LRSDP) relaxations of combinatorial quadratic problems that are equivalent to the maxcut problem. Using the Gramian representation of a positive semidefinite matrix, the LRSDP problem can be formulated as the nonconvex nonlinear programming problem of minimizing a quadratic function with quadratic equality constraints. For the solution of this problem we propose a continuously differentiable exact merit function that exploits the special structure of the constraints and we use this function to define an efficient and globally convergent algorithm. Finally, we test our code on an extended set of instances of the maxcut problem and we report comparisons with other existing codes.     

Monotone operator theory in convex optimization

In this paper, we establish tractable sum of squares characterizations of the containment of a convex set, defined by a SOS-concave matrix inequality, in a non-convex set, defined by difference of a SOS-convex polynomial and a support function, with Slater’s condition. Using our set containment characterization, we derive a zero duality gap result for a DC optimization problem with a SOS-convex polynomial and a support function, its sum of squares polynomial relaxation dual problem, the semidefinite representation of this dual problem, and the dual problem of the semidefinite programs. Also, we present the relations of their solutions. Finally, through a simple numerical example, we illustrate our results. Particularly, in this example we find the optimal solution of the original problem by calculating the optimal solution of its associated semidefinite problem.     

Backward heat equation with time dependent variable coefficient

Backward heat equation with time dependent variable coefficient is severely ill‐posed in the sense of Hadamard, so we need regularization. In this paper, we consider Backward heat equation with time dependent variable coefficient, and by small perturbing, we obtain an approximation problem. We show this approximation problem is well‐posed with small parameter. Also, we show this approximation system converges to the original problem when parameter goes to zero. Here, we use modified‐quasi boundary value method to regularize this problem. Copyright © 2016 John Wiley & Sons, Ltd.     

Heterogeneous time-harmonic Maxwell equations in bidimensional domains

This note deals with the low-frequency time-harmonic Maxwell equations for a heterogeneous media in bidimensional bounded domains. We propose a three step method to solve this problem. First, we construct an extension of the boundary data solving a scalar Neumann problem for the Laplace operator. Second, we solve a problem in the conductor with an unusual boundary condition of nonlocal type. Third, we solve a boundary value problem in the insulator using the solution calculated in the conductor. Also, this third problem can be reduced to a Neumann problem for the Laplace operator.