Questions tagged [boundary-value-problem]
For questions concerning the properties and solutions to the boundary-value problem for differential equations.
0
votes
0answers
14 views
Understanding Boundary Conditions for Approximating solutions for a PDE
Below is the Poisson Equation for $\Omega = (0,1)^2$ in which we are to approximate the solution via the Finite Difference Method:
\begin{align}
-\Delta u &= f, \ \ (x,y)\in\Omega ,\\
u &= g,...
0
votes
0answers
10 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
votes
0answers
17 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] $
1
vote
1answer
79 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 ...
0
votes
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 ...
0
votes
0answers
29 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)\...
1
vote
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:
$$u_{tt} = u_{xx} \,\,\,\,\,\,\,\, 0 < x < L$$
$$u(0,t) = u(L,t)$$
$$u_x(0,t) = u_x(L,t)$$
$$u(x,0) = sin\bigg(\frac{2\pi x}{L}\bigg) + cos\...
2
votes
1answer
29 views
Wave equation with Neumann BC on semi-infinite domain
This problem is from our recitation which I do not have solutions for, and I'm stuck on the very last part where I need to satisfy the $u_t(x,0)$ initial condition.
The problem is:
$$
\left\{
\...
0
votes
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)...
0
votes
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
vote
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.
$$\Delta u =k,\text{ $k$ constant}\\
\...
0
votes
0answers
12 views
Exist a constant that only depends of a power and dimension that bounds the euclidean norm.
I have this problem but I don't know how to solve it.
Considering the Euclidean norm |x|, in $\mathbb{R}^n$. If $a>0$, prove there exist $c>0$ depending only of $n$ and $a$ such that:
$$c^{-1}(|...
1
vote
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<...
0
votes
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 $S$...
1
vote
1answer
63 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
votes
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 $y(x_0)=y_0$ ...
0
votes
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
$$G(x,z)=\begin{...
1
vote
1answer
45 views
Boundary value problem, Laplace's equation in polar coordinates
I have trouble understanding the question and solution of a boundary value problem.
Consider Laplace's equation in polar coordinates $(r,\theta)$
$$\frac{\partial^2 \Phi}{\partial r^2} + \frac 1 r \...
1
vote
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 $$v(x,0) = ...
0
votes
0answers
28 views
2D disc-shaped region kernel density estimation boundary bias correction for Gaussian bandwidth $h$ and boundary radius $R$
I am trying to develop a closed form expression for the boundary bias correction factor for kernel density estimation in a circularly-bounded 2d region where Gaussian kernel diameter is $2h$ and ...
2
votes
0answers
30 views
Parabolic PDE with zero Neumann condition
I'm working with the following parabolic PDE:
$$u_t = \nabla\cdot(\alpha(x)\nabla u)- \beta u\\
x\in\Omega \subset \mathbb{R}^2\\
\partial_n u = 0$$
Given the Neumann boundary condition above, if I ...
1
vote
1answer
42 views
Calculus - Integral - existence problem
We are given a function $f$, $f$ is integrable (in the riemann sense) in $[a,b]$ and also $f'$ is a continuous, and $f(a)=f(b)=0$. Prove that there exists a point $c$ such that $|f'(c)| \geq \frac{4}{...
-1
votes
2answers
38 views
Laplace Equation in Rectangle: Boundary Value Problem [closed]
Solve the boundary-value problem $∆u = 0$ (by this we mean $u_{xx} + u_{yy} = 0$) in the rectangle $0 < x < π$, $0 < y < 1$, with the boundary conditions $u(0,y) = 0$, $u(π,y) = g(y)$, $u(...
2
votes
1answer
54 views
Solving a system of ODE
Solve $$\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$$
where $A,B,\eta_k$ some known constant. $e_k, k=1,2,3$ ...
0
votes
0answers
16 views
Discretization Dirichlet boundary condition for Elliptic PDE with finite volume method
I want to discretize the following equation using FMV:
$$\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
vote
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 ...
0
votes
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-...
11
votes
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 $x=0$: $\...
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 $\mu$ :
$$u_a + u_t = -\mu t u\,\,\,\,\,a,t>0$$
$$u(a,0)=u_0(a)\,\,\,a≥0$$
$$u(0,t)=F(t)=b\...
2
votes
2answers
43 views
Solving the BVP $u_{xx}+a^2u=\sin(\pi x)$ with $u(0)=1$ and $u(1)=-2$
I am trying to solve the BVP $$u_{xx}+a^2u=\sin(\pi x), \ \ \text{for} \ \ 0<x<1$$ with $u(0)=1$ and $u(1)=-2$, $\forall a\in\mathbb{R}$.
I begin by solving the homogeneous equation $u_{xx}+a^...
1
vote
0answers
84 views
Laplace equation in 3D with numerous Non-Homogeneous BC(s) [Strategy Check]
I need to solve the three-dimensional Laplace equation ($\nabla^2T = 0$) where $\nabla^2=\frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$ in the domain ...
0
votes
1answer
24 views
Laplace Equation with Inhomogeneous Boundary Condition
I'm currently learning about separation of variables as applied to situations where the boundary conditions are not homogeneous.
I'm having trouble deciding how to handle one of the boundary ...
0
votes
0answers
22 views
Matrix representation of a finite difference with Neumann boundary conditions
Given 1D data $[c_1,c_2,c_3,\cdots,c_N]$ I can represent the derivative operation as a matrix product. For example, using the central difference
$$
\left.\frac{\partial c}{\partial x}\right|_k
\...
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 $$ \frac{\...
1
vote
0answers
22 views
A sufficient condition for a Neumann problem to have solution
Suppose we have the Boundary Value Problem (BVP):
$\Delta{u}=f$ , in the domain (topos) $D$
$\partial_{n}{u}=h$, in $\partial{D}$
It is easy to prove using $1$st Green's identity that a necessary ...
0
votes
0answers
27 views
Poisson integral formula for boundary value problem
I have gotten stuck on a boundary value problem which I believe is to be solved using the Poisson Integral Formula. The problem is: $$\nabla^{2}\psi=0, \psi(x,0)=0, |x|>1 ; |\psi(x,y)|<|x|, |x|\...
0
votes
1answer
31 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 ...
2
votes
3answers
46 views
Method of Characteristics for $u_t + uu_x = -2u$
Consider the following quasi-linear PDE : $u_t + uu_x = -2u$, with the boundary condition $u(0,t) = e^{-t}$.
Show, using the method of characteristics, that the solution to this boundary value ...
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
0answers
10 views
Boundary value problem for Hemlholtz equation with pseudospectral method.
I have an equation of the form $(1-0.1 \Delta)f=1$ with boundary condition $f=0$. I need to solve it for $f$ by pseudospectral method in python. Apparently, it should be $f=\frac{1}{1-0.1\Delta}$. I ...
0
votes
0answers
7 views
Inverse of differential operator and boundary conditions
I want to clear a point that "Why boundary conditions are important in taking inverse of any differential operator (lets say Laplace operator )?". What i understood is that any transformation is ...
1
vote
0answers
28 views
Non-Dimensionalizing Separation of Variables PDE
I have a problem where I'm considering a cylinder with a closed end at the bottom (x = 0) and open end at the top (x = L).
The problem is looking at pressure disturbances throughout the air column.
...
0
votes
0answers
20 views
MATLAB solving BVP using bvp4c, variable coefficient
Say I have an bvp that involve the equation below
h''' = f(r)*h'' + h'
If I was to have enough boundary conditions to satisfy h(r) ODE, would it be possible to solve the f(r) equation ...
1
vote
1answer
38 views
Solve $u_{rr}+r^{-1}u_r+r^{-2}u_{\theta\theta}=0.$
I have two BVP's and for both of them the solution is similar, however there is one thing I can't explain. First of all let me state the BBVP's. The first one is
$$(1)=
\begin{cases}
&u_{...
0
votes
0answers
7 views
The boundary value problem has a unique solution if and only if the corresponding homogeneous system has a unique solution
I want to prove that the following claims are equivalent:
1) The boundary value problem $Ly = f, By = c$ has a unique solution for all $f \in \mathcal{C}[a,b]$ and $c \in \mathbb{C}^2$,
2) $Ly = 0, ...
0
votes
0answers
40 views
What means norm $\|u'\|$ and $\|u\|$ in Sobolev space $W^{1,2}(I)$ and why is functional continuous?
We have functional $F: W^{1,2}(I),I\langle 0,1\rangle \rightarrow \mathbb{R}$, I proved that functional is linear, I also counted that is bounded and I get this state:
$$|F(u)|\leq {C_1} \|u\|+{C_2}\|...
0
votes
0answers
22 views
Two-dimensional Laplace equation with weird Robin BC
I need to solve the steady-state heat equation a.k.a. Laplace equation over a rectangle
For $\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0$ defined on $x \in [0,a]$ and $y ...
