# 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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### Whether the solution of $\Delta u=f$ belong to $C_0^{\infty}(\Omega)$?

Assume that and $\Omega$ is a Lipschitz open bounded set , or pipe-like area such as $\{(x_1,x_2):-\infty < x_1 < +\infty, a\le x_2 \le b\}$ in ${\mathbb R}^2$. $f \in C_0^{\infty}(\Omega)$, ...
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### What are the boundary conditions on the Green's function?

Let a linear ordinary differential equation of the form $$\frac{d^2}{dx^2}f(x) = g(x)\tag{1}$$ be given to us subject to the boundary conditions on $f(x)$ and/or $f^\prime(x)$. The most general ...
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### Minimum of functional with boundary conditions over twicely differentiable functions

I found a quiz that I cannot find a way to solve, and wanted to know if anyone of you has any suggestions on how to solve it. The problem is the following: find the minimum of the following functional ...
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### Deducing Equivalent Norms on Sobolev Spaces from Boundary Value Problems

In Salsa's Partial Differential Equations in Action, 3ed page 540 the following is mentioned: Let $\Omega$ be a $C^2-$ domain and $f \in L^2(\Omega)$. The Lax-Milgram Theorem and Theorem 8.28 (...
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### Regularity of weak solution of elliptic equation with nonlinear Neumann boundary

Let $\Omega \subset \mathbb{R}^n$ be bounded smooth domain and $u \in W^{1,2}(\Omega;\mathbb{R}^m)$ be a weak solution to the following equation \begin{align*} &\int_{\Omega} \nabla u \cdot \nabla ...
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### Use method of characteristics to solve $u_x(x,y)+u_y(x,y)=(u(x,y))^2$, $u(x,0)=x$

For this, $z=u(x,y)$ So far I have the characteristic equations: \begin{align*} a&=x_t=1&\implies x(s,t)&=t + f(s),\\ b&=y_t=1&\implies y(s,t)&=t + g(s),\\ c&=z_t=z^2&\...
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### Separation of variables method with $u(0,x)=5e^{x^2}-e^{-10x^2};x>0$

\begin{cases} x\frac{\partial{u}}{\partial{t}} + \frac{\partial{u}}{\partial{x}}=0 \\ u(0,x)=5e^{x^2}-e^{-10x^2};x>0 \end{cases} Writing $u(x,t)=X(x)U(t)$: \begin{align*} &xX(x)\Big(\frac{10te^...
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### Difference of boundary conditions between elliptic and parabolic PDEs

For elliptic PDEs we know that the boundary condition no matter Dirichlet, Neumann or mixed, needs to be defined over the whole boundary of the given volume $\partial V$. Only then we will get a ...
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### Monotonicity and uniqueness of solution of boundary value problem [closed]

Consider a function u a solution of the following BVP: $-\frac{d^2u}{dx^2}-c\frac{du}{dx}=f(u) \in (-a,a). u(-a)=1,u(a)=0$, where $f(u)=u-u^2$ so that by comparison principle $0\le u\le1$, my aim is ...
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### Laplace equation in-between two non-concentric spheres

We fix two spheres $S_1$ and $S_2$ (without interior) and suppose that $S_2$ is entirely inside $S_1$. For example, $S_1 = \{x^2 + y^2 + z^2 = 25\}$ and $S_2 = \{(x-1)^2 + y^2 + z^2 = 1\}$. How to ...
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### The Variational form of a biharmonic PDE

Suppose $\Omega \subset \mathbb{R}^d$ is a $C^{1,1}$ domain. Consider the biharmonic boundary value problem (BVP): $$\begin{cases} \Delta^2 u = f \\ \nabla u \cdot \nu = g \\ u = u_D \end{cases}$$ ...
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Suppose I have a differential equation: $$\frac{\partial^2 f(x,y)}{\partial x^2}+\frac{1}{y}\frac{\partial f(x,y)}{\partial y}+\frac{\partial^2 f(x,y)}{\partial y^2}=0$$ where Boundary condition is: $$... • 585 0 votes 1 answer 73 views ### Help on transformation of boundary conditions I was working the transformation in this paper A new algorithm for solving classical Blasius equation by Lei Wang The boundary value problem is He used the transformations$$y=f''(\eta),x=f'(\eta) ...
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I am solving a version of the Laplace equation on a square ($a<x<b$, $0<y<h$) grid using finite differences. I have an analytical solution to my problem so I can easily check the ...