四阶泛函微分方程边值问题正解的存在性

利用锥拉伸与锥压缩不动点定理,研究了一类四阶泛函微分方程边值问题正解的存在性,得到其正解及多个正解存在的若干充分条件,所得结果是相应常微分方程边值问题已有结论的拓广.     

一类Caputo分数阶微分方程多点边值问题的多解性

本文讨论了一类Caputo分数阶微分方程多点边值问题的多解性,通过把分数阶微分方程的边值问题转化成与其等价的积分方程问题求出边值问题的Green函数并得到其格林函数的相关性质,最后利用锥上不动点指数定理研究分数阶微分方程多点边值问题正解和多个正解的存在性.     

一类分数阶微分方程边值问题的三个正解

利用锥上Avery-Peterson不动点定理,研究了一类分数阶微分方程积分边值问题正解的存在性,给出了该边值问题至少存在三个正解的充分条件.     

四阶微分方程三点边值问题三个正解的存在性

在四阶微分方程非线性项f中含有未知函数“的二阶导数u”的情况下,运用Avery-Peterson不动点定理,研究了一类四阶微分方程三点边值问题三个正解的存在性,得到了该类边值问题存在三个正解的充分条件．     

带p-Laplacian算子三点奇异边值问题多个对称正解的存在性

利用凸锥上的不动点定理,研究了一类带p-Laplacian算子的微分方程三点奇异边值问题对称正解的多重性,得到了这类边值问题存在多个对称正解的充分条件.     

一类分数阶微分方程m点边值问题正解的存在性

给出了一类分数阶微分方程m点边值问题的格林函数,通过讨论其性质,运用uo有界算子和不动点指数理论,在与相应的线性算子第一特征值有关的条件下获得了分数阶微分方程多点边值问题正解及多个正解的存在性结果.     

四阶非线性奇异微分方程两点边值问题的正解

利用不动点指数理论,研究四阶非线性奇异微分方程两点边值问题正解及多重正解的存在性.     

一维p-Laplacian算子方程三点奇异边值问题单调正解的多重性

收稿利用锥上的不动点指数理论,研究了类带p-Laplacian算子的微分方程三点奇异边值问题单调正解的多重性,得到了这类边值问题存在多个单调正解的充分条件.     

p-Laplacian分数阶微分方程边值问题正解的存在性

在这篇文章,我们讨论了一类p-Laplacian分数阶微分方程边值问题正解的存在性,应用凸锥上的不动点定理,我们得到了这类边值问题至少存在一个和两个正解的充分条件.     

一类带p-Laplacian算子分数阶微分方程边值问题的正解

本文应用凸锥上的不动点定理,讨论了一类带p-Laplacian算子分数阶微分方程边值问题的正解的存在性,分别得到了这类边值问题至少存在一个正解和多个正解的充分条件.最后,给出了两个具体的例子.     

一类非线性三阶微分方程边值问题解的存在唯一性

研究了一类非线性三阶微分方程边值问题解的存在唯一性.首先分析了近年来国内外三阶微分方程边值问题的研究成果,提出了边值条件中含非线性函数的非线性三阶微分方程边值问题.然后寻找相关线性问题的解决途径,利用Banach不动点定理,证明了提出的边值问题存在唯一解.最后,举例阐述了主要结果的应用.     

Computational methods of solving the boundary value problems for the loaded differential and Fredholm integro‐differential equations

The article presents a new general solution to a loaded differential equation and describes its properties. Solving a linear boundary value problem for loaded differential equation is reduced to the solving a system of linear algebraic equations with respect to the arbitrary vectors of general solution introduced. The system's coefficients and right sides are computed by solving the Cauchy problems for ordinary differential equations. Algorithms of constructing a new general solution and solving a linear boundary value problem for loaded differential equation are offered. Linear boundary value problem for the Fredholm integro‐differential equation is approximated by the linear boundary value problem for loaded differential equation. A mutual relationship between the qualitative properties of original and approximate problems is obtained, and the estimates for differences between their solutions are given. The paper proposes numerical and approximate methods of solving a linear boundary value problem for the Fredholm integro‐differential equation and examines their convergence, stability, and accuracy.     

On two-parameter spectral problems and their application

We study a spectral problem with two complex parameters for a normal linear system of second-order ordinary differential equations on a closed interval with splitting or nonlocal boundary conditions. The results of this study are used to prove the existence and uniqueness of a generalized solution of a boundary value problem in a cylinder for a class of partial differential equations.     

A parareal approach of semi‐linear parabolic equations based on general waveform relaxation

We present a parareal approach of semi‐linear parabolic equations based on general waveform relaxation (WR) at the partial differential equation (PDE) level. An algorithm for initial‐boundary value problem and two algorithms for time‐periodic boundary value problem are constructed. The convergence analysis of three algorithms are provided. The results show that the algorithm for initial‐boundary value problem is superlinearly convergent while both algorithms for the time‐periodic boundary value problem linearly converge to the exact solutions at most. Numerical experiments show that the parareal algorithms based on general WR at the PDE level, compared with the parareal algorithm based on the classical WR at the ordinary differential equations (ODEs) level (the PDEs is discretized into ODEs), require much fewer number of iterations to converge.     

Fredholm property of boundary value problems for a fourth-order elliptic differential-operator equation with operator boundary conditions

In a Hilbert space H, we study the Fredholm property of a boundary value problem for a fourth-order differential-operator equation of elliptic type with unbounded operators in the boundary conditions. We find sufficient conditions on the operators in the boundary conditions for the problem to be Fredholm. We give applications of the abstract results to boundary value problems for fourth-order elliptic partial differential equations in nonsmooth domains.     

Weak solvability of a fractional Voigt viscoelasticity model

The existence of a weak solution of a boundary value problem for a fractional Voigt viscoelasticity model is proved. The proof relies on an approximation of the original boundary value problem by regularized ones and recent results concerning the solvability of Cauchy problems for systems of ordinary differential equations in the class of regular Lagrangian flows.     

Universal boundary value problem for equations of mathematical physics

A new statement of a boundary value problem for partial differential equations is discussed. An arbitrary solution to a linear elliptic, hyperbolic, or parabolic second-order differential equation is considered in a given domain of Euclidean space without any constraints imposed on the boundary values of the solution or its derivatives. The following question is studied: What conditions should hold for the boundary values of a function and its normal derivative if this function is a solution to the linear differential equation under consideration? A linear integral equation is defined for the boundary values of a solution and its normal derivative; this equation is called a universal boundary value equation. A universal boundary value problem is a linear differential equation together with a universal boundary value equation. In this paper, the universal boundary value problem is studied for equations of mathematical physics such as the Laplace equation, wave equation, and heat equation. Applications of the analysis of the universal boundary value problem to problems of cosmology and quantum mechanics are pointed out.     

Some existence results for differential equations of second order

Existence results for initial value problem and Dirichlet boundary value problem for nonlinear differential equations of second order are obtained.     

Numerical stability of a method for transferring boundary conditions

It is proved that a previously proposed method for transferring boundary conditions as applied to a boundary value problem for a linear system of ordinary differential equations gives numerically stable results if this problem is stable with respect to small variations in the input data.     

Multipoint boundary value problems for nonlinear ordinary differential equations

In this paper we provide sufficient conditions for the existence of solutions to multipoint boundary value problems for nonlinear ordinary differential equations. We consider the case where the solution space of the associated linear homogeneous boundary value problem is less than 2. When this solution space is trivial, we establish existence results via the Schauder Fixed Point Theorem. In the resonance case, we use a projection scheme to provide criteria for the solvability of our nonlinear boundary value problem. We accomplish this by analyzing a link between the behavior of the nonlinearity and the solution set of the associated linear homogeneous boundary value problem.