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.
1,634
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Techniques for solving the Navier-Cauchy equations of linear elastostatics
I'm trying to get an exact solution to a specific boundary value problem in linear elasticity. Standard techniques one would use to solve the heat or wave equations, such as separation of variables ...
- 111
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1
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Proving a regularity estimate for $−εu′′ + bu′ = f $
I've been given:
$$−εu′′ + bu′ = f ,x ∈ (0, 1), u(0) = u(1) = 0$$
and have been asked to prove the regularity estimate
$$∥u′′∥_{L^2(0,1)} ≤ C_R∥f∥_{L^2(0,1)}$$
I normally try to provide some working ...
- 171
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1
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Method of characteristics - inhomogeneous pde
The boundary conditions problem:
$2\frac{\partial u}{\partial x}-14\frac{\partial u}{\partial y}=3(x+y)$
Where the boundary conditions are:
$u(x,-6x)=Cos((2x)^2) \quad \forall x \in \mathbb{R}$
How ...
0
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1
answer
23
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boundary conditions - inhomogenius liniar pde
I have this boundary conditions question:
$2\frac{\partial u}{\partial x}-14\frac{\partial u}{\partial y}=3(x+y)$
Where the boundary conditions are:
$u(x,-6x)=Cos((2x)^2) \quad \forall x \in \mathbb{R}...
0
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1
answer
38
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Determining the bilinear form a(. , .) is bounded.
I am trying to derive a weak formulation of the boundary value problem
$-\epsilon u′′ + bu′ = f $ for $x \in (0, 1)$, $u(0) = u(1) = 0$,:
where $\epsilon \in \mathbb{R}$, is such that $\epsilon > 0$...
- 171
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26
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Boundary conditions for weight function in weak form of an ODE or PDE
I am really quite confused by how to pick a weight function's boundary conditions when evaluating the weak form of a differential equation.
For example, given this ODE in the domain $0<x<1$:
$$u'...
- 322
3
votes
1
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2D Laplacian with 1 Non-Homogeneous Dirichlet, 1 Homogeneous Dirichlet and 2 Homogeneous Neumann BCs, Rectangular Domain
To preface, I am very new to PDEs and not overly familiar with hyperbolic trig so forgive me if there are obvious mistakes - I have tried my best to research this problem before asking this question. ...
- 306
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Can you bound the H1 norm of the solution of a Poisson problem with its L2 norm?
Let $u \in H^1(\Omega)$ be such that
$$
\int_{\Omega} \nabla u \cdot \nabla v + \int_{\Omega} u v
= \int_{\partial\Omega} q v
\qquad \forall v \in H^1(\Omega)
$$
for a given boundary datum $q \in H^{-...
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2
answers
29
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Integral of a two-variable function with boundary condition [closed]
A function $f(x,y)$ with $f(x=0,y)=0$.
We define $g(x,y) = \int f(x,y)dy$.
Does $g(x=0,y)=0$?
How does one prove this?
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2
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Condition on Sobolev function on the boundary of a domain
I encountered the following kind of a condition when studying boundary value problems:
Let $\Omega \subset \mathbb{R}^2$ be a domain. Let $\varphi \in H^1
(\Omega)$ be such that $\varphi \ge 0 $ on ...
- 1,262
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0
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I cant solve a problem based on using central difference approximation to discretize a boundary value problem [closed]
problem statement:
I tried solving this problem a couple of days ago,but my answer didnt match any of the available options.Can someone show me how to solve this kind of problem,and give me the answer?...
- 1
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Variational formulation, weak formulation
I'd like to find the weak formulation of the problem
$-u''+au=f$ on $(0,1)$
$u(0)=0$
$u'(1)=b$
$a>0$
and show that there exists a unique solution using Lax-Milgram.
What I did:
By multiplying ...
- 331
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0
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49
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Green function for Homogeneous diffusion equation in the semi infinite domain with inhomogenous neumann boundary conditions
The Initial Boundary Value-Problem in the semi infinite domain $0 \leq x < \infty$ is given by
$$
k u_{xx} - u_{t} = 0 \\
u(x,0) = u_0 \\
-u_x (0,t) = f(t) \\
-u_x (x \rightarrow \infty , t) = ...
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Using finite element method to study a second order diff. equation with b. c., which includes a value of a solution at a certain point.
I am looking into a certain analytical result numerically. I am using FreeFem and Mathematica, but have trouble with implementing a part of differential operator. Can you recommend a different library ...
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29
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How to determine the boundary layer based on the PDE?
My professor solved a problem and I don't understand how they explained the location of the boundary layer of the following PDE:
$$\epsilon y'' -(2x+1)y' + 2y=0$$
$$y(0)=y(1)=1$$
As $\epsilon → 0$, a ...
- 471
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1
answer
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What are the boundary conditions on Green's function and why?
The equation $$\frac{d^2}{dx^2}\phi(x)=f(x)\tag{1}$$ has the general solution $$\phi(x)=\phi_0(x)+\int G(x,x')f(x')dx^\prime\tag{2}$$ where $$\frac{d^2}{dx^2}G(x,x^\prime)=\delta(x-x^\prime)\tag{3}$$ ...
- 593
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Existence of solution to linear boundary value problem
Consider a differential equation of the form
$$
\dot \gamma(t)=F_t(\gamma(t))
$$
For $\gamma:[0,1]\to \mathbb R^{2n}$ and a smooth $F:[0,1]\times \mathbb R^{2n}\to \mathbb R^{2n}$ such that every $F_t$...
- 596
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Analytical Equation of the gaussian 1D wave equation with periodic Boundary condition
I am trying to validate the 1D analytical wave equation with a numerical solution with periodic boundary conditions. I have implemented the periodic boundary condition for the numerically calculated ...
- 111
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0
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Solving IBVP with Laplace Transform
It is given the following BVP
\begin{align}
u_{t} &= u_{xxxx},\quad\quad x \in (0, \pi), t >0 \tag1\\
u(0,t) &= u(\pi,t) = 0 ,\quad t > 0 \tag 2\\
u_{x}(0,t) &= e^{-4t},\quad t > ...
2
votes
0
answers
152
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Eigenvalues for hypergeometric differential equation using asymptotic analysis (or WKB approximation)
I am studying a problem in high energy physics and have reduced it to the following problem regarding the confluent hypergeometric differential equation:
$$
\left(\gamma \partial_\gamma^2 + \partial_\...
- 268
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0
answers
12
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Under what condition on the convection term is the 2nd order BVP $-(au')' + b u' + cu = f$ the critical point of a convex energy functional?
Consider the second-order elliptic boundary value problem
$$-(a(x) u(x)')' + b(x) u' + c(x)u = f(x)$$
over interval $I = [0,1]$, together with homogeneous Dirichlet boundary conditions $u(0)=u(1) = 0$....
- 7,078
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0
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is my analytical solution for Heat Equation correct?
Could you please check my analytical solution for accuracy and correctness?
Consider the heat equation: $$ u_t(x, t)=u_{x x}(x, t) $$
where $$ 0<x<1, \quad t>0 $$ Neumann Boundary condition:
$...
- 11
1
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0
answers
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Example of Green's function solution to linear homogeneous differential equation subject to inhomogeneous boundary conditions
Context
Part A
Generally, I like Handbook on Differential Equations by Zwillinger [1]. It seems to meet my needs most of the time. For example, from this book I was able to learn how to solve linear ...
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Continuity of the branch function at the boundary of the subdomains
I solved this exercise, but I am insecure of myself so I would like some comment from you about the correctness, or not, of my reasoning.
Consider the set $A$ and the function $f(x, y)$ such that
$$A =...
- 2,051
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votes
1
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How do I prove that the BVP $y'' = y\sin\left(x\right), y\left(0\right) = a, y\left(1\right) = b$ has a unique solution?
I want to prove that the BVP $$ y'' = y\sin\left(x\right), y\left(0\right) = a, y\left(1\right) = b $$ has a unique solution.
I know how to prove uniqueness:
Suppose there are two different solutions $...
- 78
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1
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General compatibility condition for pure Neumann problem
It is known than for a second order elliptic boundary-value problem with pure Neumann conditions, a certain compatibility condition between the data must be satisfied.
For example, in the case of the ...
- 889
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Transformation of boundary conditions at Laplace Equation in polar Coordinates
It is given the following BVP
\begin{align}
\Delta u &=0 , \quad 0\leq r <R,\quad 0<\phi<\alpha\leq 2\pi \tag1\\
u(r,0)&=0,\quad 0\leq r <R \tag 2\\
u(r,\alpha)&=\alpha,\quad 0\...
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0
answers
24
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FEM code for PDE subdomain interface condition
I have a simple problem to solve. I have two pde, one for $u_{1}$ and one for $u_{2}$, where the domain in space for $u_{1}$ is $[-L_{0}, 0] $ , and for $u_{2}$ the space domain is $ [0,L_{0}] $.
I ...
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1
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How do I solve $0= \frac{d}{dx}(x^2u'(x))$ in $C^1[-2,1]$ with boundary conditions
I am trying to solve the following ODE with boundary conditions in the set $C^1[-2,1]$, with $u(-2)=-1/2$ and $u(1)=1$. that is I must tell if there exist a solution $u$ in $C^1[-2,1]$
$$0= \frac{d}{...
- 2,103
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1
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Finding the value of a constant in a Boundary Value Problem
Let $t \to y(t)$ be a real-valued smooth function on any open interval containing $[0,1]$. Suppose that $y(t)$ satisfies the differential equation:
$$y''(t) + w(t)y(t) = \lambda y(t)$$
where $\lambda$ ...
- 395
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vote
1
answer
85
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Invert Laplace Transform with Heaviside function
I'm solving the following boundary value problem
$$
y \frac{\partial u}{\partial y}+\frac{\partial u}{\partial x}=1, \quad u(x, 1)=1=u(0, y) .
$$
I've derived that
$\bar{u}(p, y)=p^{-2}+p^{-1}-p^{-2} ...
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0
answers
41
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How can I solve $u_t = u_x + u_{xx}$ on an isolated half-rod (Robin conditions)?
This question seems so simple, yet the answer eludes me..
I'm trying to solve the advection-diffusion equation on the half-line x > 0:
$$ u_t = u_x + u_{xx} $$
The advection is in the direction of ...
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0
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The solution of a set of equations is about to appear but still elusive.
A set of equation is derived from an elastic problem with axisymmetry. The solution is assumed in a serial form, whose coeffecients are almost found but still remain elusive because of the expansion ...
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Initial boundary value problem for the 1-D Schrödinger equation
Consider the following initial boundary value problem for the 1-D Schrödinger equation for a function $u(t,x)$ in the domain $\Omega=[0,T]\times[0,L]$:
$$
\begin{cases}
iu_{t}(t,x)+\Delta u(t,x)=0,\...
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1
answer
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How are higher order boundary conditions handled in a weak formulation?
I am reading Gazzola et al's book and there for a space $H^{m}(\Omega)$ they seem to allow boundary conditions that are of degree as high as $2m-1$. How is this reconciled with the fact that ...
- 1,741
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1
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What is the exact solution of this stationary convection-diffusion problem (1d)
I need some assistance with the following problem.
I am trying to find the exact solution of a convection-diffusion equation as follows:
$$-Du''(x)+0.5u'(x)=1, 0<x<\pi$$
$$u(0)=u(\pi)=0$$
where $...
- 13
3
votes
1
answer
125
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Polyharmonic interpolation with mixed boundary conditions, variational formulation?
I am reading up on polyharmonic interpolation, though I want to use a different domain and boundaries than what is used by Duchon and Meinguet (notably the Green's functions do not even have a closed ...
- 1,741
1
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0
answers
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Solvability of elliptic PDE with Neumann boundary condition
I'm trying to better understand the theory of second-order elliptic PDEs on a smooth compact region $\Omega \in \mathbb{R}^N$ with boundary conditions. For the Dirichlet case, there seems to be a ...
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0
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Heat Wave problem
In case 2 I have stuck in the second case what do I have to do? Have I to use hyperbolic function?
$$
\begin{align}
u_{t}&=k u_{xx} \quad -l \leq x \leq l,\ t \geq0, k>0 \tag1 \\
u(-l,t)&...
0
votes
0
answers
24
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PDE with resetting/self referencial boundary conditions
I am trying to solve the pde of the form
$$
-\rho v(x,t) - \frac{\partial v(x,t)}{\partial t} -\nu\frac{\partial v(x,t)}{\partial x}+\frac{\sigma^{2}}{2} \frac{\partial^{2} v(x,t)}{\partial x^{2}}=0
...
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1
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1
answer
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How to transform the boundary conditions between formulations of PDE? 2D fluid flow
Question: How do I transform the boundary conditions of the speed formulation $(u, \ v)$ into the vortex-stream formulation $(\psi, \ w)$?
Cavity flow problem:
A incompressible Newtonian fluid is in ...
- 1,136
3
votes
1
answer
148
views
Laplace’s equation (using separation of variables)-- Is the solution correct?
I am trying to solve Laplace’s equation (using the separation of variables). I found a result below. But for some reason does not seems right to me. Any suggestions? Thank you in advance
$$
\begin{...
0
votes
0
answers
65
views
How to "numerically" calculate eigenvalues of differential operator using Finite difference?
Consider a simple problem $$ \frac{d^2y}{dx^2} y = -\lambda^2 y.$$ I would like to calculate the eigenvalues of operator $\, d_x^2\, $ using finite difference, but I am not sure how to do it. ...
0
votes
0
answers
41
views
Time-evolution of a matrix of vector fields: well-defined?
Consider a $2 \times 2$ matrix of smooth vector fields on a smooth manifold $M$:
$$X =
\begin{pmatrix}
X_{11} & X_{12} \\
X_{22} & X_{21} \\
\end{pmatrix}$$
I'm thinking about $X$ as an ...
- 21.5k
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votes
1
answer
80
views
Numerically solving a BVP for a 2nd order ODE with mixed Dirichlet and Neumann boundary conditions with the finite difference method
I would like to solve a simple BVP for a second order differential equation on the domain $I=[0,\pi/2]$ using the finite difference method.
$$
u''(x)+u(x)=0\\
u(0)=0\\
u'(\pi/2)=0
$$
By construction, $...
- 57
1
vote
1
answer
60
views
Calculus of Variations with first derivative boundary conditions
I am self teaching calculus of variations and have a seemingly basic question, which I cannot find the answer to. I understand that to obtain the shortest path between two points, $A$ and $B$, one can ...
- 11
0
votes
1
answer
43
views
Fourier series for a potential with boundary conditions on cylinder
I'm working with a problem from Electromagnetics where I'm supposed to calculate the potential distribution in two different intervals. However, I'm very unfamiliar with the boundary conditions and I ...
- 1,789
1
vote
1
answer
72
views
How to find Eigenfunction of this Sturm-Liouville problem
My question what do I have to do to find the eigenfunction in the first case
It is given the following problem
\begin{align}
xy''(x)+y'(x)+ \frac{\lambda}{x}y(x) &=0, & &x\in(1,e^{2\pi}) ...
0
votes
1
answer
29
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Difference between Neumann condition and Robin Condition
I'm a bit confused by the difference between a Neumann condition and a Robin condition. In the textbook a Neumann condition is defined as a normal derivative that is specified on the boundary. It's in ...
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votes
0
answers
12
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Can I use this condition to solve a radially summetric equation with the finite difference method
Consider the symmetric radial heat equation:
$$ f_t = \nabla^2f \Leftrightarrow f_t = f_{rr} + {1 \over r}f_r \tag 1$$
To solve this with the finite difference method using the FTCS scheme, we need ...
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