This tag is for "partial differential equations". As opposed to "ordinary differential equations".
1
vote
1answer
27 views
Let $u(x,t)$ be the bounded solution of $u_t-u_{xx}=0$
I am stuck on the following problem:
Let $u(x,t)$ be the bounded solution of $u_t-u_{xx}=0$ with $u(x,0)=\frac{e^{2x}-1}{e^{2x}+1}$.Then $\lim_{t \to +\infty} u(1,t)=?$
Can someone point me ...
2
votes
1answer
19 views
The extension of smooth function under the restriction of its Laplacian
$u$ is a smooth bounded function on $\Omega-\{0\}$ where $\Omega$ is an open neighborhood of $0$ in $\mathbb R^n$. If $\Delta u$ is a bounded function on $\Omega-\{0\}$, then can we extend $u$ to be a ...
1
vote
1answer
37 views
PDEs: subsequence converges to solution, so whole sequence does too
Suppose we want existence of a function $u$ for the PDE
$$(\frac{d}{dt}u,v) = b(u,v)$$
for all $v$ in a test space.
Sometimes in PDE you have use a Galerkin approximation, so say $u_n$ is a sequence ...
1
vote
1answer
67 views
solution to heat equation in a particular case
$$\frac{\partial u}{\partial t}(t,x)-\frac{1}{2}\frac{\partial^2 u}{\partial x^2}(t,x)=0,\ \ t>0, x\in\mathbb{R}$$
$$u(0,x)=\max(x,0)$$
$$\frac{\partial v}{\partial ...
1
vote
0answers
39 views
Black-76 pde hedging argument wrong
I want to obtain the PDE for the Black-76 model. I believe it has to be the following PDE:
$$\left(\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}F^{2}\frac{\partial^{2} V}{\partial ...
3
votes
1answer
41 views
Regularity for solutions of $-\operatorname{div}(g(|\nabla u|^2)\nabla u)=f$.
Let $\Omega\subset\mathbb{R}^N$ and suppose that $g\in C^{1,\alpha}(\mathbb{R},\mathbb{R})$, $f\in C^{0,\alpha}(U)$ for every open $U$ with $\overline{U}\subset\Omega$, $\alpha\in (0,1)$ and $g\geq ...
0
votes
0answers
33 views
laplace equation in a rectangle with boundary condition
$u_{xx}+u_{yy}=0 \quad in \quad the \quad rectangle \quad 0<x<a \quad 0<y<1$
$u=0 \quad on \quad y=1$
$u=j(y) \quad on \quad x=0 $
$u_y +u=0 \quad on \quad y=0$
$u_x=0 \quad on ...
3
votes
1answer
57 views
Weak solution of a non-linear problem with Lipschitz functions
I'm trying to solve the problem 9.5 in Evans PDE book. The statement goes as follows:
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a $\Lambda$-lipschitz bounded function with $f(0)=0$ and ...
0
votes
0answers
34 views
Laplace equation PDE with boundary condition
$u_{xx}+u_{yy}=0$ in the rectangle $0<x<a , 0<y<b$ with the following boundary conditions:
$u_x=-a \quad on \quad x=0$
$u_x=0 \quad on \quad x=a$
$u_y=b \quad on \quad y=0$
$u_y=0 ...
2
votes
1answer
22 views
The definition of $p$ capacity of a set $A\subset\mathbb{R}^n$
I am having a bit of difficult understanding the definition of the $p$-capacity of a set $A\subset\mathbb{R}^n$ and I was wondering if anyone would be able to clarify whether I have the right idea or ...
3
votes
1answer
45 views
Elliptic regularity - nonlinear case
Let's say I have a weak solution $u \in H^1(\Omega)$, $\Omega \subset \mathbb{R}^n$ open, to the equation
$$
\Delta u = e^u,
$$
let's also assume that $e^u \in L^\infty(\Omega)$.
Does it follow that ...
3
votes
2answers
62 views
Different types of domains $\Omega \subset \mathbb{R}^n$ in PDEs
In PDEs I often read things like:
Let $\Omega$ be a bounded
Lipschitz or
$C^1$ or
$C^2$ or
$C^\infty$
domain
But I have no clue what this means in real life. I understand ...
3
votes
1answer
31 views
Regularization of solutions from a quasilinear equation.
Let $\Omega\subset\mathbb{R}^N$ be a bounded regular domain and $p\in (1,\infty)$. Let $-\Delta_p :W_0^{1,p}\to W^{-1,p'}$ be the $p$-Laplace operator, i.e. $$\langle-\Delta_pu,v\rangle=\int_\Omega ...
-4
votes
0answers
78 views
burgers equation [closed]
This project deals with the partial differential equation (PDE)
$$u_t (x,t)+u(x,t) \cdot u_x(x,t)=0, x \in \mathbb{R}, t \geq 0 \qquad(1)$$
together with the initial condition
$$u(x,0)=u_0(x), x ...
1
vote
1answer
43 views
Laplace Equation PDE
$$U_{XX}+U_{YY}=1 $$ in the annulus $ a<r<b$ with u vanishing on both parts of the boundary r=a and r=b
What I have done is that $u_{xx}+u_{yy}=0$ has only 0 due to the fact that u is vanishing ...
-1
votes
0answers
20 views
PDE question on transition probabilty and option pricing [closed]
Does anyone know how I can do this question, its nothing on solving the PDE i think all I need to do is just differentiate the integral but I am stuck with that any suggestions
0
votes
2answers
39 views
Continuation of smooth functions on the bounded domain
Given a bounded domain $\Omega\subset \mathbb{R}^n$ and a smooth function $f$ with bounded derivatives on $\Omega$, is it possible to extend $f$ to $\tilde{f} : \mathbb{R}^n \to \mathbb{R}$ such that ...
0
votes
1answer
47 views
Divergence and regularity result on bounded domains
Let $\Omega\subset\mathbb{R}^2$ a bounded set with Lipshitz continuous boundary.
If $$z\in L_0^2(\Omega)=\{v\in L^2(\Omega):\int_\Omega v\,dx=0\},$$
it is true that exists $\phi\in ...
0
votes
0answers
15 views
tool to compute a rigidity matrix?
I have the following mesh and I have to compute the associated rigidity matrix.
I have computed it manually but I need to be sure it's correct. Is there a tool or something simple to help me ...
1
vote
2answers
37 views
Diffusion process. Distribution vs transition probability.
I need confirmation on the following problem: Take a SDE of the form:
\begin{equation}
dX_t=a(X_t,t)dt+b(X_t,t)dW_t
\end{equation}
where all the conditions, such that the solution $X_t$ is defined ...
0
votes
1answer
24 views
question on the weak formulation of a pde
I am trying to find a function u such that it solves equation 1 and 2 here:
Can someone explain to me how 1 and 2 implies the last equation? (the vector n here is the normal that goes "out" of the ...
1
vote
1answer
30 views
Sobolev spaces - embedding - exercise
I have to show that $H^{2}(\Omega) \subset \subset H^1(\Omega)$. I think the Arzela - Ascoli theorem can help.. .I dont know how to start this exercise . I am beginner in Sobolev spaces.. someone can ...
1
vote
0answers
36 views
Nonhomogeneous PDE for the 2-dimensional heat flow
Consider the BVP for the two-dimensional heat flow (the pde is nonhomogeneous) in a square plate. The side $x=20$ is insulated while the other sides are kept at 0 temperature. There is a heat source ...
1
vote
0answers
34 views
Smooth Approximation of $L^p$ function
Given a bounded domain $\Omega \subset \mathbb{R}^n$, is it possible to approximate every $L^p(\Omega)$ function (where $1\leq p < \infty$) by smooth functions $\mathit{C}^{\infty}$ ?
2
votes
1answer
52 views
Concerning the noncompactness of the map, $I: W^{1, 2}(\mathbb{R}^n)\rightarrow L^{2^{\ast}}(\mathbb{R}^n)$
Consider the identity map $I:W^{1,2}(\mathbb{R^n})\rightarrow L^{2^{\ast}}(\mathbb{R}^n)$ where $n\geq 3$. Suppose that this map is not compact that is given some bounded sequence of functions ...
0
votes
1answer
37 views
Calculus of variation,minimizing sequence exists.
Let $F(x,z,p)$ be $C^1$ in $\Omega\times R\times R^n $ where $\Omega$ is bounded domain.
Assume that $F(x,z,p)>\Phi(|p|)-C$ for some continuous $\Phi(s)$ satisfying that $\Phi(s)/s\rightarrow ...
1
vote
0answers
40 views
How to prove the function is a fundamental solution of the operator?
Show that the function $u(x_1,x_2)=\left\{\!\! \begin{aligned} & \frac{1}{2}, if \left| {x_1 - \xi _1 } \right| < \xi _2 - x_2 \\ &0,otherwise \end{aligned} \right.$ is a fundamental ...
1
vote
0answers
51 views
Physical interpretation of a wave equation
Do exist any physical interpretation of (several) boundary conditions like :
$$y=g_1\ne 0,\; \Delta y=g_2,\; ..=\Delta^{k-1} y =g_k,\;
\mbox{on} \; \partial \Omega,\; k\ge 2$$
for the wave equation ...
5
votes
1answer
71 views
Why are weak solutions to PDEs good enough?
Obviously solutions in $C^k$ are nicer but it seems people are happy with obtaining weak solutions in some Sobolev space that only satisfy the weak formulation. Why should this, in the real world, be ...
2
votes
0answers
56 views
When $ A \int_0^{\infty} e^{-\lambda t}S(t)u dt = \int_0^{\infty} e^{-\lambda t}S(t)Au dt$?
I have real Banach space $X$ and a bounded linear operator $S: X \to X$ which satisfy:
1) $S(0)u = u$ $\text{ }$ for all $u \in X$
2) $S(t+s)=S(t)S(s)u = S(s)S(t)u $ $\quad$($ t,s \geq 0$, $u \in X ...
4
votes
1answer
100 views
Tough Inverse Fourier Transform
In reference to this answer I gave the other day, I came across a very interesting function whose IFT would be nice to evaluate as part of completing the solution to the problem I answered. The ...
0
votes
1answer
35 views
A second-order non-linear differential equation
Which are the possible solutions for this second-order non-linear différential equation :
$$\frac{\partial Z}{\partial q} \frac{\partial^2 Z}{\partial p \space \partial t} - \frac{\partial ...
0
votes
1answer
36 views
Is there an elliptic regularity for the whole space?
Is there a $W^{2,p}$ estimate or $C^{2,\alpha}$ estimate for Poisson equations in the whole space $\mathbb{R}^n$? That is, suppose $f\in L^{p}(\mathbb{R}^n),1<p\leq\infty(\in C^{\alpha})$, and ...
1
vote
1answer
34 views
Steady Temperature Distribution Pipe
I was wondering if anyone can show me what approach to finding the steady state temperature distribution in this problem. The image is in the link below.
...
3
votes
1answer
46 views
Change of variables, chain rule
I'm having a bit of trouble getting my head around some notation in a question. I'm told that $u$ satisfies the heat equation $u_t - u_{xx} = 0$ and I'm asked to find the equation satisfied by $v(y, ...
1
vote
1answer
38 views
Simplifying a double integral
How can I evaluate this? Is there a trick or simplification that will make it nicer?
$$ \frac{1}{4\pi} \frac{\partial}{\partial t}\left(2t\int_0^{2\pi}\int_0^1{(x+tr\cos\theta)^2 ...
0
votes
0answers
36 views
about a proof in chap 5 in PDE evans book
in the proof of the theorem 5 page 281 of Evans ( PDE - Evans ) , he write the Morrey estimate. he writes in the estimate : $y \in B(x,r)$
After in the proof we have
$$ |v(y) - v(x)| = |u(y) - ...
0
votes
1answer
23 views
Splitting a 2-D wave equation into two 1-D equations
In finding a solution to $$\begin{cases}
u_{tt}=\nabla^2u, & x\in\mathbb R^2, \,t\gt0 \\
u(x,0)=x_1^2, u_t(x,0)=x_1^2+x_2^2, & x\in\mathbb R^2
\end{cases} $$
a hint was given to split it ...
1
vote
1answer
24 views
Sequence of solutions to heat equation
I'm trying to solve the following question for homework:
Let $U \subset \mathbb{R}^n$ be bounded and open and let $u_i$ satisfy the heat equation $\partial_t u_i - \Delta u_i = 0$ in $U_T$ := $U ...
3
votes
1answer
52 views
Finite difference implicit schema for wave equation 1d not unconditionally stable?
The wave equation 1D with constant density is defined as:
\begin{equation}
\frac{\partial^2 U}{\partial t ^2} = V^2 \frac{\partial^2 U}{\partial x ^2}
\label{eqa}
\end{equation}
And the implicit ...
2
votes
0answers
30 views
Riemann mapping between arbitrary triangles
Question---Is there nice formula for Riemann mapping between arbitrary triangles with vertices (a_1,a_2,a_3) and (b_1,b_2,b_3)?
Comment---I look for the conformal equivalence of interiors promised ...
7
votes
2answers
103 views
Mean Value Property of Harmonic Function on a Square
A friend of mine presented me the following problem a couple days ago:
Let $S$ in $\mathbb{R}^2$ be a square and $u$ a continuous harmonic function on the closure of $S$. Show that the average of ...
2
votes
1answer
44 views
Fourier Transform on Infinite Strip Poisson Equation
Im trying to solve the following Poisson equation:
$$u_{xx} + u_{yy} = \exp(-x^2)\ \text{for}\ x \in (-\infty, \infty)\ \text{and}\ y \in (0,1)$$
$$u(x,0) = 0,\ u(x,1) = 0$$
$$u(x,y) \to 0\ ...
3
votes
2answers
45 views
Sobolev space boundary value in PDE
I often read this:
Let $\Omega$ be a open bounded set. There is a unique $u \in H^1(\Omega)$ such that
$$-\Delta u = f \text{ on $\Omega$}$$
$$u|_{\partial \Omega} = g$$
But how can we write ...
-1
votes
2answers
46 views
When does $v \in H_0^1$ and $|v|_{L^2}=0$ imply that $\|v\|_{H_0^1}=0$?
Let us assume that the boundary of the domain in the definition of the Sobolev spaces $L^2$ and $H_0^1$ is sufficiently smooth.
Let $|\cdot |$ denote the norm in $L^2$. Then for a function $v$ in ...
1
vote
0answers
35 views
Hodograph transformation and implicit solution of a non-linear PDE
I am trying to understand how can one apply the Hodograph transformation to a non-linear PDE. I read that this transformation implies the representation of the solution in the implicit form . So, if I ...
1
vote
2answers
86 views
sobolev space-equivalence of scalar product
Let $f \in L^2(\mathbb{R}^n).$ Why the equation $\Delta u - u = \dfrac{\partial f}{\partial x_i}$ admits a unique solution $u \in H^1(\mathbb{R}^n)$?
2- Prove that there exist a constant $C \geq 0$ ...
1
vote
1answer
39 views
Problem related to wave equation
I am stuck with the following problem:
If $u(x,t)$ satisfies the wave equation:
$u_{tt}=c^2u_{xx}, x \in \Bbb R,t>0$ ,with initial conditions
$u(x,0)=\begin{cases}\sin\dfrac{\pi x}{c}&0\leq ...
0
votes
1answer
44 views
How do I solve the following PDE?
How do I solve the following PDE for $y(x,t)$ and the given auxiliary
conditions for $0≤x≤1$ and $t≥0$?
$$y_{tt}=4y_{xx}$$
$$y(0,t)=y(1,t)=0$$
$$y(x,0)=0$$
$$y_t(x,0)=\sin(\pi x)-\sin(3\pi x)$$
I ...
0
votes
2answers
129 views
Solving Burgers' Equations
I am difficulty with this assignment:
This project deals with the partial differential equation (PDE)
$$
u_t(x, t) + u(x, t)u_x(x, t) = 0, \quad x \in \mathbb{R} \text{ and } t \geq 0 \tag{1}
$$
...
