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.
1,790
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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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1
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Laplace's Equation on an annulus with Dirichlet/Neumann boundary conditions
We're given general solutions to Laplace's equation of the form $ \phi(r,\theta) = Ar^\alpha \cos(\beta \theta) $, and we're asked to find specific solutions given the boundary conditions: $ a \leq r \...
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1
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Solve the PDE using Direct Integration
Question: Solve for $z(x,y)$ using direct integration
$\frac{\partial{^2z}}{\partial{x^2}} = a^2z$
with the conditions
$\frac{\partial{z}}{\partial{x}}(0,y)=a\sin (y)$
$\frac{\partial{z}}{\partial{y}}(...
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0
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Do additional boundary conditions affect the compact embedding of Sobolev spaces?
Suppose $\Omega \subset \mathbb{R}^2$ open bounded and sufficiently smooth with outward unit normal $\mathbf{n}$.
It is known by the Rellich-Kondrachov theorem that
$$ H^2(\Omega) \hookrightarrow H^1(\...
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36
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Green Function Expansion Theorem Proof
Let $G(t,\tau,l)$ be the Green Function associated to a differential self-adjoint eigenvalue problem:
$$
\hat{L} \psi = l \psi \hspace{2mm} \text{on} \hspace{2mm} [a,b] \\
\underline{\hat{U}} \psi = \...
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Reduction of periodic boundary conditions to Dirichlet boundary conditions
Let us consider the following heat equation with periodic boundary conditions, which is nothing but the problem on the flat torus:
$$ \partial_t u(x,t) - \partial_x^2 u(x,t) = f(x,t),(x,t) \in [0,1] \...
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Solving a boundary value problem with an integral constraint.
EDIT1: Initial post left as it, but modified below to answer comments
EDIT2: the proper problem (see EDIT1) was ill-posed close to the right boundary, preventing convergence of the BVP. Changing the ...
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Has anyone seen PDE involving determinant of Hessian?
Travelling through long and convoluted mathematical tracks, I have stumbled upon the following PDE. All I'm really asking is whether someone has seen this PDE before, and if it has a name. So here is ...
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1
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Sturm-Liouville problem: $\frac{d}{dx}\Big(e^{2x}\frac{dy}{dx}\Big)+e^{2x}(1+\lambda)y=0$
I'm trying to solve the Sturm-Liouville problem
$$\frac{d}{dx}\Big(e^{2x}\frac{dy}{dx}\Big)+e^{2x}(1+\lambda)y=0$$
with boundary conditions $y(0)=y(\pi)=0$ but i'm stuck because every reference on SL ...
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1
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Determining weak solution for Dirichlet problem
Let $D$ be the unit disk in the plane and let $\Omega= D\setminus\{0\}$. The Dirichlet problem
\begin{cases}
Δu = 1 & \text{in } \Omega \newline
u=0 & \text{on } \partial \Omega
\end{cases}
...
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1
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Eliminating Neumann boundary condition for elliptic PDE
In his PDE book, Evans demonstrates that for elliptic PDEs with Dirichlet boundary condition, the boundary term can be eliminated:
I am now wondering if this also works with Neumann boundary ...
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Help to solve the integro-differential equation $y'(x)=-k\frac{y^2(x)}{x^2}\int_0^xt^2y(t)\,dt$
I have this differential integral equation from a physics problem
$y'(x)=-k\frac{y^2(x)}{x^2}\int_0^xt^2y(t)\,dt$ and i dont know how to solve such equation.
Edit 3 (the third time's a charm):
Comment:...
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Questions regarding $C_n^1(\overline\Omega)$, the space of functions with normal derivatives
The definition of which functions have normal derivatives, and to which we can apply Green's First Identity to, seems to be very delicate. Let $\Omega$ be a $C^1$ domain in $\mathbb{R}^d$ with $d\geq ...
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Domain Truncation Error for Semi-Infinite BVP
Suppose I have a BVP defined on a semi-infinite domain, of a form that looks something like $$ N[f] = 0 \\ f(0) = a \\f'(0) =b \\ f'(\infty) = c$$ where $f$ is some (generically nonlinear) third order ...
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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 ...
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28
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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 ...
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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}...
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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 ...
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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
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0
answers
75
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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\...
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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 ...
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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.$$
A numerical approach is to almost first write as ...
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A boundary value problem for differential inclusions
Given a finite family $(\varphi_k)_{1 \le k \le K}$ of smooth scalar functions on $\mathbb{R}^n$ and two points $a, \, b \in \mathbb{R}^n$, I am interested in conditions such that there exists a ...
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Centered-difference left Neumann boundary condition discretization when integrating equation inward
I want to discretize the following nonlinear equation on a domain $[0, R]$:
$\frac{dA}{dr} + F(A) = 0$,
where $F$ is a nonlinear operator, with the left boundary condition at large finite radius $r = ...
- 1
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Variational calculus with constraints of boundary conditions. How to take into account
I am looking for references to understand the following.
I've recently solved the thin plate functional minimization subjected to interpolation constraints.
The calculations I did are mostly here: ...
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Error in the energy norm for inhomogeneous Dirichlet boundary conditions
Short description
When conducting FEM analysis with inhomogeneous Dirichlet boundary conditions, I compute the error in the energy norm with an expression that should only work for problems with ...
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1
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Solution of the parabolic PDE using Green's function
Green's function for the parabolic PDE is defined as:
$$\Delta G(\vec{x},t,\vec{\xi},\theta)=\delta(\vec{x}-\vec{\xi},t-\theta)$$
Where $G$ satifies the homogeneous initial and boundary conditions. ...
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Troubles with solving a Laplace equation
I'm struggling in solving an exercise about the Laplace equation over the domain $[\frac{\pi}{2}, \pi] \times [0, \pi]$ with the boundary conditions:
$f(\frac{\pi}{2},y)=f(\pi,y)=f(x,\pi)=0$
and
$f(x,...
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Existence of Fokker-Planck equation under Cauchy boundary condition.
A 1D Fokker-Planck equation within a constrain region is uniquely characterized by three functions and a boundary conditon:
A drifting term $\mu(x,t)$.
A diffusion term $\sigma(x,t)$.
An initial ...
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0
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Solving Dirichlet problem on the unit disc. Is it correct?
I would like to solve the Dirichlet problem in $\Omega = B(i,2)$ and with boundary function $\varphi(x+iy) = x^2y^2$.
Attempt
I first consider the conformal map $f(z) = \frac{z-i}{2}$, with inverse $f^...
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Möbius transformation and Poisson kernel
Let's say $T(x,y)$ measures the temperature in degrees Celsius at points $(x,y)$ of a region $A$:
$$A= \{ z \in \mathbb{C}: \text{Re } z> 0 , |z-2| > 1\}$$
If I put some ice at air pressure at ...
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0
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Solvability of the oblique derivative boundary value problem
Let $\Omega$ be a bounded $C^{2,\alpha}$ domain in $\mathbb{R}^n$ that satisfies an interior sphere condition at each point of $\partial \Omega$, i.e., for each $x_0\in \partial \Omega$, there exists ...
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If $f(x) = Ae^{x} + Be^{-x}$ and $f(1) = 0$, then $f(x) = C\sinh(x - 1)$
I need to find a function $f(x)$ of the form $$ f(x) = Ae^{x} + Be^{-x} \;\; A,B \in \mathbb{R} $$ with $f(1) = 0$
The professor immediately concluded that $$ f(x) = C \sinh(1-x) \;\;\; C \in \mathbb{...
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1
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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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0
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How many Boundary and initial conditions are needed for nonlinear coupled PDEs? Is there a theorem?
I would like to solve the following coupled PDEs, and I wonder how many initial and boundary conditions I need.
$$ \partial_t z(x,t) + \partial_x (z(x,t) b(x,t))=0 $$
$$ \partial_x (z(x,t) |b_x|^{m-1}...
1
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1
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Confusion about one initial/boundary value problem for heat equation
This is a follow-up question to this. The referenced question arose while I was trying to solve
$$
\begin{cases}
u_t = \frac{1}{2} \Delta u, & x \in X, \\
u ( 0, x ) = 1, & x \in X, \\
u ( t, ...
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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&\...
user1172934
1
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1
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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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0
answers
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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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1
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Compatibility of Initial/Boundary Conditions in a Convection-Diffusion Problem?
So, I'm reading a book that numerically solves the following convection-diffusion problem
$$\dfrac{\partial u}{\partial t} + c\dfrac{\partial u}{\partial x} = \alpha\dfrac{\partial^2 u}{\partial x^2} \...
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0
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How do we rigorously eliminate $r^n$ and $\log r$ terms in a Fourier series (solving the polar Laplace equation) which is bounded at infinity?
In my PDE module, the general solution to Laplace's equation $\nabla^2 T=0$ in the plane (in polar coordinates) was shown to be $$T(r,\theta)=A_0+B_0\log r+\sum_{n=1}^\infty(A_nr^n+B_nr^{-n})\cos(n\...
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1
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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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0
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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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votes
2
answers
270
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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):
$$
\begin{cases}
\Delta^2 u = f \\
\nabla u \cdot \nu = g \\
u = u_D
\end{cases}
$$
...
- 1,682
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votes
0
answers
29
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Can separation of variable be used for mixed boundary conditions where boundary conditions are dependent on one another?
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:
$$...
- 585
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1
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Help on transformation of boundary conditions
I was working the transformation in this paper
A new algorithm for solving classical Blasius equation by Lei Wang
The boundary value problem is
He used the transformations
$$y=f''(\eta),x=f'(\eta)$$
...
- 1,266
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0
answers
33
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Application of boundary condition finite difference scheme
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 ...
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