Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

learn more… | top users | synonyms (2)

0
votes
0answers
89 views

Discrete Harmonic Functions

Let $\Omega$ be a Lipschitz domain, $\Gamma\subset\partial\Omega$ with Lebesgue measure $>0$. Let $u_\Gamma\in V^h|_\Gamma$ be the trace of a finite element function. Then the following holds: $$ ...
1
vote
1answer
73 views

Homework: Solve the poisson equation in the outer sphere

Our teacher asked us to solve the poisson equation: \begin{eqnarray}\left\{\begin{array}{ccc}\Delta u &= &0 \\ u|_{\partial \overline{B(0,R)}} & = & g \\ \lim_{|\vec{x}|\to\infty} ...
0
votes
1answer
39 views

An estimate of the heat kernel on a bounded doain

I'm interesting to PDE, and I'm asking if the heat kernel $p_{D}(t,x,y)$ with Dirichlet boundary conditions on $[0,1]^{d}$, where $d\geq 1$ satify $\int_{0}^{t}\int_{D}p_{D}(s,x,y)dyds =c_{0} > 0$ ...
0
votes
1answer
47 views

Different definitions of $e^{t\Delta}$

All function spaces are over $\mathbb{R}^n$. We have 3 different ways to define $e^{t\Delta}$: (1) For $f\in H^2$, we have $\widehat{\Delta f} (\xi)=-4\pi^2|\xi|^2\hat{f}(\xi)$. Define ...
1
vote
2answers
101 views

Weak convergence in $H^1_0(U)$ implies convergence in $L^2(U)$

Recall some definitions : $ H^1_0(U) =W_0^{1,2}(U)$ and $u\in W^{k,p}_0(U)$ i there exists $u_m\in C_c^\infty(U)$ s.t. $u_m\rightarrow u$ in $W^{k,p}(U)$ question For $1\leq p \leq \infty$, ...
0
votes
0answers
78 views

Integration by part on weak derivative

As you know if $f,\ g\in C^1(\overline{U})$, where $U$ is bounded in ${\bf R}^n$ and if $f,\ g$ have compact support in $U$, then $$ \int_{{\bf R}^n} f(x)_{x_i}g(x)\ dx =-\int_{{\bf R}^n} f(x) ...
1
vote
1answer
105 views

Laplace boundary value problem

I came across the following Laplace bvp: $u_{xx}+u_{yy}=0\ $ for $\ 0<x<1,\ 0<y<2$ $u(x,0)=u(x,2)=0$ $u(0,y)=0$ $u(1,y)=y(2-y)$ I didn't have any problems solving it. It was quite ...
2
votes
2answers
113 views

closed form solution to the heat equation

Let smooth functions $f(x) , g(t)$ are given solve the heat equation on the semi infinite domain $(a,\infty) \times (0,T)$. for simplicity, we can let $a = 0$. \begin{eqnarray} &&u_t(x,t) = ...
1
vote
2answers
252 views

Change of Coordinate in Differential Equation

I'm sorry, it's probably a very simple question but I'm confused between change of variable and change of coordinate in a differential equation. To take a very simple example, let's start with this ...
9
votes
2answers
217 views

Wave-Particle Duality in PDE?

I am reading Arnold's Lectures on Partial Differential Equations. It is definitely a good book, yet sometimes I am a little bit confused. One theme of the first chapter seems to be From the ...
1
vote
1answer
116 views

I don't understand this PDE solution involving Fourier coefficients and orthonormal eigenfunctions.

$u_{xx} + u_{yy} = 0$ with $x \in (0,\pi)$ and $y \in (0, \pi)$ Initial Conditions: $$ u(x,0) = x^2 $$ $$ u(x,\pi) = 0 $$ Boundary conditions: $$ u_{x}(0,y) = 0 = u_{x}(\pi, y) $$ I performed ...
0
votes
1answer
60 views

Solve PDE by seperation of variables?

I'm trying to solve the following PDE by separation of variables: $\nabla^2 T=0$ $T(0,y)=300 \\ T(4,y)=600 \\ \frac{\partial T}{\partial y}(x,0)=0 \\ \frac{\partial T}{\partial y}(x,2)=0$ $0\leq x ...
1
vote
2answers
105 views

characterization of uniform ellipticity

Let $B$ be a $n\times n$ matrix over $\mathbb{R}$ and define $A:=BB^*$. I read in a paper that the following two statements are equivalent: (1) the matrix $A$ is uniformly elliptic; i.e. for all ...
2
votes
0answers
71 views

transformation of variables in a fokker planck PDE

I am trying to solve the following Fokker Planck PDE: $$ \dfrac{\partial u(t,x)}{\partial t} = -\dfrac{\partial u(t,x)}{\partial x} + \dfrac{1}{2}\dfrac{\partial^2 }{\partial x^2}[ 3 x^2 u(t,x)]. $$ ...
0
votes
1answer
49 views

Why is $\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\cos\left[ a\xi \right]\hat{f}(\xi)d \xi = f(a)$?

Background: We are looking at the wave equation on $\mathbb{R}^n$ via the Fourier transform. If $u(x,t)$ solves $\Delta u = u_{tt}$ in $\mathbb{R}^n$, with $u(x,t) = f(x)$ at $t=0$ and $u_t(x,t) = ...
1
vote
1answer
53 views

Series of complex Fourier coefficients.

I've been trying to figure this out for days now, but I have no idea how to show this. It's from Partial Differential Equations: An Introduction by Walter A. Strauss. Suppose $\int_{-\pi}^{\pi} [ ...
0
votes
1answer
134 views

Deduce 1d wave equation from 2d by method of descent

Suppose we take as given that the formula for the wave equation on $\mathbb{R}^2$ is $$u(x,t) = \frac{1}{2\pi} \int_{|y-x|<t} \frac{g(y)}{\sqrt{t^2-|y-x|^2}}dy+ \frac{d}{dt}\left\{ ...
0
votes
1answer
96 views

How to obtain $\frac{\partial C}{\partial \sigma}$ from Black-Scholes PDE?

The famous Black-Scholes PDE is: $ \frac{\partial C}{\partial t} + rS\frac{\partial C}{\partial S} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 C}{\partial S^2}=rC$ Then, how should I find another PDE ...
3
votes
0answers
278 views

What is compensated compactness?

As the title says, what is compensated compactness? I see people talk about it in the books and papers I am reading but I can only find hand wavy definitions when I look online. Is there a definition ...
2
votes
0answers
91 views

Parabolic PDEs Maximum Principle

Consider diffusion equation for t>0 and $ \boldsymbol{x} $ in a bounded doman $ D$ in $\mathbb{R}^n$, and a given scalar field $a(\boldsymbol{x}) >0$ that is uniformly bounded and continuously ...
2
votes
1answer
78 views

Differential forms on the torus correspond to periodic forms on $\Bbb{R}^n$?

Let $T^n=\Bbb{R^n/Z^n}$ be the torus. Is it possible say that forms on the torus bijectively correspond to forms on $\Bbb{R}^n$ invariant under translations by integers?
0
votes
1answer
228 views

Solving a quasi-linear PDE using the method of characteristics

How does one solve the following equation by the method of characteristics? $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=-u^2~\\ $$ with initial data $~u(x,0)=f(x)=x$ What I have ...
0
votes
1answer
56 views

About convergence in Sobolev space

I have $u_{n}$ sequence of $H^{1}_{0}(\Omega)$ where $\Omega$ is open bounded and connected domain in $\mathbb{R}^{n}$ with $n>1$. $u_{n}\rightarrow u$ in $H^{1}_{0}(\Omega)$ norm. Let ...
1
vote
1answer
81 views

PDE using Fourier Transform

Using the Fourier Transform, solve: $u_t=u_{xx}+\alpha u$ with $\alpha>0$, for $x \in \mathbb{R}, t>0$ with initial data $u(x,0)=f(x)$, with $f$ continuous in $\mathbb{R}$ Apllying Fourier ...
1
vote
1answer
200 views

Verifying a solution from Strauss' “Partial Differential Equations - An Introduction”, 2nd edition

In section 9.2 on page 241, question #12 is given as follows: "Solve the three-dimensional wave equation in $\{r\ne0,t>0\}$ with zero initial conditions and with the limiting condition ...
0
votes
1answer
94 views

Navier-Stokes equation apriori estimate

Assume a Navier-Stokes system of incompressible fluid flow. Let the velocity vector has components $\vec{\mathbf{v}} = (u, v, w)$. Are there any ways to obtain apriori estimate of the form $\| w_x \| ...
1
vote
0answers
21 views

PDE oscillation problem

How do I make a start to this question? I am unsure how the given system of equations relate to the pde. Once I know that it would be a trivial to find the eigenvalues and use the given criteria. ...
1
vote
0answers
130 views

Solution to heat equation, differentiation under integral sign

For $f\in L^1(\mathbb{R}^n)$ a solution to the heat equation $\frac{1}{2}\Delta u=\frac{\partial}{\partial t} u$ is given by $$u(x,t)=(2\pi t)^{-n/2} \int\exp{\left(-\frac{\lVert ...
1
vote
1answer
43 views

Separating Partial Differential Eq

I have a PDE: $$ \frac{\partial^2\phi(r,\theta)}{\partial r^2} + \frac{1}{r}\frac{\partial\phi(r,\theta)}{\partial r} + \frac{1}{r^2}\frac{\partial^2\phi(r,\theta)}{\partial\theta^2} + ...
2
votes
1answer
45 views

Show $\int_{\mathbb{R}^n}\Delta_x \Phi(x-y)f(y)dy = \int_{\mathbb{R}^n}\Delta_y \Phi(x-y)f(y)dy.$

I read in an article about Laplace's equation that $$-\int_{\mathbb{R}^n}\Delta_x \Phi(x-y)f(y)dy = -\int_{\mathbb{R}^n}\Delta_y \Phi(x-y)f(y)dy.$$ Could someone explain to me why this is? I ...
5
votes
2answers
277 views

Analytic solutions to a nonlinear second order PDE

I am trying to solve the following 2nd order PDE analytically, but haven't succeeded so far. I have tried to separate the variables, but it doesn't work here. I would be very grateful for any ...
1
vote
0answers
39 views

Estimate starting with variational formula

I'm working on an a priori estimate, using equality's like Young, Cauchy,... But I'm stuck with my testfunction. I've got the following problem: $\frac{\partial u}{\partial t} - \Delta u + \int_\Omega ...
2
votes
1answer
109 views

Composition of Differential Operators

If I have: $A=\partial_x^2+u(x)$ $B=u(x)\partial_x$ How do I compose: $AB$ and $BA$?
2
votes
1answer
39 views

Elliptic approximation of a $L^2$ function

Let $\Omega$ an open and bounded set of $\mathbb{R}^N$. Consider $f\in L^2(\Omega)$, and $\varepsilon >0$. Let us call $u_\varepsilon$ the unique weak solution in $H^1_0(\Omega)$ of the problem ...
1
vote
0answers
148 views

Fundamental solution of wave equation in 3D

I want to ask for assistance in verifying the fundamental solution of the wave equation in $\mathbb{R}^{3}$. Here the fundamental solution is given by $$\frac{1}{2\pi}H(t)\delta(t^{2}-|x|^{2})$$which ...
2
votes
0answers
36 views

Change of variables in PDE's

How the change of variables in PDE effects the initial and boundary condition? will the condition be accordingly changed with the substitution use for change of variables? any example will be ...
1
vote
1answer
229 views

uniqueness of the solution of the beam equation

Does anyone know how to prove there exists at most one smooth solution $ u $ of the following problem for the beam equation? $ u_{tt} + u_{xxxx} = 0 $ in $ (0,1) \times (0, T) $ $ u(0,t) = u(1,t) = ...
1
vote
2answers
107 views

Partial Differential Equation $u_t+u_x=\cos(c-t)$

Given $u_t+u_x=\cos(c-t)$ where $u(x,0)=\dfrac{1}{1+x^2}$ . Find the solution $u(x,t)$ using characteristic method. I have found $\dfrac{dt}{ds}=1$ and $t(0)=0\implies t=s$ $\dfrac{dx}{ds}=1$ and ...
3
votes
0answers
44 views

Finite Difference Methods for arbitrary elliptic PDE

I am looking for textbook references that describe lattice numerical methods for arbitrary elliptic PDEs, particularly finite difference schemes and particularly in 2d. The few references that I have ...
1
vote
0answers
40 views

Estimative in Hs spaces

Consider $W(t)$ defined by $\widehat{W(t)} = (2 \pi)^{\frac{-n}{2}} \cos (t|\xi|)$. Consider $g \in H^{s-1} (R^{n-1})$. Show that: $$ \| W(t ) *g\|_{H^s} \leq c \sqrt2 (1 + |t|)\| g\|_{H^{s-1}}$$ ...
1
vote
0answers
21 views

Largest classes of singularities

As seen, for instance, in "Survey on Singularities and Differential Algebras of Generalized Functions : A Basic Dichotomic Sheaf Theoretic Singularity Test", ...
3
votes
0answers
27 views

Convergence of the series $\sum_{\xi\in\mathbb Z^n} e^{2\pi ix\cdot \xi} a(x, \xi)\hat{f}(\xi)$?

I need some help with the following problem: let $a:\mathbb R^n\times \mathbb R^n\rightarrow \mathbb C$ be a smooth function and suppose there are constantes $C_{\alpha, \beta}$ and $M(\alpha, \beta)$ ...
2
votes
1answer
103 views

Are there any other materials on Hamilton-Jacobi equation besides the book by Evans?

I've been recently reading the book on PDE by Evans,and I had a hard time understanding the part about Hamilton-Jacobi equation in chapter 3.I wonder if there are any other materials on this ...
0
votes
2answers
99 views

Boundary value problem without separation of variables

It is well-known that the separation of variables method is useful for boundary value PDE problems. For example, it is usual to use separation of method for this problem $$V:\mathbb{R}^2\to ...
2
votes
0answers
58 views

The help for Max Principle for the heat equation

Give a direct proof that if U is bounded and $ u \in C_{1}^2(U_{T}) \cap C(\overline U_{T})$ solves heat equation, then $\max_{\overline U_{T}} u$= $\max_{\tau_{T}}u$ I appreciate it Thanks
2
votes
1answer
262 views

How to properly prepare for a graduate level PDE course using the books by Evans and Strauss

For my undergrad background, I have Calculus 1-3, Linear Algebra, one semester of ODE, one semester of real analysis. Never had any PDE before. Thus I know this background is hardly enough to do well ...
1
vote
0answers
41 views

Canonical form of the PDE $u_{xx}+2u_{xy}+2u_{yz}+u_{zz}=0$

Find the canonical form of the PDE: $$u_{xx}+2u_{xy}+2u_{yz}+u_{zz}=0$$ I know how to do that in the "normal" way: finding the the actual transformation using the eigenvectors of the appropriate ...
2
votes
1answer
44 views

estimate on $| \nabla (u |u|^2) - \nabla(w|w|^2)|$ for $u,w \in H^1$

suppose $u, w \in H^1 (R^2)$. I'd like to know where does the following inequality come from (it appears in a proof I've been reading and I can't figure it out) $$ | \nabla (u |u|^2) - \nabla(w|w|^2)| ...
0
votes
2answers
85 views

What is the closure of $ C^\infty_c(\mathbb{R}^n\setminus\{0\})$ in Sobolev $ W^{1,p} $ norm?

For $1 \leq p < \infty, n\geq 1 $ my guess of the answer was $ W^{1,p}(\mathbb{R}^n)$ but I can't prove the inclusion $ \overline{C^\infty_c(\mathbb{R}^n\setminus\{0\})} \subseteq ...
6
votes
4answers
664 views

Why separation of variables works in PDEs?

The method of separation of variables is used in many occasions in the upper level physics courses such as QM and EM. But when it is used there is no clear reason why using it is permitted it except ...