# Questions tagged [boundary-value-problem]

For questions concerning the properties and solutions to the boundary-value problem for differential equations. By a Boundary value problem, we mean a system of differential equations with solution and derivative values specified at more than one point. Most commonly, the solution and derivatives are specified at just two points (the boundaries) defining a two-point boundary value problem.

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### Find curve that minimizes lenght, with integral constraint

I'm interested in finding the curve $q(t):[0,1] \rightarrow \mathbb{R}^+$ that satisfies the boundary conditions $q(0)=q(1)=0$, the integral condition $\int_0^1q(t)dt=a>0$, and that minimizes the ...
17 views

### Effect of boundary conditions on general solution

I am having problems integrating given boundary conditions on a wave-equation. The problem is as stated below. I am no expert in solving PDE's, so please forgive if I oversee something obvious or &...
1 vote
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### Show that is continuous and coercive

I have the following boundary value problem, where $k, q \in \mathbb{R}$ are given. $$-u'' + ku' +qu = f,\qquad u(0)=u(1)= 0$$ What I want to prove is that the bilinear form, associated with this BVP,...
1 vote
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### Find harmonic $u : \mathbb{R}^2 \to \mathbb{R}$ such that $u(x, 0) = u_y(0,0) = 0.$

Find harmonic $u : \mathbb{R}^2 \to \mathbb{R}$ such that $u(x, 0) = u_y(0,0) = 0.$ Without the last condition, we have $u = y.$ I'm trying to prove that if in addition $u>0$ on the upper half-...
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### Parabolic Pde with unkown boundary conditions

I have the following parabolic partial differential equation: \begin{equation} \frac{\partial^2 \phi}{\partial x^2} - \alpha \sin{x} \frac{\partial \phi}{\partial t} + \beta(\cos{x} - \gamma) \phi = ...
1 vote
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### Lax-Milgran theorem with the Poisson equation

Let $\Omega$ be a bounded domain in $\mathbb R^3$ with smooth boundary. Consider the Poisson equation $$-\Delta u=f$$ where $f\in C_0^{\infty}(\Omega)$ and $f$ is null outside $\Omega$. I'm not ...
37 views

### A boundary value problem of a Harmonic potential

A 2D electrostatic (i.e. harmonic potential) boundary value problem is shown in the figure. The solid lines are conductors (all are parallel), the two conductors with potential $V$ are infinitely long,...
1 vote
36 views

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### Inhomogeneous Boundary Value Problem. How would I solve?

The heat equation: \begin{align} \frac{\partial u}{\partial t} &= {9} \frac{\partial^2 u}{\partial x^2}\,, \qquad 0<x<{3}, \quad t \gt 0\, \\ \end{align} Has boundary and initial ...
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### Solving a Fourth-Order Linear Homogeneous Differential Equation

I like to solve the ordinary fourth order homogeneous differential equation given by $\displaystyle \frac{d^{4}\theta}{d z^{4}} + \lambda \cdot \theta = 0$ with a constant coefficient $\lambda$. ...
1 vote
103 views

### Need help - Robin condition for a 1d wave equation on the first quardant

Given $\alpha \neq 0$ and $$u_{tt}-c^2u_{xx} =0 \quad x,t>0,$$ $$u(x,0) = f(x), \quad x\geq 0,$$ $$u_t(x,0)=g(x), \quad x\geq 0,$$ $$u_x(0,t)+\alpha u(0,t)=0, \quad t\geq 0.$$ I know it’s a Robin ...