Exact and approximation product solutions form of heat equation with nonlocal boundary conditions using Ritz–Galerkin method with Bernoulli polynomials basis

In this article, a new method is introduced for finding the exact solution of the product form of parabolic equation with nonlocal boundary conditions. Approximation solution of the present problem is implemented by the Ritz–Galerkin method in Bernoulli polynomials basis. The properties of Bernoulli polynomials are first presented, then Ritz–Galerkin method in Bernoulli polynomials is used to reduce the given differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the techniques presented in this article for finding the exact and approximation solutions. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1143–1158, 2017     

含积分边界条件的分数阶微分方程边值问题的正解的存在性

研究了含积分边界条件的分数阶微分方程的边值问题，首先给出格林函数及性质，其次将问题转化为一个等价的积分方程，最后应用Krasnoselkii及Leggett-Williams不动点定理得到了一个及多个正解的存在性，推广了以往的结果．     

一类具非局部边界条件的四阶非线性微分方程的对称正解

本文考虑形如的非线性四阶微分方程非局部边值问题,这里a,b∈L~1[0,1],g:(0,1)→[0,∞)在(0,1)上连续、对称,且可能在t=0和t=1处奇异．f:[0,1]×[0,∞)→[0,∞)连续且对所有x∈[0,∞],f(·,x)在[0,1]上对称．在某些适当的增长性条件下,应用Krasnoselskii不动点定理证明了对称正解的存在性和多重性．     

Existence of positive solutions for singular impulsive differential equations with integral boundary conditions

In this paper, we study the existence of positive solutions for singular impulsive differential equations with integral boundary conditions where the nonlinearity f(t,u,v) may be singular at u = 0 and v = 0. The proof is based on the theory of Leray–Schauder degree, together with a truncation technique. Some recent results in the literature are generalized and improved. Copyright © 2014 John Wiley & Sons, Ltd.     

A strong solution of a high-order mixed type partial differential equation with integral conditions

In this article, we study a mixed problem with integral boundary conditions for a high-order partial differential equation of mixed type. We prove the existence and uniqueness of a strong solution. The proof is based on energy inequality and on the density of the range of the operator generated by the considered problem.     

Existence of positive solutions for fourth‐order impulsive integral boundary value problems on time scales

This paper is devoted to the existence of positive solutions for a fourth‐order impulsive boundary value problem with integral boundary conditions on time scales. Existence results of at least two and three positive solutions are established via the double fixed point theorem and six functionals fixed point theorem, respectively. Also, an example is given to illustrate the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.     

具有积分边界条件的四阶奇异特征值问题的正解

本文研究一类具有积分边界条件的四阶奇异特征值问题正解的存在性,非线性项f(t,u)允许在t=0和/或t=1和u=0处奇异.首先给出一个新的比较定理,然后构造奇异特征值问题的上下解,最后运用Schauder不动点定理获得了当f(t,u)关于u是减的情况下正解的存在性,给出了处理f(t,u)允许在u=0处奇异的方法,可以处理f(t,u)在u=0处奇异的方法并不多见.     

含积分边界条件的二阶边值问题的两个正解的存在性

利用Avery-Henderson不动点定理,研究了含积分边界条件的二阶边值问题正解的存在性,推广了以往的结果.     

一类具积分边界条件种群细胞迁移算子的本质谱

本文在L^1空间上，研究一类具积分边界条件种群细胞迁移方程，利用泛函分析中构造算子和比较算子方法及相关半群知识证明了迁移算子A_H产生的G_0半群V_H（t）的Dyson-Phillips展开式的n阶余项R_n（t）（n≥1）的弱紧性及V_H（t）和U_H（t）（streaming算子B_H产生）具有相同的本质谱及一致的本质谱型，得到了在区域Г中迁移算子A_H仅由有限个具有限代数重数的离散本征值组成及迁移方程解的渐近稳定性．     

Banach空间二阶非线性混合型脉冲微分-积分方程边值问题的解

在较弱的条件下,利用MSnch不动点定理,研究了Banach空间中二阶非线性混合型脉冲微分-积分方程边值问题解的存在性,推广和改进了某些已有的结果.