Spectral methods for some singularly perturbed third order ordinary differential equations

Spectral methods with interface point are presented to deal with some singularly perturbed third order boundary value problems of reaction-diffusion and convection-diffusion types. First, linear equations are considered and then non-linear equations. To solve non-linear equations, Newton’s method of quasi-linearization is applied. The problem is reduced to two systems of ordinary differential equations. And, then, each system is solved using spectral collocation methods. Our numerical experiments show that the proposed methods are produce highly accurate solutions in little computer time when compared with the other methods available in the literature.      

Local existence of multiple positive solutions to a singular cantilever beam equation

By constructing suitable cone and control functions, we prove some local existence theorems of positive solutions for a singular fourth-order two-point boundary value problem. In mechanics, the problem is called cantilever beam equation. Furthermore, we improve a famous method appeared in the studies of singular boundary value problems. The approximation theorem of completely continuous operators and the Guo-Krasnosel'skii fixed point theorem of cone expansion-compression type play important parts in this work.     

Positive solutions for boundary value problem of nonlinear fractional differential equation

In this paper, we investigate the existence and multiplicity of positive solutions for nonlinear fractional differential equation boundary value problem:

     

On positive solutions of boundary value problems for second-order functional differential equations on infinite intervals

We study the existence of positive solutions for the following boundary value problem on infinite interval for second-order functional differential equations:

     

Exact number of positive solutions for a class of semipositone problems

This paper is concerned with the boundary value problems y″+λ(y^p−y^q)=0 and y(−1)=y(1)=0, where p>q>−1 and λ>0 is a positive parameter. We discuss the existence of positive solutions and give a complete study.     

二阶时滞微分方程边值问题的正解

本文利用锥映射不动点指数定理研究了二阶时滞微分方程边值问题正解的存在性,其中τ>0.推广了Liu和Li关于常微分方程边值问题的工作.     

Similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition

The steady laminar boundary layer flow over a permeable flat plate in a uniform free stream, with the bottom surface of the plate is heated by convection from a hot fluid is considered. Similarity solutions for the flow and thermal fields are possible if the mass transpiration rate at the surface and the convective heat transfer from the hot fluid on the lower surface of the plate vary like x^−1/2, where x is the distance from the leading edge of the solid surface. The governing partial differential equations are first transformed into ordinary differential equations, before being solved numerically. The effects of the governing parameters on the flow and thermal fields are thoroughly examined and discussed.     

Exact number of positive solutions for semipositone problems

This paper is concerned with the exact number of positive solutions for the boundary value problem ^′(|y^′|^p−2y^′)+λf(y)=0 and y(−1)=y(1)=0, where p>1 and λ>0 is a positive parameter. We consider the case in which both f(u) and g(u)=(p−1)f(u)−uf^′(u) change sign exactly once from negative to positive on (0,∞).     

Blowup of solutions for improved Boussinesq type equation

The paper studies the existence and uniqueness of local solutions and the blowup of solutions to the initial boundary value problem for improved Boussinesq type equation u_tt−u_xx−u_xxtt=σ(u)_xx. By a Galerkin approximation scheme combined with the continuation of solutions step by step and the Fourier transform method, it proves that under rather mild conditions on initial data, the above-mentioned problem admits a unique generalized solution u∈W^2,∞([0,T];H²(0,1)) as long as . In particular, when σ(s)=as^p, where a≠0 is a real number and p>1 is an integer, specially a<0 if p is an odd number, the solution blows up in finite time. Moreover, two examples of blowup are obtained numerically.     

On the structure of similarity solutions of a differential equation arising from boundary layer problem

We consider an ordinary differential equation with f(0)=a, f^′(0)=1, f^′(∞):=limt→∞f^′(t)=0, where β is a real constant. The given problem may arise from the study of steady free convection flow over a vertical semi-infinite flat plate in a porous medium, or the study of a boundary layer flow over a vertical stretching wall. In this paper, the structure of solutions for the cases of β?−2 is studied. Combining the results of [B. Brighi, T. Sari, Blowing-up coordinates for a similarity boundary layer equation, Discrete Contin. Dyn. Syst. 5 (2005) 929-948; J.-S. Guo, J.-C. Tsai, The structure of solution for a third order differential equation in boundary layer theory, Japan J. Indust. Appl. Math. 22 (2005) 311-351; J.-C. Tsai, Similarity solutions for boundary layer flows with prescribed surface temperature, Appl. Math. Lett. 21 (1) (2008) 67-73], we conclude that the given problem may possess at most two types solutions for β∈R. Moreover, multiple solutions are also verified for various pairs of (a,β).     

On a boundary value problem for delay differential equations of population dynamics and Chebyshev approximation

In this paper, a boundary value problem for delay differential equations of population dynamics is considered. We obtain approximate solutions by using Chebyshev polynomial series and Newton–Raphson's procedure and give the error estimation. The method of the error estimation has been obtained in an existence theorem proved by a part of the authors. We carry out some numerical experiments by a computer language MATLAB.     

Green's function for a third-order generalized right focal problem

We determine the Green's function for the third-order three-point generalized right focal boundary value problem

     

Eigenvalue criteria for existence of multiple positive solutions to boundary value problems of second-order delay differential equations

Criteria are established for existence of multiple positive solutions to the boundary value problems expressed by second-order delay differential equations of the form

     

Banach空间中一类分数阶微分方程边值问题

研究Banach空间中一类非线性分数阶微分方程边值问题.构建此类方程的Green函数,利用非紧测度和相关的不动点定理,得到了此类方程的mild解存在的几个充分条件,所得结果改进和推广了一些已有的结论.     

一类带奇异性的两点边值问题

对一类带有奇异性的两点边值问题讨论正解的存在性,在很一般的条件下,建立了摄动问题的可解性与原问题的可解性之间的关系,做为此结论的应用,对某些特殊情形,给出正解存在的充分必要条件。