On a family of extremum problems and the properties of an integral

The following extremum problem is studied:

Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems

We consider initial value/boundary value problems for fractional diffusion-wave equation: , where 0<α?2, where L is a symmetric uniformly elliptic operator with t-independent smooth coefficients. First we establish the unique existence of the weak solution and the asymptotic behavior as the time t goes to ∞ and the proofs are based on the eigenfunction expansions. Second for α∈(0,1), we apply the eigenfunction expansions and prove (i) stability in the backward problem in time, (ii) the uniqueness in determining an initial value and (iii) the uniqueness of solution by the decay rate as t→∞, (iv) stability in an inverse source problem of determining t-dependent factor in the source by observation at one point over (0,T).     

Existence and uniqueness of positive solutions of boundary value problems for coupled systems of singular second-order three-point non-linear differential and difference equations

Using the mixed monotone method we establish the existence and uniqueness of positive solutions for coupled systems of second-order three-point non-linear differential and difference equations. The theorems obtained are very general and complement previously known results.     

Nonlocal overdetermined boundary value problem for stationary Navier-Stokes equations

A stationary system of Stokes and Navier-Stokes equations describing the flow of a homogeneous incompressible fluid in a bounded domain is considered. The vector of the flow velocity and a finite number of nonlocal conditions are defined at a part of the domain boundary. It is proved that, in the linear case, the problem has at least one stable solution. In the nonlinear case, the local solvability of the problem is proved.     

Non-existence and uniqueness results for boundary value problems for Yang-Mills connections

We show uniqueness results for the Dirichlet problem for Yang-Mills connections defined in -dimensional () star-shaped domains with flat boundary values. This result also shows the non-existence result for the Dirichlet problem in dimension 4, since in 4-dimension, there exist countably many connected components of connections with prescribed Dirichlet boundary value. We also show non-existence results for the Neumann problem. Examples of non-minimal Yang-Mills connections for the Dirichlet and the Neumann problems are also given.

     

Symmetry of heat and mass transfer equations in case of dependence of thermal diffusivity coefficient either on temperature or concentration

This paper describes the solution of group classification problem for heat and mass transfer equations with respect to 3 transport coefficients. Two coefficients depend on temperature and concentration, and the thermal diffusivity coefficient is the function of only one of these state parameters. The forms of the arbitrary elements providing the additional transformations are found. Examples of exact solutions of the governing equations are constructed.     

The forward and inverse problems for a fractional boundary value problem

In this paper, the authors study the forward and inverse problems for a fractional boundary value problem with Dirichlet boundary conditions. The existence and uniqueness of solutions for the forward problem is first proved. Then an inverse source problem is considered.     

Some existence and uniqueness results for boundary value problems for difference equations

Existence and uniqueness results for bvp problems for difference equations are discussed. The weighted norm technique and the Banach contraction mapping principle are employed     

Existence and uniqueness results for boundary value problems of higher order fractional integro-differential equations involving gronwall's inequality in banach spaces

We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α ∈ (n-1,n) in Banach spaces. Existence and uniqueness results of solutions are established by virtue of the Holder's inequality, a suitable singular Gronwall's inequality and fixed point theorem via a priori estimate method. At last, examples are given to illustrate the results.     

Existence and Ulam's type stability results for a class of fractional boundary value problems on a star graph

This paper is concerned with a nonlinear fractional boundary value problem on a star graph. By using a transformation, the suggested problem is converted into an equivalent system of fractional boundary value problem. Schaefer's fixed point theorem and Banach's contraction principle is used to establish its existence and uniqueness results. Further, different kinds of Ulam's type stability results for the proposed problem have been discussed. Finally, two examples are presented to illustrate the application of the obtained results.     

Two-point and three-point boundary value problems for n th-order nonlinear differential equations

In this article, we are concerned with existence and uniqueness of solutions of four kinds of two-point boundary value problems for nth-order nonlinear differential equations by “Shooting” method, and studied existence and uniqueness of solutions of a kind of three-point boundary value problems for nth-order nonlinear differential equations by “Matching” method.     

Uniqueness implies existence and uniqueness criterion for nonlocal boundary value problems for third order differential equations

For the third order differential equation, we consider uniqueness implies existence results for solutions satisfying the nonlocal -point boundary conditions, Uniqueness of solutions of such boundary value problems is intimately related to solutions of the third order equation satisfying certain nonlocal -point boundary conditions. These relationships are investigated as well.

     

Existence theorems for boundary value problems in difference equations

In this research, we study linear difference equations with constant coefficients subject to boundary conditions. Necessary and/or sufficient conditions for the existence of a unique solution will be established. The proofs of the existence and uniqueness theorems are established by means of special types of determinants called Mosaic Vandermonde determinants.     

Existence and uniqueness of a classical solution for the coupled heat and mass transfer during the freezing of high‐water content materials

A recent model for the coupled problem of heat and mass transfer during the solidification of high‐water content materials like soils, foods, tissues and phase‐change materials was developed. This model takes into account the role played by material properties and process variables on the advance of freezing and sublimation fronts, temperature and water vapour profiles and weight loss. The goal of this paper is to determine the existence of a unique local classical solution for the corresponding two‐phase coupled free boundary problem in an adequate functional space. Copyright © 2011 John Wiley & Sons, Ltd.     

Higher-order Lidstone boundary value problems for elliptic partial differential equations

The aim of this paper is to show the existence and uniqueness of a solution for a class of 2nth-order elliptic Lidstone boundary value problems where the nonlinear functions depend on the higher-order derivatives. Sufficient conditions are given for the existence and uniqueness of a solution. It is also shown that there exist two sequences which converge monotonically from above and below, respectively, to the unique solution. The approach to the problem is by the method of upper and lower solutions together with monotone iterative technique for nonquasimonotone functions. All the results are directly applicable to 2nth-order two-point Lidstone boundary value problems.     

高阶微分积分方程的单调迭代法及其应用

首先利用上下解方法以及微分不等式理论给出了n阶微分积分方程的初值问题解的存在性及其单调迭代法,然后将所得结果应用到n阶微分方程的两点边值问题,得到了n阶非线性两点边值问题解的存在性及其单调迭代法,所得结果推广了已有的结果．