Periodic boundary value problems for impulsive fuzzy differential equations

The peculiarity of the Hukuhara derivative makes it impossible to find periodic solutions for fuzzy differential equations with the exception of very restrictive situations. In this work, we consider a boundary value problem associated with an impulsive fuzzy differential equation and approximate the extremal solutions in a fuzzy functional interval using the monotone method. Fuzzy comparison results are useful in our procedure and the expression of the solution for some impulsive periodic ‘linear’ differential problems is also provided.     

A posteriori error majorants of the modeling errors for elliptic homogenization problems

In this paper, we derive new two-sided a posteriori estimates of the modeling errors for linear elliptic boundary value problems with periodic coefficients solved by homogenization. Our approach is based on the concept of functional a posteriori error estimation. The estimates are obtained for the energy norm and use solely the global flux of the non-oscillatory solution of the homogenized model and solution of a boundary value problem on the cell of periodicity.     

Keplerian Action,Convexity Optimization,and the 4-Body Problem

In this paper we introduce a method to construct periodic solutions for the n-body problem with only boundary and topological constraints. Our approach is based on some novel features of the Keplerian action functional, constraint convex optimization techniques, and variational methods. We demonstrate the strength of this method by constructing relative periodic solutions for the planar four-body problem within a special topological class, and our results hold for an open set of masses.     

Numerical Computation of Connecting Orbits in Delay Differential Equations

We discuss the numerical computation of homoclinic and heteroclinic orbits in delay differential equations. Such connecting orbits are approximated using projection boundary conditions, which involve the stable and unstable manifolds of a steady state solution. The stable manifold of a steady state solution of a delay differential equation (DDE) is infinite-dimensional, a problem which we circumvent by reformulating the end conditions using a special bilinear form. The resulting boundary value problem is solved using a collocation method. We demonstrate results, showing homoclinic orbits in a model for neural activity and travelling wave solutions to the delayed Hodgkin–Huxley equation. Our numerical tests indicate convergence behaviour that corresponds to known theoretical results for ODEs and periodic boundary value problems for DDEs.     

An iterative method based on equation decomposition for the fourth‐order singular perturbation problem

In this article, we propose an iterative method based on the equation decomposition technique ( 1 ) for the numerical solution of a singular perturbation problem of fourth‐order elliptic equation. At each step of the given method, we only need to solve a boundary value problem of second‐order elliptic equation and a second‐order singular perturbation problem. We prove that our approximate solution converges to the exact solution when the domain is a disc. Our numerical examples show the efficiency and accuracy of our method. Our iterative method works very well for singular perturbation problems, that is, the case of 0 < ε ? 1, and the convergence rate is very fast. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Estimates of modeling errors for linear elliptic problems

We derive new estimates of modeling errors for linear elliptic boundary value problems with periodic coefficients solved by homogenization method. Our approach is based on the concept of functional a posteriori error estimation. The estimates are obtained for the energy norm and use solely the global flux of the non-oscillatory solution of the homogenized model and solution of a boundary value problem on the cell of periodicity. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Positive solutions of impulsive boundary value problems for first-order nonlinear functional differential equations

In this paper, we study the problem on the existence of positive solutions for a class of impulsive periodic boundary value problems of first-order nonlinear functional differential equations. By using the fixed point theorem in cones and some analysis techniques, we present some sufficient conditions which guarantee the existence of one and multiple positive solutions for the impulsive periodic boundary value problems. Our results generalize and improve some previous results. Moreover, our results show that positive solutions for the impulsive periodic boundary value problems may be yielded completely by some proper impulsive conditions (see Example 4.1 and Remark 4.2 in Sect. 4), and also implies that proper impulsive conditions are of great significance to simulate processes, optimal control, population model and so on.     

Eigenfrequencies of a tube bundle immersed in a fluid

In this paper we study a simplified version of a mathematical model that describes the eigenfrequencies and eigenmotions of a coupled system consisting of a set of tubes (or a tube bundle) immersed in an incompressible perfect fluid. The fluid is assumed to be contained in a rectangular cavity, and the tubes are assumed to be identical, and periodically distributed in the cavity. The mathematical model that governs this physical problem is an elliptic differential eigenvalue problem consisting of the Laplace equation with a nonlocal boundary condition on the holes, and a homogeneous Neumann boundary condition on the boundary of the cavity. In the simplified model that we study in this paper, the Neumann condition is replaced by a periodic boundary condition. Our goal in studying this simple version is to derive some basic properties of the problem that could serve as a guide to envisage similar properties for the original model. In practical situations, this kind of problem needs to be solved for tube bundles containing a very large number of tubes. Then the numerical analysis of these problems is in practice very expensive. Several approaches to overcome this difficulty have been proposed in the last years using homogenization techniques. Alternatively, we propose in this paper an approach that consists in obtaining an explicit decomposition of the problem into a finite family of subproblems, which can be easily solved numerically. Our study is based on a generalized notion of periodic function, and on a decomposition theorem for periodic functions that we introduce in the paper. Our results rely on the theory of almost periodic functions, and they provide a simple numerical method for obtaining approximations of all the eigenvalues of the problem for any number of tubes in the cavity. We also discuss a numerical example.     

Boundary element method for spatially-periodic potential problems

In this paper, we propose a boundary element method for two-dimensional potential problems with one-dimensional spatial periodicity, which have been difficult to be solved by the ordinary boundary element method. In the presented method, we reduce the potential problems with Dirichlet and Neumann boundary conditions to integral equation problems with the periodic fundamental solution of the Laplace operator and, then, obtain approximate solutions by solving linear systems given by discretizing the integral equations. Numerical examples are also included. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of a time periodic solution for the compressible Euler equation with a time periodic outer force

We are concerned with a time periodic supersonic flow through a bounded interval. This motion is described by the compressible Euler equation with a time periodic outer force. Our goal in this paper is to prove the existence of a time periodic solution. Although this is a fundamental problem for other equations, it has not been received much attention for the system of conservation laws until now.When we prove the existence of the time periodic solution, we face with two problems. One is to prove that initial data and the corresponding solutions after one period are contained in the same bounded set. To overcome this, we employ the generalized invariant region, which depends on the space variables. This enable us to investigate the behavior of solutions in detail. Second is to construct a continuous map. We apply a fixed point theorem to the map from initial data to solutions after one period. Then, the map needs to be continuous. To construct this, we introduce the modified Lax–Friedrichs scheme, which has a recurrence formula consisting of discretized approximate solutions. The formula yields the desired map. Moreover, the invariant region grantees that it maps a compact convex set to itself. In virtue of the fixed point theorem, we can prove a existence of a fixed point, which represents a time periodic solution. Finally, we apply the compensated compactness framework to prove the convergence of our approximate solutions.     

分数阶微分方程的一些新结果

第一部分,介绍分数阶导数的定义和著名的Mittag—Leffler函数的性质．第二部分,利用单调迭代方法给出了具有2序列Riemann—Liouville分数阶导数微分方程初值问题解的存在性和唯一性．第三部分,利用上下解方法和Schauder不动点定理给出了具有2序列Riemann—Liouville分数阶导数微分方程周期边值问题解的存在性．第四部分,利用Leray—Schauder不动点定理和Banach压缩映像原理建立了具有n序列Riemann—Liouville分数阶导数微分方程初值问题解的存在性、唯一性和解对初值的连续依赖性．第五部分,利用锥上的不动点定理给出了具有Caputo分数阶导数微分方程边值问题,在超线性（次线性）条件下C310,11正解存在的充分必要条件．最后一部分,通过建立比较定理和利用单调迭代方法给出了具有Caputo分数阶导数脉冲微分方程周期边值问题最大解和最小解的存在性．     

Eventual periodicity of forced oscillations of the Korteweg-de Vries type equation

In this paper, we study the eventual periodicity of the initial boundary value problem (IBVP) for Korteweg-de Vries equation posed on a bounded domain. We show that if the boundary forcing is periodic of period τ, then the solution u of the IBVP at each spatial point becomes eventually time-periodic of period τ. In order to exhibit eventual periodicity, we approximate the solution of the IBVP using the Adomian decomposition method. We compare our work with the approximate solution of IBVP obtained by the homotopy perturbation method and present numerical experiments using Mathematica.     

脉冲微分方程的上、下解及其在常微分方程上的应用

In this paper, we establish the existence of upper and lower solutions for a periodic boundary value problems (PBVP for short) of impulsive differential equations. which guarantees the existence of at least one solution for the problem. As an application, these results are applied to PBVP of ODE and some examples are given to illustrate our results.     

Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients

We propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients. The main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution. This is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions. As a consequence, the base functions are adapted to the local properties of the differential operator. In this paper, we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain a useful asymptotic solution structure. The issue of boundary conditions for the base functions will be discussed. Our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution, independently of the small scale in the homogenization limit. Application of our method to problems with continuous scales is also considered.

     

On Riemann problems for single‐periodic polyanalytic functions

In this article, Riemann‐type boundary‐value problem of single‐periodic polyanalytic functions has been investigated. By the decomposition of single‐periodic polyanalytic functions, the problem is transformed into n equivalent and independent Riemann boundary‐value problems of single‐periodic analytic functions, which has been discussed in details according to two growth orders of functions. Finally, we obtain the explicit expression of the solution and the conditions of solvability for Riemann problem of the single‐periodic polyanalytic functions.     

Singular boundary method for 3D time-harmonic electromagnetic scattering problems

A frequency domain singular boundary method is presented for solving 3D time-harmonic electromagnetic scattering problem from perfect electric conductors. To avoid solving the coupled partial differential equations with fundamental solutions involving hypersingular terms, we decompose the governing equation into a system of independent Helmholtz equations with mutually coupled boundary conditions. Then the singular boundary method employs the fundamental solutions of the Helmholtz equations to approximate the scattered electric field variables. To desingularize the source singularity in the fundamental solutions, the origin intensity factors are introduced. In the novel formulation, only the origin intensity factors for fundamental solutions of 3D Helmholtz equations and its derivatives need to be considered which have been derived in the paper. Several numerical examples involving various perfectly conducting obstacles are carried out to demonstrate the validity and accuracy of the present method.     

Biharmonic equation in a highly oscillating domain and homogenization of an associated control problem

We consider a Dirichlet boundary control problem posed in an oscillating boundary domain governed by a biharmonic equation. Homogenization of a PDE with a non-homogeneous Dirichlet boundary condition on the oscillating boundary is one of the hardest problems. Here, we study the homogenization of the problem by converting it into an equivalent interior control problem. The convergence of the optimal solution is studied using periodic unfolding operator.     

Elastostatic inverse formulation

In this paper an inverse method for solving elastostatic problems with incomplete boundary conditions is presented. In general, inverse problems are ill-posed boundary value problems whose stability and uniqueness of solution and sensitivity-based formulations require additional constraints. In the development we use the Betti-reciprocal theorem to represent the boundary traction field in terms of the boundary and field displacements in an integral form. Initially, we assume the unknown boundary conditions and deformations required to solve the problem. In this way we equate the work done by the exact solution (unknown) to the work done by an assumed solution. Discretizing the resulting equations and using an iterative procedure each step in the solution process becomes the solution to a well-posed problem. Thus, with sufficient perturbations the correct boundary conditions are reconstructed.