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.
1,787
questions
1
vote
1
answer
49
views
Sturm-Liouville problem: $\frac{d}{dx}\Big(e^{2x}\frac{dy}{dx}\Big)+e^{2x}(1+\lambda)y=0$
I'm trying to solve the Sturm-Liouville problem
$$\frac{d}{dx}\Big(e^{2x}\frac{dy}{dx}\Big)+e^{2x}(1+\lambda)y=0$$
with boundary conditions $y(0)=y(\pi)=0$ but i'm stuck because every reference on SL ...
- 15
-1
votes
0
answers
18
views
what is called this sub space $ H^{1}_{0,p} $ [closed]
What's the name of this sub sobolev space $ H^{1}_{0,p}= \left\{ u\in AC\left( \left[0,+\infty \right), \mathbb{R} \right) : u\left( 0 \right) = u\left( +\infty \right) =0 , \sqrt{p} \acute{u} \in ...
3
votes
1
answer
51
views
Determining weak solution for Dirichlet problem
Let $D$ be the unit disk in the plane and let $\Omega= D\setminus\{0\}$. The Dirichlet problem
\begin{cases}
Δu = 1 & \text{in } \Omega \newline
u=0 & \text{on } \partial \Omega
\end{cases}
...
- 387
3
votes
1
answer
67
views
Eliminating Neumann boundary condition for elliptic PDE
In his PDE book, Evans demonstrates that for elliptic PDEs with Dirichlet boundary condition, the boundary term can be eliminated:
I am now wondering if this also works with Neumann boundary ...
- 105
5
votes
1
answer
107
views
Help to solve the integro-differential equation $y'(x)=-k\frac{y^2(x)}{x^2}\int_0^xt^2y(t)\,dt$
I have this differential integral equation from a physics problem
$y'(x)=-k\frac{y^2(x)}{x^2}\int_0^xt^2y(t)\,dt$ and i dont know how to solve such equation.
Edit 3 (the third time's a charm):
Comment:...
0
votes
0
answers
26
views
Questions regarding $C_n^1(\overline\Omega)$, the space of functions with normal derivatives
The definition of which functions have normal derivatives, and to which we can apply Green's First Identity to, seems to be very delicate. Let $\Omega$ be a $C^1$ domain in $\mathbb{R}^d$ with $d\geq ...
- 45
1
vote
0
answers
29
views
Domain Truncation Error for Semi-Infinite BVP
Suppose I have a BVP defined on a semi-infinite domain, of a form that looks something like $$ N[f] = 0 \\ f(0) = a \\f'(0) =b \\ f'(\infty) = c$$ where $f$ is some (generically nonlinear) third order ...
- 2,426
2
votes
0
answers
41
views
Existence of an unique solution of an ODE without boundary conditions
I would like to ask how to determine if there is a unique solution to an ODE that does not have any boundary conditions, nor initial conditions. It sounds weird but I wanted to know if there is a case ...
- 49
0
votes
0
answers
26
views
High order finite difference schemes for boundary value problems on a finite interval
I have some questions. I'm going to assume everything is in 1d with a Laplacian operator. If I discretize the Laplacian operator using $p = 2a+1$ grid points with periodic boundary conditions, I ...
- 533
0
votes
0
answers
18
views
Elliptic boundary value problem with time dependency
I am looking at an elliptic boundary value problem for an open set $\Omega\subset \mathbb{R}^3$ that is solved over a time interval $(0,T)$ with $T>0$
\begin{equation*}
\begin{cases}
\begin{aligned}...
- 1
0
votes
0
answers
25
views
Required number of boundary conditions for a partial differential equation
Consider an ordinary differential equation of order $N$ for a function $u(x)$, of the form
$\dot{u} = f\left(u, \frac{du}{dx}, \frac{d^2 u}{dx^2}, ..., \frac{d^N u}{dx^N} \right)$.
The number of ...
- 161
0
votes
0
answers
27
views
How to justify the way the ghost nodes are applied in Finite Difference Method?
Boundary problem consists of:
PDE which is fulfilled for the points inside some domain D, and
boundary conditions that apply to the points on the boundary.
In other words, the equation describes ...
- 1
0
votes
0
answers
56
views
Uniqueness of the exterior Neumann problem
I have a question regarding the uniqueness of the exterior Neumann problem, as stated by Lemma 2.4 in this paper.
Let $\Omega \subset \mathbb{R}^3$ be a bounded domain with $C^2$ boundary $\partial\...
- 71
0
votes
0
answers
37
views
Relations between boundary and inner probabilities for 2D finite state Markov chains
In calculus the Green's theorem relates an integral around a curve to a double integral over the plane region bounded by that curve.
Does there exist an analogy of this theorem for 2-dimensional ...
- 511
5
votes
0
answers
200
views
Reference for Shooting Method
Consider the following setup. We have a second order boundary value problem:
$$\dfrac{d^2y}{dx^2}=F(x,y,dy/dx);\qquad y(x_0)=y_0,\quad y(x_f)=y_f.$$
A numerical approach is to almost first write as ...
- 7,789