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.
1,352
questions
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0answers
12 views
Non-Linear ODE with some consideration
π(π‘)πβ³(π‘)+πΌπβ²(π‘)βπ½π(π‘)πβ²(π‘)βπΎπ(π‘)2+π=0
π(0)=0πβ²(0)=0πβ³(0)=0
All constant is positive and π is just depending on time.
This equation I have although is a little bit weird but I feel ...
0
votes
1answer
27 views
Do solutions to the forced Stokes equations for zero-Reynolds number flows always exist?
I'm in the process of solving a forced Stokes flow problem in two dimensions, the governing equations of which are given by
$$
\nabla p = \eta \Delta \mathbf{v} + \mathbf{f}, \quad \nabla\cdot \mathbf{...
1
vote
1answer
22 views
Laplace's equation on the plane: how much boundary data must be specified to guarantee existence and uniqueness?
My question stemmed from a specific problem, so let's jump right in. I want to solve Laplace's equation in plane polars $(r, \theta)$ on the domain $(r,\theta) \in [1,\infty)\times[0,2\pi)$ subject to ...
0
votes
0answers
20 views
Discretization of the Neuman boundary condition.
I am trying to derive the approximation
$$
u_x(0,t) \approx \frac{-3u(0,t) + 4 u(h,t) - u(2h,t)}{2h}
$$
and
$$
u_x(1,t) \approx - \left(\frac{-u(1-2h,t) + 4 u(1-h,t) - 3u(1,t)}{2h}\right)
$$
I ...
0
votes
0answers
32 views
What is the period of the solution of a wave equation boundary value problem
In my studies of numerical PDEs, I was given this problem
We consider a vibrating string that satisfies the wave equation $u_{tt}=u_{xx}$ on the unit interval with boundary conditions $u(0,t)=0$, $u(...
0
votes
1answer
35 views
Are these boundary conditions consistent $y(x,0)=\cos(\pi x)$ and $y(0,t)=0$
I was asked to solve
$$\dfrac{\partial^2 y}{\partial t^2}=9\dfrac{\partial^2 y}{\partial x^2}$$
with the boundary conditions
$y(x,0)=\cos(\pi x)$,$\dfrac{\partial y}{\partial t}(x,0)$ and $y(0,t)=y(4,...
0
votes
0answers
16 views
Existence of weak solutions to initial value problems of second-order hyperbolic differential equations on unbounded domains?
Evans gives a theorem for the existence of weak solutions for second-order hyperbolic differential equations. Specifically, he says if $U$ is an open, bounded subset of $\mathbb{R}^n$, we call $U_T = (...
3
votes
1answer
50 views
Why is this operator called a lift in a PDE context?
Let $L$ be some differential operator and consider the PDE $Lu = 0$ with some boundary conditions. Let $W$ denote the space that solutions live in and $B$ the space that boundary conditions live in.
...
1
vote
0answers
37 views
Holomorphic function with given boundary values
I have a complex valued function on the boundary of a set, and want to know if it has a holomorphic extension to the entire set (holomorphic in interior, continuous up to the boundary).
In other words:...
1
vote
0answers
14 views
finding variational equivalent of a BVP
Find the variational equivalent of the BVP $$-au''(x)+xu'(x)+u(x)=f(x)$$ subject
to $u(0)=u'(L)=0$ .
We start with $$\delta\int_0^Lfudx=\int_0^Lf\delta udx=\int_0^L(-au''+xu'+u)\delta udx$$ By parts ...
1
vote
2answers
33 views
Solving a wave equation with boundary conditions on a circle?
How might one solve a wave equation which had boundary conditions on a circle?
i.e. given $$(\partial^2_x+\partial^2_y)\phi(x,y)=0$$
And known values, $f$ on a circle:
$$\phi(\cos(\theta),\sin(\theta))...
0
votes
0answers
23 views
How to solve a boundary value problem?
for this question I'm having trouble with finding a particular equation, and solving this equation. Here is what I have so far. Can anyone please help me out?
Solve $y'' + \pi^2y = 6x, 0<x<1$
$y(...
1
vote
1answer
35 views
Name of vector which includes value and its derivatives
What would I call a vector that describes a quantity (Temperature in this case) and its first and second derivatives at a specific location? It seems that a vector like this would have a formal name.
...
-1
votes
1answer
11 views
Boundary condition for derivative of trigonometric behaving function
I want to find a trigometric function for which:
$u(0)=u(1)=0$
and
$\frac{\partial u}{\partial y}(y=0)=\frac{\partial u}{\partial y}(y=1)=0$.
Lets look at e.g. $u=C1 cos(a y+\phi_0)+C2 sin(a y +\phi_1)...
2
votes
0answers
13 views
Classic reference text for studying the Dirichlet to Neumann map of an elliptic operator.
I was wondering if anyone could provide good reference texts for studying the Dirichlet to Neumann map of an elliptic operator?
Ideally it would be one of the later chapters of an introductory/general ...
1
vote
0answers
23 views
Does this second-order system of nonhomogeneous ODE have bounded solutions?
The equations of motion for a bead on a smooth, freely rotating rod with unit length are
$$
\left\{\begin{aligned}
0&= \omega_0^2\sin\theta - x\dot{\theta}^2+\ddot{x}\\
0&=3\omega_0^2\cos\...
0
votes
0answers
18 views
Wave equation on half space with nonzero Dirichlet boundary condition
We consider the wave equation for $u(t,x)$
$$
\partial_t^2 u - \Delta u = 0 \, \text{ in } (0 , \infty) \times \mathbb R^n_+,
$$
where $\mathbb R^n_+ := \{ (x_1, \cdots , x_n ) \in \mathbb R^n: x_n &...
1
vote
0answers
27 views
Solving inhomogeneous ODE - using Duhamels principle
I want to solve following ODE:
$a (-b u(x_2)+\frac{\partial^2 u(x_2)}{x_2^2})+c x_2 (-b u(x_2)+\frac{\partial^2 u(x_2)}{x_2^2})-d(\frac{\partial^4 u(x_2)}{x_2^4}-2b\frac{\partial^2 u(x_2)}{x_2^2}+b^2u(...
0
votes
0answers
13 views
Creating a score that fits within the value boundaries of an existing model
First let me say that I'm sorry if I don't explain this in the clearest way. I'm a researcher, and not trained in mathematics. I'm looking at an existing algebraic model and trying to create an ...
0
votes
0answers
43 views
Vibrating Circular Membrane with Vibrating Boundary
Looking online, I have found many solutions to variations of 'vibrating circular disc problem', that is solving the following PDE
\begin{align*}
\begin{cases}
u_{tt} - c^2\nabla^2
u = 0\\
u(R, \varphi,...
1
vote
0answers
35 views
Is this BVP solvable using Neumann boundary conditions?
I have the BVP
$$u_{xx}+sin(x) = 0 \qquad \forall x\in[0,2\pi]$$
with dirichlet boundary conditions
$$u(0)=u(2\pi)=0$$
Obviously $sin(x)$ is a solution. But if we instead have Neumann boundary ...
1
vote
3answers
32 views
Analytic solution of the heat equation with a source term
I have the heat equation with Dirichlet boundary conditions
$$u_t(t,x)=u_{xx}(t,x)+\sin(x)$$
$$u(t,0)=u(t,2\pi)=0$$
$$u(0,x)=u_0(x)$$
Now, without the source term I could write the solution as
$$u(t,x)...
0
votes
0answers
21 views
How to solve this Boundary Value Problem? (Using Fourier)
$u_{tt} = u_{xx} - u$
Boundary condition: $u(x,0)=f(x), u_t(x,0)=0,$ $f(x)$ is a Schwartz function.
I tried to make $u(x,t)=X(x)T(t)$, then I get $\frac{X}{X''}=\frac{T}{T'' + T}$.
Let $\frac{X}{X''}...
0
votes
0answers
9 views
Control chart boundaries for the proportion of words incorrectly typed by a typist per hour
This a problem presented in Hoel's Probability Book Chapter 3.
Suppose you wish to construct a control chart for the proportion of words incorrectly typed by a typist per hour. If she typed 1200 words ...
0
votes
0answers
19 views
Harmonic function with Dirichlet and Neumann boundary
Suppose $D^+_1 \subset \mathbb{R}^2$ is the upper half part of the unit disk and $u$ solves
$$\left\{
\begin{aligned}
\Delta u &=0&
\quad &\text{ in $D^+_1$};\\
\partial_2 u&= f(x,u)&...
1
vote
1answer
48 views
Verify that integral satisfies boundary conditions
Consider $$u(x_0,y_0,z_0)=\frac{z_0}{2\pi}\int_{\mathbb R^2}\frac{f(x,y)}{\left[(x-x_0)^2+(y-y_0)^2+z_0^2\right]^{3/2}}\,dx\,dy.$$
Show that
(i) $u\to f(x_0,y_0)$ in the limit as $z_0\to0$ and (ii) $u\...
0
votes
0answers
23 views
Diffusion (Heat) equation at the equilibrium
Let's consider the diffusion (heat) equation at the equilibrium:
$D \rho''\left(x\right)= \phi \rho\left(x\right)+\eta\left(x\right)$.
On a domain $x\in\left[-L,L\right]$, for instance $L=\infty$.
My ...
0
votes
2answers
102 views
Show that this ODE with boundary conditions has a unique solution
We have the ODE $$-\ddot{x}+q(t)x=g(t),\quad x(a)=x_a,\, x(b)=x_b$$
with $g\in C([a,b],\mathbb{R}$ and $q\in C([a,b],\mathbb{R}_0^+)$. Show that there is a unique solution.
First of all, I ...
2
votes
0answers
18 views
Poisson equation in a cylinder
I need to solve the problem $\nabla^{2} u(r,\theta,z)=Q(r,\theta,z)$
inside a circular cylinder $(0 < r < a, 0 < \theta < 2\pi, 0 < z < H)$ subject to $u = 0$ on the sides.
I'm ...
1
vote
0answers
21 views
Wave equation near two boundaries in n dimensions
Consider a function $u(t, x, y)$ defined on $t \in (t_i, t_f) \ , x > 0 \ , y \in \mathbb{R}^{n - 2}$ satisfying the wave equation (here $\Delta_y$ is the Laplace operator w.r.t. the variable $y$)
\...
0
votes
0answers
32 views
Coupled reaction diffusion equation ODE.
How would one go about solving this with the B.C.s considering the derivatives are not defined at the boundary?
1
vote
1answer
18 views
Find the Eigenvalues and Eigenfunctions for the Boundary problem
I recently found this answer to a similar problem I'm currently working on.
The problem is the following...
Find the eigenvalues and eigenfunctions for
$y^{\prime \prime}+\lambda y=0$
with the ...
0
votes
0answers
21 views
Asymptotic expansion linear elliptic PDE in 2D
I am looking for the asymptotic expansion of a solution to the boundary value problem
$$
\Delta u + \varepsilon^\alpha f(u)=0
$$
with $\alpha \geq 1$ and $f$ polynomial (e.g. $f(u)=u$, $f(u)=u^2$, ...)...
4
votes
1answer
101 views
uniqueness of system of PDE using Lax Milgram
Consider the following question.
(5) Consider the system
(1)
$$
\begin{array}
-\Delta u(x)-v(x)&=&f(x) && x \in \Omega \\
u(x)-\Delta v(x) & = & g(x) && x \in \Omega \\...
0
votes
0answers
28 views
Sturm-Liouville Form with boundary conditions
I am having some problems understanding the Sturm-Liouville Theory for ordinary values when we have a non-symmetric system and non-homogenous conditions. I understand that we use a weighting function ...
3
votes
2answers
79 views
Prove that $f\left ( x \right )- x^{2021}$ always has at least one root $x_{0}\in\left ( 0, 1 \right )$
Given positive continuous function $f\left ( x \right )$ on the interval $\left [ 0, 1 \right ]$ so that $\int_{0}^{1}f\left ( x \right ){\rm d}x< \frac{1}{2022}.$ Prove that $f\left ( x \right )- ...
0
votes
2answers
46 views
1D Wave PDE with Nonzero Initial and Boundary Conditions
I'm not sure how to start this PDE since the initial and boundary conditions are nonzero. May someone point me in the right direction?
This is the problem:
$$u_{tt} = u_{xx}$$
$$u(x,0) = \frac{1}{2+ \...
1
vote
0answers
31 views
Solution of the 2D diffusion equation in a rectangular domain
I have a very specific problem concerning the solution of the two-dimensional diffusion equation in a rectangular domain. The problem I have is that I don't understand the way to get to the final ...
0
votes
0answers
23 views
Finding norm of operator on $W^1_2 [0;1]$
Let $H = W^1_2[0;1]$ (Sobolev space) and operator $A: H \to H$, $(Ax)(t) = t\cdot x(t)$. I'm trying to find $||A||$ w.r.t norm $||x||^2 = \int\limits_0^1 \left( x^2(t) + \dot{x}^2(t)\right) dt$.
What ...
0
votes
0answers
16 views
Using Lax Milgram lemma and energy estimates on the real line
I just want to check something. I want to use the energy estimates on the real line for an elliptic operator $L$ acting on $L^2(\mathbb{R})$. (The energy estimates are related to the Lax-Mi
https://...
10
votes
2answers
232 views
Impose PDE itself as Boundary Condition?
Consider, for example, the elliptic PDE $u_{x}+u_y+u_{xx}+u_{yy}=0$ for $(x,y)\in[0,\infty)^2$.
Solution methods often require me to impose boundary conditions. Often, these arise naturally from ...
0
votes
0answers
7 views
How to generalize a boundary value problem?
A boundary value problem can be of k-order with k conditions where it could be linear and non-linear equations. Can you give me a neat way to represent the equation or a good free/open source?
write a ...
0
votes
1answer
32 views
Minimizing $|\frac{f'}{f}|$ and $|\frac{f''}{f}|$ under constraints.
Let's give constraints first.
$f$ is a $C^2$ function on $[0,K]$, and we require that $f(0)=f(K)=1$, $f'(0)=C$, $C>0$ is a given constant. The goal is to minimize the quantities $$\left|\frac{f'}{f}...
0
votes
0answers
8 views
What does it mean agreement in norm mean in differential equations?
I am taking a course in numerical analysis in physical systems.
Trying to understand the next definition:
Absorbing boundary condition:
Suppose $
u$
solves the well-posed differential equation
$$
Lu=f,...
1
vote
1answer
70 views
How to maximize $\| {\bf U x}\|^2$ when ${\bf U}$ is a upper triangular matrix
When ${\bf U}$ is an $N$-by-$N$ complex-valued upper triangular matrix whose diagonal elements are positive real values, how to obtain an $N$-by-$1$ vector ${\bf x}=[x_1\cdots x_N]^T$ with $|x_n|=1$, ...
0
votes
0answers
46 views
Can I use separation of variables to solve the heat equation on an infinitely long rod
The heat equation in 1D is:
$$
\frac{\partial T(x,t)}{\partial t} = \alpha \,\frac{\partial^2T(x,t)}{\partial x^2} \, , \quad \alpha > 0
$$
My IC's and BC's
\begin{cases}
T'_x(0,t) = T'_x(L,t) = ...
3
votes
1answer
55 views
Is this Laplace BVP well posed? If not, why?
Consider the boundary-value problem for the Laplacian $\nabla^2\phi(x,y)=0$ within a semi-infinite strip $0<x<a$, $0<y<\infty$, with the following boundary conditions
$$\partial_x\phi(0,y)=...
2
votes
1answer
69 views
Conceptual Question About the Nature of PDE Boundary Conditions
In this $1D$ boundary value explanation, it is stated that "Taking the example of a straight line, whose slope at the boundary points is decidedly not $0$ [the animation shows a mostly-unlabeled ...
2
votes
1answer
32 views
Applying simple boundary conditions to a multi-variable function.
If I have the following general differential equation solution (to Laplace equation):
$$
\phi(r,\theta)= A+B\ln(r)+\sum_{n=1}^{\infty}\left(C_nr^n+\frac{D_n}{r^n}\right)\left( E_n \cos(n\theta) + F_n \...
2
votes
1answer
43 views
Integral of Legendre polynomials with tangent
I have encountered the following relationship$^{[1][2]}$, stated without proof both times
$$\int_0^\gamma dt \tan(t/2)\cdot [P_n(\cos(t))+P_{n-1}(\cos(t))]=\frac{1}{n}[P_{n-1}(\cos(\gamma))-P_{n}(\cos(...
