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Questions tagged [boundary-value-problem]

For questions concerning the properties and solutions to the boundary-value problem for differential equations.

4
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0answers
954 views

Weak form of steady Navier-Stokes equations with special boundary condition

Suppose we want to solve the steady low-Mach-number Navier-Stokes equations coupled with a passive scalar $\xi$, which read: \begin{align} \nabla \cdot \rho \mathbf{v} & = 0,\\ \rho \mathbf{v} \...
1
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1answer
86 views

1D wave equation with Boundary Conditions: Fourier Transform solution

I am considering the 1D wave equation with $c=1$ for the sake of simplicity: $$u_{tt}-u_{xx}=0,\quad \forall x\in\mathbb R,\; \forall t\in\mathbb R\tag{1}\label{eq:1}$$ with the following boundary ...
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0answers
11 views

Green's function for homogeneous PDE

I was looking for some Green's function method to solve a homogeneous PDE with nonhomogeneous boundary conditions (i.e., $Lu=0$ in $D$ with $u=f(\mathbf{x})$ in $\partial D$), but most of the ...
1
vote
1answer
43 views

Weak Form versus Boundary Integral Form

I am considering a fairly basic Boundary Value Problem to expose my question. Boundary Value Problem (1) local equation: $u_{xx}(x)=0$, $\forall x \in [0,1]$ boundary conditions: $u(0)=0$ and $u(1)=...
11
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2answers
269 views

Robin BC in the 1D wave equation

The problem of interest is as follows: the quantity of interest: $u(x,t)$ the wave equation: $\partial_2^2u(x,t)-c^2\partial_1^2u(x,t)=0$ where $c>0$ one Robin boundary condition at $x=0$: $\...
-1
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0answers
19 views

Can I solve the following problem in integration along with constraints? [closed]

max $\int (x-f(x)dx$ such that $f(0) =a$ $f(1) = b$ $f'(0) = c$ $f'(1) =d$ $f''(0.5) = 0$ $f'(x) >0 $ $ \forall x \in[0,1] $
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1answer
1k views

Compatibility Condition of the Poisson Equation with Neumann Boundary Conditions

I am trying to solve the following general Poisson equation with homogeneous Neumann boundary conditions in a rectangular domain ($0 \le x \le L$ and $0 \le y \le H$). $$ \frac{\partial^2 p(x,y)}{\...
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0answers
24 views

Periodic boundary conditions, general dimension, sets and spectral properties of $-\Delta$ - reference recommendation

Let's consider the eigenvalue problem $-\Delta u = \lambda u$ on the interval $[0,1]$ with periodic boundary conditions: $u(0)=u(1),$ $\frac{du}{dx}(0) = \frac{du}{dx}(1).$ Similar conditions could be ...
1
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1answer
676 views

Solution of boundary value problem using Fourier series

I want to solve the following PDE using Fourier series: $u(x,y): \Omega \to \mathbb{R}$, $\Omega=(0,\pi)\times (0,2\pi)$ $u-3u_{xx}-u_{yy}= 3\sin(2x)-\sin(5x)$ $u_{xx}$ and $u_{yy}$ are second ...
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1answer
688 views

How to solve [K]{U}={F} FEM where some values of {U} are known.

I am trying to solve the 2D-Poisson equation using the FEM. I have created the global stiffness matrix and I know my boundary conditions. I am looking at $0<x<4$ and $0<y<2$ with boundary ...
1
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1answer
1k views

2D finite difference boundary conditions for radial direction

I am trying to solve Poisson's equation in an axisymmetric cylindrical domain using finite difference. So I start with my differential equation and boundary conditions and discretize them. However, I'...
2
votes
1answer
29 views

Wave equation with Neumann BC on semi-infinite domain

This problem is from our recitation which I do not have solutions for, and I'm stuck on the very last part where I need to satisfy the $u_t(x,0)$ initial condition. The problem is: $$ \left\{ \...
0
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0answers
30 views

Proof of reduction of non-homogeneous ODE problem to homogeneous

Consider the following boundary value problem: $$-(p(x)u')'+q(x)u=g(x) \hspace{35pt} (1)\\ 0<x<1, u(0)=u_l, u(1) = u_r\hspace{35pt}$$ where $p\in C^1[0,1], p(x) \ge p_0 >0; q\in C[0,1], q(x)\...
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1answer
21 views

Solving Wave Equation on $0 < x < L$ with initial condition containing both sine and cosine terms

I am trying to solve the wave equation as follows: $$u_{tt} = u_{xx} \,\,\,\,\,\,\,\, 0 < x < L$$ $$u(0,t) = u(L,t)$$ $$u_x(0,t) = u_x(L,t)$$ $$u(x,0) = sin\bigg(\frac{2\pi x}{L}\bigg) + cos\...
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0answers
19 views

Boundary 'smoothness' for partial differential equations

I'm struggling to understand how and why the smoothness of the boundary affects the existence and uniqueness of partial differential equation solutions. For example, for Laplace's equation $\Delta u(x)...
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0answers
19 views

Coercivity of mapping A , sequence bounded

in Hess' article I don't understand why the sequence $u_{n\epsilon}$ is bounded (and $\frac{\partial u_{u\epsilon}}{\partial t}$ too) $Q=\Omega \times (0,T)$ (A2) There exist constants $q \quad(1<...
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0answers
25 views

Precise definition of the solution of a differential equation on a compact set (short question)

Let $f : (0,T) \rightarrow \mathbb{R}$ be differentiable and $u_0 \in \mathbb{R}$. Consider the differential problem $(DP)$ given by $$ u^{\prime} (t) = f(u(t)), $$ $$ u (0) = u_0 .$$ What is the ...
1
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1answer
45 views

Boundary value problem, Laplace's equation in polar coordinates

I have trouble understanding the question and solution of a boundary value problem. Consider Laplace's equation in polar coordinates $(r,\theta)$ $$\frac{\partial^2 \Phi}{\partial r^2} + \frac 1 r \...
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0answers
17 views

Existence and uniqueness of solution of discretized Poisson equation

I'm discretizing the following Poisson equation using FVM where the domain $\Omega$ of the solution is a regular hexagon of side $1$ centered about the origin. $$\Delta u =k,\text{ $k$ constant}\\ \...
0
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0answers
12 views

Exist a constant that only depends of a power and dimension that bounds the euclidean norm.

I have this problem but I don't know how to solve it. Considering the Euclidean norm |x|, in $\mathbb{R}^n$. If $a>0$, prove there exist $c>0$ depending only of $n$ and $a$ such that: $$c^{-1}(|...
1
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1answer
64 views

Validity of solution to PDE $xu_x - uu_t = t$ obtained from characteristics

I have a boundary-value problem: $$ xu_x - uu_t = t $$ with boundary conditions: $$ u(1, t)= t, -\infty<t<\infty $$ Finding the characteristic equations is no problem, and I get a general ...
0
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1answer
22 views

Proving an inequality using the divergence theorem

I have a question that reads $$\frac{\partial}{\partial t}w=\nabla^2w$$ for $w(t,\vec x)$ with initial condition $w(0,\vec x) = w(0,x)$ in $V$ , boundary condition $w(t,\vec x) = f(\vec x)$ on $S$...
0
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0answers
11 views

When Linearised Model of Finite Depth Waves is Not a Sufficient Model

I have been investigating the linearised model of water wave motion in a finite depth fluid. In my particular case the flow is Inviscid, Irrotational and Incompressible and surface tension effects are ...
1
vote
1answer
29 views

$\frac{d^2y}{dx^2}=f(x)$ with boundary conditions, how to find integration bounds

Given $$\frac{d^2y}{dx^2}=f(x),\quad y(-1)=y(1)=0,$$ I used $u=y'$ and $u(x_0)=u_0$ to get $$ u(x)=u_0+\int_{x_0}^xf(\xi)d\xi. $$ Then we have $y'=u$, which we can integrate again using $y(x_0)=y_0$ ...
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0answers
26 views

Solution to Green's function second degree differential equation

I'm given a differential equation in the form of $$y''+P(x)y'+Q(x)y=R(x)$$ with the specified boundaries $y(a)=y(b)=0$ I've already shown that the Green function of this ODE is $$G(x,z)=\begin{...
0
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0answers
28 views

2D disc-shaped region kernel density estimation boundary bias correction for Gaussian bandwidth $h$ and boundary radius $R$

I am trying to develop a closed form expression for the boundary bias correction factor for kernel density estimation in a circularly-bounded 2d region where Gaussian kernel diameter is $2h$ and ...
1
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1answer
26 views

How do I solve the non-homogeneous wave equation with homogeneous boundary and initial conditions?

I want to solve $$v_{tt}(x,t) - v_{xx}(x,t) = -\left(\frac{3}{4} \cos(t) + \frac{1}{4} \cos(3t)\right) \sin(x)$$ with boundary conditions $$v(0,t) = v(\pi,t) = 0$$ and initial conditions $$v(x,0) = ...
1
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0answers
84 views

Laplace equation in 3D with numerous Non-Homogeneous BC(s) [Strategy Check]

I need to solve the three-dimensional Laplace equation ($\nabla^2T = 0$) where $\nabla^2=\frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$ in the domain ...
2
votes
0answers
30 views

Parabolic PDE with zero Neumann condition

I'm working with the following parabolic PDE: $$u_t = \nabla\cdot(\alpha(x)\nabla u)- \beta u\\ x\in\Omega \subset \mathbb{R}^2\\ \partial_n u = 0$$ Given the Neumann boundary condition above, if I ...
1
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1answer
42 views

Calculus - Integral - existence problem

We are given a function $f$, $f$ is integrable (in the riemann sense) in $[a,b]$ and also $f'$ is a continuous, and $f(a)=f(b)=0$. Prove that there exists a point $c$ such that $|f'(c)| \geq \frac{4}{...
6
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0answers
353 views

Green Solution to Laplace Equation with Robin Boundary Conditions

Let's say that I know a solution for the Laplace equation in the whole plane: $$\nabla^2u(\mathbf{x})=0\quad \mathbf{x}\in\mathbb{R}^2$$ And I need a solution for the laplace equation in the ...
-1
votes
2answers
38 views

Laplace Equation in Rectangle: Boundary Value Problem [closed]

Solve the boundary-value problem $∆u = 0$ (by this we mean $u_{xx} + u_{yy} = 0$) in the rectangle $0 < x < π$, $0 < y < 1$, with the boundary conditions $u(0,y) = 0$, $u(π,y) = g(y)$, $u(...
2
votes
1answer
55 views

Solving a system of ODE

Solve $$\eta_k\frac{d^2C_k}{dz}(z)=-e_k, k = 1,2,3$$ $$C_1(0)=0, C_2(0)=A, C_3(0)=0$$ $$C_1(L)=B, \frac{dC_2}{dz}(L)=0, \frac{dC_3}{dz}(L)=0$$ where $A,B,\eta_k$ some known constant. $e_k, k=1,2,3$ ...
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0answers
16 views

Discretization Dirichlet boundary condition for Elliptic PDE with finite volume method

I want to discretize the following equation using FMV: $$\nabla \cdot (a(x)\nabla u)=f(x)\\x\in \Omega \subset \mathbb{R}^2 \\u_{|\partial\Omega}=g$$ To this end, let $V_i \subset\Omega$, $i=1,\dots,N$...
3
votes
0answers
27 views

Curious Question on Inhomogeneous Boundary Conditions of a PDE

I am given the PDE, $\ u_t=u_{xx},$ with boundary conditions $u(0,t)=A, \ u(1,t)=B$ and $u(x,0)=f(x)$. I have found the solution of this PDE is $$u(x,t)=A+(B-A)x+\sum_{n=1}^{\infty}B_ne^{n^2\pi^2 t}\...
1
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0answers
58 views

Solving definite integral in two variables.

Solving a PDE with the following boundary problem with arbitrary constant $b$: $$u(0,t)=F(t)=b\int_0^\infty u(a,t)\mathrm{d}a$$ Hint given in the question is as follows: Split this integral in two ...
2
votes
2answers
86 views

Solve PDE using method of characteristics with non-local boundary conditions.

Given the population model by the following linear first order PDE in $u(a,t)$ with constants $b$ and $\mu$ : $$u_a + u_t = -\mu t u\,\,\,\,\,a,t>0$$ $$u(a,0)=u_0(a)\,\,\,a≥0$$ $$u(0,t)=F(t)=b\...
0
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0answers
12 views

REFEREENCE REQUEST for Non-Local Boundary Value problems

It would be really helpful if someone could suggest me any reference (Books or Papers) where I would find worked-out examples of Elliptic Boundary value problems (especially Laplace equation) with non-...
1
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1answer
725 views

Converting Dirichlet Boundary Conditions to Neumann Boundary Conditions for the Heat Equation

I'm solving the heat equation on a two dimensional square domain. The problem is defined as: $$ u_{xx}+u_{yy} = 0 \hskip{0.5cm}\text{for} \hskip{0.5cm} 0 \leq x \leq 1, 0 \leq y \leq 1 $$ with the ...
0
votes
1answer
710 views

2d laplace equation with neumman boundary condition

$$\Delta u(x,y)=0$$ $$x,y\in(0,1),$$ $$\frac{\partial u(0,y)}{\partial x}=0,\quad \frac{\partial u(1,y)}{\partial x}=0,\quad\frac{\partial u(x,0)}{\partial y}=0,\quad\frac{\partial u(x,1)}{\partial y}=...
2
votes
2answers
43 views

Solving the BVP $u_{xx}+a^2u=\sin(\pi x)$ with $u(0)=1$ and $u(1)=-2$

I am trying to solve the BVP $$u_{xx}+a^2u=\sin(\pi x), \ \ \text{for} \ \ 0<x<1$$ with $u(0)=1$ and $u(1)=-2$, $\forall a\in\mathbb{R}$. I begin by solving the homogeneous equation $u_{xx}+a^...
0
votes
1answer
24 views

Laplace Equation with Inhomogeneous Boundary Condition

I'm currently learning about separation of variables as applied to situations where the boundary conditions are not homogeneous. I'm having trouble deciding how to handle one of the boundary ...
0
votes
1answer
28 views

Separating variables in a PDE with multiple constants

My question is: How do you use separation of variables on a PDE that has more than one constant in it? All the examples I can find in my book/online only have one constant in it, like $$ \frac{\...
0
votes
0answers
23 views

Matrix representation of a finite difference with Neumann boundary conditions

Given 1D data $[c_1,c_2,c_3,\cdots,c_N]$ I can represent the derivative operation as a matrix product. For example, using the central difference $$ \left.\frac{\partial c}{\partial x}\right|_k \...
4
votes
2answers
707 views

Laplace Equation on the Corners and Boundary of a Rectangle?

Consider for some rectangle $[a,b] \times [c,d] \in \mathbb{R}^2$, we have a generic boundary value problem: \begin{equation*} \begin{cases} \frac{\partial ^2 u}{\partial x ^2}+\frac{\partial ^2 u}{\...
1
vote
0answers
22 views

A sufficient condition for a Neumann problem to have solution

Suppose we have the Boundary Value Problem (BVP): $\Delta{u}=f$ , in the domain (topos) $D$ $\partial_{n}{u}=h$, in $\partial{D}$ It is easy to prove using $1$st Green's identity that a necessary ...
0
votes
0answers
27 views

Poisson integral formula for boundary value problem

I have gotten stuck on a boundary value problem which I believe is to be solved using the Poisson Integral Formula. The problem is: $$\nabla^{2}\psi=0, \psi(x,0)=0, |x|>1 ; |\psi(x,y)|<|x|, |x|\...
0
votes
1answer
32 views

Helmholtz equation with robin boundary condition

Consider the differential equation $(\nabla^2+\frac{1}{R^2})\psi(\bar{r}) = 0$ in 2 dimensions, with the boundary condition $\partial_r\psi(R)+ \kappa \psi(R) = 0$, on unit disk of radius R. What is ...
1
vote
0answers
28 views

Finding the equation from boundary values [closed]

I have two equations like $$\frac{\partial Y}{\partial x}=A(L-2x)\;\text{ at }(x,0)$$ And $$\frac{\partial Y}{\partial t}=0\;\text{ at }(0,0)$$ Is it possible to find out the real equation? (That is ...
0
votes
1answer
33 views

Proving that the eigenvalues of the Airy problem are positive

I am solving an exercise concerning the Airy eigenvalue problem $$ -y''+xy =\lambda x, \quad y(0)=y(1)=0, \quad (*) $$ which (among other things) asks me to prove that all eigenvalues are positive. I ...