scholarly journals Singular Dirichlet Boundary Value Problem for Second Order Ode

2007 ◽  
Vol 14 (2) ◽  
pp. 325-340
Author(s):  
Irena Rachůnková ◽  
Jakub Stryja
Keyword(s):  
Dirichlet Problem ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Value Problem ◽  
Existence Of A Solution ◽  
Absolutely Continuous ◽  
First Derivative ◽  
Carathéodory Conditions ◽  
Time Singularities ◽  
Existence Principle ◽  
Singular Dirichlet Problem

Abstract This paper investigates the singular Dirichlet problem –𝑢″ = 𝑓(𝑡, 𝑢, 𝑢′), 𝑢(0) = 0, 𝑢(𝑇) = 0, where 𝑓 satisfies the Carathéodory conditions on the set and . The function 𝑓(𝑡, 𝑥, 𝑦) can have time singularities at 𝑡 = 0 and 𝑡 = 𝑇 and space singularities at 𝑥 = 0 and 𝑦 = 0. The existence principle for the above problem is given and its application is presented here. The paper provides conditions which guarantee the existence of a solution which is positive on (0; T) and which has the absolutely continuous first derivative on [0, 𝑇].

Multiple solutions for semi-linear corner degenerate elliptic equations with singular potential term

2016 ◽  
Vol 19 (04) ◽  
pp. 1650043 ◽  
Author(s):  
Hua Chen ◽  
Shuying Tian ◽  
Yawei Wei
Keyword(s):  
Dirichlet Problem ◽  
Variational Method ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Singular Potential ◽  
Degenerate Elliptic Equations ◽  
Dirichlet Boundary Value Problem ◽  
Potential Term ◽  
Degenerate Elliptic

The present paper is concern with the Dirichlet problem for semi-linear corner degenerate elliptic equations with singular potential term. We first give the preliminary of the framework and then discuss the weighted corner type Hardy inequality. By using the variational method, we prove the existence of multiple solutions for the Dirichlet boundary-value problem.

scholarly journals Variational Method to p-Laplacian Fractional Dirichlet Problem with Instantaneous and Noninstantaneous Impulses

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Yiru Chen ◽  
Haibo Gu ◽  
Lina Ma
Keyword(s):  
Dirichlet Problem ◽  
Variational Method ◽  
Boundary Value Problem ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Major Result ◽  
Dirichlet Boundary Value Problem ◽  
Fractional Differential ◽  
Noninstantaneous Impulses

In this paper, a research has been done about the existence of solutions to the Dirichlet boundary value problem for p-Laplacian fractional differential equations which include instantaneous and noninstantaneous impulses. Based on the critical point principle and variational method, we provide the equivalence between the classical and weak solutions of the problem, and the existence results of classical solution for our equations are established. Finally, an example is given to illustrate the major result.

scholarly journals About the Dirichlet Boundary Value Problem using Clifford Algebras

2018 ◽  
Vol 15 ◽  
pp. 8098-8119
Author(s):  
Johan Ceballos
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problem ◽  
Clifford Algebras ◽  
Dirichlet Boundary ◽  
Relevant Literature ◽  
Monogenic Functions ◽  
Dirichlet Boundary Value Problem ◽  
Dirichlet Problems ◽  
Initial Value ◽  
Do So

This paper reviews and summarizes the relevant literature on Dirichlet problems for monogenic functions on classic Clifford Algebras and the Clifford algebras depending on parameters on. Furthermore, our aim is to explore the properties when extending the problem to and, illustrating it using the concept of fibres. To do so, we explore ways in which the Dirichlet problem can be written in matrix form, using the elements of a Clifford's base. We introduce an algorithm for finding explicit expressions for monogenic functions for Dirichlet problems using matrices in Finally, we illustrate how to solve an initial value problem related to a fibre.

scholarly journals Dirichlet Problem for Maximal Graphs of Higher Codimension

2020 ◽  
Author(s):  
Yang Li
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Dirichlet Boundary Value Problem ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
General Existence

Abstract We consider the Dirichlet boundary value problem for graphical maximal submanifolds inside Lorentzian-type ambient spaces and obtain general existence and uniqueness results that apply to any codimension.

scholarly journals The Dirichlet boundary value problem for two non-overlapping spheres

1970 ◽  
Vol 2 (2) ◽  
pp. 237-245 ◽  
Author(s):  
Dieter K. Ross
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problem ◽  
Hypergeometric Functions ◽  
Dirichlet Boundary ◽  
Linear Equations ◽  
External Potential ◽  
New Method ◽  
Dirichlet Boundary Value Problem ◽  
Overlapping Spheres ◽  
Infinite Set

A new method is found for solving the general Dirichlet problem for two non-overlapping spheres of different radius. The expression for the external potential involves hypergeometric functions and is obtained from an infinite set of linear equations. In essence the method makes use of the fact that (I–A)–1 = I + A + A2 + A3 + …, where A belongs to a certain class of infinite matrices and I is the unit matrix.

Second order BVPs with state dependent impulses via lower and upper functions

2014 ◽  
Vol 12 (1) ◽  
Author(s):  
Irena Rachůnková ◽  
Jan Tomeček
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problem ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Second Order ◽  
Dirichlet Boundary Value Problem ◽  
State Dependent ◽  
Lower And Upper Functions

AbstractThe paper deals with the following second order Dirichlet boundary value problem with p ∈ ℕ state-dependent impulses: z″(t) = f (t,z(t)) for a.e. t ∈ [0, T], z(0) = z(T) = 0, z′(τ i+) − z′(τ i−) = I i(τ i, z(τ i)), τ i = γ i(z(τ i)), i = 1,..., p. Solvability of this problem is proved under the assumption that there exists a well-ordered couple of lower and upper functions to the corresponding Dirichlet problem without impulses.

About Asymptotic and Oscillation Properties of the Dirichlet Problem for Delay Partial Differential Equations

2003 ◽  
Vol 10 (3) ◽  
pp. 495-502
Author(s):  
Alexander Domoshnitsky
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problem ◽  
Dirichlet Boundary ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Asymptotic Properties ◽  
Dirichlet Boundary Value Problem ◽  
The Maximum Principle ◽  
Unbounded Solutions ◽  
All Solutions

Abstract In this paper, oscillation and asymptotic properties of solutions of the Dirichlet boundary value problem for hyperbolic and parabolic equations are considered. We demonstrate that introducing an arbitrary constant delay essentially changes the above properties. For instance, the delay equation does not inherit the classical properties of the Dirichlet boundary value problem for the heat equation: the maximum principle is not valid, unbounded solutions appear while all solutions of the classical Dirichlet problem tend to zero at infinity, for “narrow enough zones” all solutions oscillate instead of being positive. We establish that the Dirichlet problem for the wave equation with delay can possess unbounded solutions. We estimate zones of positivity of solutions for hyperbolic equations.

scholarly journals On the existence of a solution for some distributed optimal control hyperbolic system

2000 ◽  
Vol 23 (5) ◽  
pp. 297-311 ◽  
Author(s):  
Dariusz Idczak ◽  
Stanislaw Walczak
Keyword(s):  
Optimal Control ◽  
Hyperbolic System ◽  
Convex Set ◽  
Linear Time ◽  
Time Varying ◽  
Optimal Process ◽  
Existence Of A Solution ◽  
Distributed Optimal Control ◽  
Absolutely Continuous ◽  
State Variable

We consider a Bolza problem governed by a linear time-varying Darboux-Goursat system and a nonlinear cost functional, without the assumption of the convexity of an integrand with respect to the state variable. We prove a theorem on the existence of an optimal process in the classes of absolutely continuous trajectories of two variables and measurable controls with values in a fixed compact and convex set.

Refined Boundary Behavior of the Unique Convex Solution to a Singular Dirichlet Problem for the Monge–Ampère Equation

2018 ◽  
Vol 18 (2) ◽  
pp. 289-302
Author(s):  
Zhijun Zhang
Keyword(s):  
Dirichlet Problem ◽  
Crucial Role ◽  
Blow Up ◽  
Boundary Behavior ◽  
Structure Condition ◽  
Bounded Smooth Domain ◽  
Convex Solution ◽  
Bounded Smooth ◽  
Monge Ampere ◽  
Singular Dirichlet Problem

AbstractThis paper is concerned with the boundary behavior of the unique convex solution to a singular Dirichlet problem for the Monge–Ampère equation\operatorname{det}D^{2}u=b(x)g(-u),\quad u<0,\,x\in\Omega,\qquad u|_{\partial% \Omega}=0,where Ω is a strictly convex and bounded smooth domain in{\mathbb{R}^{N}}, with{N\geq 2},{g\in C^{1}((0,\infty),(0,\infty))}is decreasing in{(0,\infty)}and satisfies{\lim_{s\rightarrow 0^{+}}g(s)=\infty}, and{b\in C^{\infty}(\Omega)}is positive in Ω, but may vanish or blow up on the boundary. We find a new structure condition ongwhich plays a crucial role in the boundary behavior of such solution.

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