Browsing by Subject "boundary value problem"

Now showing 1 - 9 of 9

Access status: Open Access ,
About sign-constancy of Green's functions for impulsive second order delay equations
(2014) Domoshnitsky, Alexander; Landsman, Guy; Yanetz, Shlomo
We consider the following second order differential equation with delay $\begin{cases} (Lx)(t)\equiv{x''(t)+\sum_{j=1}^p {b_{j}(t)x(t-\theta_{j}(t))}}=f(t), \quad t\in[0,\omega],\\ x(t_j)=\gamma_{j}x(t_j-0), x'(t_j)=\delta_{j}x'(t_j-0), \quad j=1,2,\ldots,r. \end{cases}$ In this paper we find necessary and sufficient conditions of positivity of Green's functions for this impulsive equation coupled with one or two-point boundary conditions in the form of theorems about differential inequalities. By choosing the test function in these theorems, we obtain simple sufficient conditions. For example, the inequality $\sum_{i=1}^p{b_i(t)\left(\frac{1}{4}+r\right)}\lt \frac{2}{\omega^2}$ is a basic one, implying negativity of Green's function of two-point problem for this impulsive equation in the case $0\lt \gamma_i\leq{1}$, $0\lt \delta_i\leq{1}$ for $i=1,\ldots ,p$.
Access status: Open Access ,
Application of Green's operator to quadratic variational problems
(2006) Azbelev, Nikolay V.; Tsalyuk, Vadim Z.
We use Green’s function of a suitable boundary value problem to convert the variational problem with quadratic functional and linear constraints to the equivalent unconstrained extremal problem in some subspace of the space $L_{2}$ of quadratically summable functions. We get the neccessary and sufficient criterion for unique solvability of the variational problem in terms of the spectrum of some integral Hilbert–Schmidt operator in $L_{2}$ with symmetric kernel. The numerical technique is proposed to estimate this criterion. The results are demonstrated on examples: 1) a variational problem with deviating argument, and 2) the problem of the critical force for the vertical pillar with additional support point (the qualities of the pillar may vary discontinuously along the pillar’s axis).
Access status: Open Access ,
Fundamental solution of the problem describing ship motion in waves
(2006) Jankowski, Jan
The problem describing a ship motion in waves comprises the Laplace equation, boundary condition on wetted surface of the ship, condition on the free surface of the sea in the form of a differential equation, the radiation condition, and a condition at infinity. This problem can be transformed to a Fredholm equation of second kind, and then numerically solved using the boundary element method, if the fundamental solution of the problem is known. This paper presents the derivation of the fundamental solution. In physical interpretation, the fundamental solution represents the moving and pulsating source under free surface of the sea. The free surface elevation, generated by the source for different forward speed and frequency of pulsation, is presented in this paper.
Access status: Open Access ,
On nonexistence of global in time solution for a mixed problem for a nonlinear evolution equation with memory generalizing the Voigt-Kelvin rheological model
(Wydawnictwa AGH, 2017) Pukach, Petro; Il'kiv, Volodymyr; Nytrebych, Zinovii; Vovk, Myroslava
The paper deals with investigating of the first mixed problem for a fifth-order nonlinear evolutional equation which generalizes well known equation of the vibrations theory. We obtain sufficient conditions of nonexistence of a global solution in time variable.
Access status: Open Access ,
Polynomials on the space of ω-ultradifferentiable functions
(2007) Grasela, Katarzyna
The space of polynomials on the space $D_{\omega}$ of $\omega$-ultradifferentiable functions is represented as the direct sum of completions of symmetric tensor powers of $D^{\prime}_{\omega}$.
Access status: Open Access ,
Positive solutions to a third order nonlocal boundary value problem with a parameter
(Wydawnictwa AGH, 2024) Szajnowska, Gabriela; Zima, Mirosława
We present some sufficient conditions for the existence of positive solutions to a third order differential equation subject to nonlocal boundary conditions. Our approach is based on the Krasnosel'skiĭ-Guo fixed point theorem in cones and the properties of the Green's function corresponding to the BVP under study. The main results are illustrated by suitable examples.
Access status: Open Access ,
Right focal boundary value problems for difference equations
(2010) Henderson, Johnny; Liu, Xueyan; Lyons, Jeffrey W.; Neugebauer, Jeffrey T.
An application is made of a new Avery et al. fixed point theorem of compression and expansion functional type in the spirit of the original fixed point work of Leggett and Williams, to obtain positive solutions of the second order right focal discrete boundary value problem. In the application of the fixed point theorem, neither the entire lower nor entire upper boundary is required to be mapped inward or outward. A nontrivial example is also provided.
Access status: Open Access ,
The use of integral information in the solution of a two-point boundary value problem
(2007) Drwięga, Tomasz
We study the worst-case ε-complexity of a two-point boundary value problem $u^{\prime\prime}(x)=f(x)u(x)$, $x \in [0,T]$, $u(0)=c$, $u^{\prime}(T)=0$, where $c,T \in \mathbb{R}$ ($c \neq 0$, $T \gt 0$) and $f$ is a nonnegative function with $r$ ($r\geq 0$) continuous bounded derivatives. We prove an upper bound on the complexity for linear information showing that a speed-up by two orders of magnitude can be obtained compared to standard information. We define an algorithm based on integral information and analyze its error, which provides an upper bound on the $\varepsilon$-complexity.
Access status: Open Access ,
Weak solutions for nonlinear fractional differential equations with integral boundary conditions in Banach spaces
(2012) Benchohra, Mouffak; Mostefai, Fatima-Zohra
The aim of this paper is to investigate a class of boundary value problems for fractional differential equations involving nonlinear integral conditions. The main tool used in our considerations is the technique associated with measures of weak noncompactness.