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

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Unicity of solution for a parabolic problem?

How can I show that the parabolic problem $$ \begin{cases} \partial_xu- \Delta u=0 & \mathbb{R} \times (0,+ \infty)\\ u(x,0)=f(x) & \mathbb{R} \end{cases} $$ has a unique solution? Can ...
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39 views

How to “uniformly” mollify a sequence without knowing the higher derivative.

The motivation and previous discussion of some related ideas can be seen here. My question: given a sequence $u_n\in u_0+H^1_0(\Omega)$ where $u_0\in H^1(\Omega)$ and $\Omega\subset \mathbb R^N$ is ...
1
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1answer
30 views

Evans PDE, Problem 8 Chapter 2 clarification on $|x-y|$

Hi I am attempting problem 8 (Chapt2 Evans PDE). Again I found the solution on the internet. enter link description here I understood much of everything of the proof except for one line. " Since ...
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+50

Integrating an infinite sum (statement from paper)

Let us work on a bounded domain $\Omega$ with Neumann BCs, with $\varphi_k$ and $\lambda_k$ being the orthonormalised eigenvectors and eigenvalues of the Neumann Laplacian. Define $$v(x,y) = \sum_{k ...
2
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1answer
57 views

How do you read a partial differential equation?

In calculus we can read the "normal derivative", $\frac {df}{dx}$, as the rate of change of our function $f$ with respect to $x$. With partial derivatives of multivariate functions it is very much the ...
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Find the general solution of $u_{ttxx}(x,t)=(u_{tt}(t,x))^2$

Find the general solution of the equation $$u_{ttxx}(x,t)=(u_{tt}(t,x))^2$$ Let set $v(x,t)=u_{tt}(x,t)$. Then $$v_{xx}(x,t)=(v(x,t))^2$$ What should I do next? Any help would be greatly appreciated.
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1answer
44 views

analytic solution poisson equation spherical coordinates

I'm trying to analitically solve a poisson equation $\nabla^{2}p(r,\theta,\phi)=f(r,\theta)$ in spherical coordinates; the boundary condition is $p(\infty,\theta,\phi)$=0. I'm quite used to the ...
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0answers
21 views

If $\partial\Omega\in C^{2+\alpha}$ and $-\Delta\Theta=f\text{ in }\Omega$ with $f\in C_0^\infty(\Omega)$, then $\Theta\in C^{2+\alpha}$

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain with $\partial\Omega\in C^{2+\alpha}$ for some $\alpha>0$ $f\in C_0^\infty(\Omega)$ $\Theta\in C^0(\overline{\Omega})\cap C^2(\Omega)$ be the ...
0
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1answer
30 views

Example of an $H^{-1}$ function that isn't $L^2$

I'm going back over some PDE and Sobolev space theory, and the following is puzzling to me. Consider a nice domain $\Omega$ and the space $H^1_0(\Omega)$ of functions with $L^2$ first derivatives, ...
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1answer
24 views

$u=\sum (u,\varphi_k)_{L^2}\varphi_k$ vs $u(x)=\sum (u,\varphi_k)_{L^2}\varphi_k(x)$ and related questions

I have some doubts after reading several threads. Let us work on a bounded domain $\Omega$ with Neumann BCs, with $\varphi_k$ and $\lambda_k$ being the orthonormalised eigenvectors and eigenvalues of ...
1
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1answer
26 views

Meaning of “$\triangledown$u*ñ=0 on the boundary”

I'm doing homework for my PDE class, I'm coming across this notation and I don't what the ñ means: $\triangledown$u* ñ=0. I have tried to google it, but unfortunately questions like this don't really ...
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0answers
27 views

Minimal boundary conditions for divergence theorem

I've noticed that some domain conditions of questions here were only supposed to be finite dimensional and bounded. And then the divergence theorem was applied in the answers. But if I'm not mistaken, ...
0
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1answer
33 views

how to calculate integral of product of exponential function and trigonometry function?

Let $x_0$ and $\sigma$ are constants. How to calculate this? $$ \int^{L}_{-L}e^{-\frac{(x-x_0)^2}{2\sigma^2}}\cos x dx $$ I think i can solve that with integration by parts. But I'm confused how to ...
2
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1answer
27 views

Why can $D^ku(x) := \{D^\alpha u(x) \mid |\alpha| = k \}$ be regarded as a point in $\mathbb{R}^{n^k}$?

This is a comment made in the Appendix of Evans's Partial Differential Equations. He defines the set of all $k$ order partial derivatives as $D^ku(x):= \{D^\alpha u(x) \mid |\alpha| = k \}$ (using ...
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2answers
32 views

What do authors usually mean from a geometrical interpretation when “radially symmetric” is mentioned?

I was reading up on a text on an investigated case on heat flow in an annulus. The author mentioned "radially symmetric" and consequently proceeded to deduce that the variable 'theta' can be done away ...
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28 views

How to justify that the following system of PDE only admits linear solutions?

I have the following system of partial differential equations: $$ \frac{\partial\Lambda^{\mu}}{\partial x^{\nu}}= $$ $$ \small \begin{pmatrix} ...
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1answer
23 views

Absolute Continuity defined by Necas

I read a definition of the absolute continuity in Necas' book "Direct Methods in the Theory of Elliptic Equations": Let $\Omega$ be a domain in $\mathbb{R}^n$ , $P$ a line verifying ...
2
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1answer
30 views

Eigenvalue of Laplacian with Robin boundary condition

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial \Omega$ and let $\nu$ denote the outer unit normal. Let $u$ be an eigenfunction of $-\Delta$ in $\Omega$ satisfying ...
2
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1answer
45 views

Conditions under which a function vanishing on the boundary belongs to $H_0^1$

Let $\Omega \subseteq \mathbb{R}^n$ be a bounded open set ,and $u \in C(\overline{\Omega}) \cap C^1(\Omega) \cap H^1(\Omega) $ be a function such that $u \big|_{\partial \Omega}=0 $.Prove that $ u ...
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0answers
61 views

How to modify a $H^1$ weak convergence sequence so that I have the $L^2$ equi-integrability of gradient?

Assume $u_n\to u$ weakly in $H^1(\Omega)$ where $\Omega\subset \mathbb R^N$ is open bounded Lipschitz boundary. My goal is to find a new sequence $\bar u_n$ and a new function $\bar u$ such that ...
2
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1answer
20 views

Derive diffusion coefficient for heat equation from random walk simulation

I want to simulate the underlying stochastic process of diffusion on a microscopic level and compare the result with the solution of the heat equation. However, I'm not able to match the solution of ...
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1answer
23 views

Laplace equation in a circle with non-continuous Dirichlet boundary conditions

I have to solve: $$ \begin{cases} u_{rr}+\frac{1}{r}u_{r}+\frac{1}{r^2}u_{\theta \theta}=0 & [0,1) \times [-\pi,\pi] \\ u(1,\theta)=0 & (-\pi,0) & (1) \\ u(1,\theta)=1 & (0,\pi) ...
3
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1answer
37 views

If the weak derivative $\nabla u$ of $u\in L^2(\Omega)$ exists, then $\int_\Omega|\nabla u|^2=\int_\Omega|\nabla u^+|^2+\int_\Omega|\nabla u^-|^2$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded $u\in \mathcal{L}^2(\Omega)$ be weakly differentiable, i.e. $$\int_\Omega u\nabla\psi\;d\lambda^n=-\int\psi\nabla u\;d\lambda^n\;\;\;\text{for all ...
2
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1answer
41 views

How can I solve: $u_{xx} + u_{yy} = g(x,y)$ numerically?

If $u(x,y)$ is defined in $\mathcal{R} = {(x,y): 0 \leq x \leq a, 0 \leq y \leq b}$ $$ u(x,0) = 10 \mbox{ and } u(x,b) = 90, \mbox{ for } 0 \leq x \leq a \\ u(0,y) = 70 \mbox{ and } u(a,y) = ...
1
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1answer
57 views

A integral in Evans PDE chapter 2 problem 2 (straight calculus problem)

Here I am attempting problem 3 in Chapter 2 of the book PDE by Evans. Thankfully I have found the solution on the internet enter link description here However, there is one line confuses me I was ...
1
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1answer
40 views

Finding the eigenvalues and eigenfunction (tricky)

I'm given $$X"- vX' +X \lambda=0$$ (v is a constant) I have worked x' to be: X'(x) = $$\frac{1}{2} B v e^{\frac{v x}{2}} \sin \left(\frac{1}{2} x \sqrt{v^2-4 \beta ^2}\right)+\frac{1}{2} B ...
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Question on Littlewood-Paley trichotomy

In proving the product estimate, we need the Littlewood-Paley trichotomy. See http://www.math.ucla.edu/~tao/254a.1.01w/notes3.ps. In the decomposition $$P_k (fg)=\sum_{k',k''\in Z} P_k (P_{k'} f ...
1
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1answer
24 views

Is this a regular Sturm-liouville problem

Given $$C_{t} + VC{x}=C_{xx}$$ with $$0<x<L and t>0$$ with BC: $$C(0,t) = 0 , C_{x}(L,t)= 0 $$ Then the associated Eigenvalue problem is $$X" - vX'=- \lambda X$$ With BC $$X(0) = X'(L) ...
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0answers
16 views

Sources and sinks for parabolic PDE algorithm

I am given a very basic fortran program (View here) and asked 1st to investigate its accuracy and stability, for various values of $ \Delta t $ and lattice spacings. The program is an implementation ...
2
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1answer
48 views

Reference about Sobolev spaces

I'm looking some book from which I could learn more about Sobolev spaces. I'm interested rather in abstract theory: some topics which I would like understand in detail include: general construction ...
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21 views

Is it possible to construct a green function of the Dirichlet problem from the green function of the Cauchy problem?

For the heat equation. Is there a method to obtain the green function of the Dirichlet problem in a rectangular 2D domain from the green function of the Cauchy problem (infinite domain) PDE's?
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43 views

How to motivate those expansions?

I've been reading a paper where the author needs to solve the biharmonic equation on the plane. In truth, the function being saught is a function $v$ such that $v = \nabla \times U$ and $\nabla^4 U = ...
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14 views

$\nabla\phi_k\stackrel{L^2}{\to}\nabla\phi\Rightarrow\langle\nabla u,\nabla\phi_k\rangle\stackrel{L^2}{\to}\langle\nabla u,\nabla\phi\rangle$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded, $f\in L^2(\Omega)$ and $(\phi_k)_{k\in\mathbb{N}}\subseteq L^2(\Omega)$ with ...
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0answers
13 views

Checking boundary conditions for 3 dimensional wave equation

I'm studying pde's and right now I'm trying to understand the derivation of Kirchhoffs formula. In the derivation we use d'Alemberts formula for the reflected problem in the case where $0\leq r \leq ...
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1answer
39 views

my PDE solution not matching with text book solutio

A small question from text book; my solution does not match with text book solution. Please help. $$4 \frac {\partial u} {\partial x} + \frac {\partial u} {\partial y} = 3u$$ given $u(0,y) = \Bbb e ...
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0answers
50 views

Is $C^\infty_{0}(\bar{\Omega})$ dense in the hilbert space $W^{2,2}(\Omega)\cap W^{1,2}_{0}(\Omega)$

Assume $\Omega$ is open bounded domain in $\mathbb R^n$ Is $C^\infty_{0}(\bar{\Omega})$ dense in the hilbert space $W^{2,2}(\Omega)\cap W^{1,2}_{0}(\Omega)$ with inner product ...
0
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1answer
45 views

How to Invert the Euler Lagrange Equations?

Suppose I have a functional L. For example $L = y+3y'$. Where y is itself a function of real variable x It's easy for me to evaluate the Functional Derivative of L via the Euler Lagrange Equations: ...
0
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1answer
18 views

Using asymptotic form of Bessel Functions to find three unknown coefficients

I'm studying from Applied Partial Differential Equations (Haberman) and got stuck on the last question (c) shown below. How do I find the coefficients $a_1, a_2, b_1$ and $b_2$? Problem My Work ...
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19 views

Transfer a partial differential expression by substituting $u = xz$ and $v = yz$

Let $A = (\frac{\partial{z}}{\partial{x}})^2 + (\frac{\partial{z}}{\partial{y}})^2$. Transfer $A$ when considering $x$ as a function and $u = xz$ and $v = yz$ as independent variables. It is an ...
3
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1answer
142 views

Damped Wave Problem

I understand what the general solution is to a wave equation, but am unsure of the general solution for a damped wave equation. If someone knows what that is, or the steps to find it, that would be ...
2
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1answer
26 views

Problem understanding “formal proof” of Duhamel principle

I am now studying PDEs. My teacher referred to these notes. In particular, I'm having trouble understanding the proof on pages 20-21 of this part. This is a non-rigorous, formal proof of Duhamel's ...
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35 views

Transform the equation of $z(x,y)$ into an equation of $w(u.v)$

$$ z_{xx} - 2z_{xy} + z_{yy} = 0 $$ Transform the equation of $z(x,y)$ into an equation of $w(u.v)$, where $w = \frac{z}{x}$ and $$ u = x + y $$ $$ v = \frac{y}{x} $$ Is there any easy way to do ...
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54 views

resonance in pde

(I migrated the question from overflow b/c I think it belongs here instead.) About a year back I was in Professor Henry Mckean's office and he told me alot of interesting things. One thing he told me ...
2
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1answer
74 views

Determining the shock solutions to a PDE.

I'm confused by the question below. Particularly, sketching the base characteristics at the discontinuities in $u(x,0)$ and thus finding the shock solutions. Some advice would be appreciated. Problem ...
2
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1answer
40 views

Forced string problem with Forcing $\sin(\pi x) \cos(\pi t)$

the problem states to solve the forced string problem with $$\frac{\partial^2u}{\partial t^2}=\frac{\partial^2u}{\partial x^2} - \cos( \pi t)\sin( \pi x)$$ the boundary conditions for the string are ...
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0answers
43 views

asking a way to prove an inequality

Assume $\Omega$ is a bounded smooth domain in $\mathbb R^N $ with $N \ge 5 $ and $u \in C^2(\Omega)$ . I want to proof $$\int_{\Omega}\frac{|\nabla u|^2}{|x|^2}d{x} \;\ge\; ...
2
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1answer
30 views

A classical solution of Poisson's equation is also a weak solution

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain and $u\in C^0(\overline{\Omega})\cap C^2(\Omega)$ be a solution of $$\left\{\begin{matrix}-\Delta u&=&f&&\text{in }\Omega\\ ...
3
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1answer
51 views

Prelim problem $\sup_{\bar\Omega}|\nabla u|^2\leq C(\sup_{\Omega}u^2+\sup_{\Omega}f^2+\sup_{\Omega}|\nabla f|^2)+\sup_{\partial\Omega}|\nabla u|^2$

This is an old prelim problem. Let $\Omega \subset \mathbb{R}^n$ open and bounded with smooth boundary. If $u\in C^3(\bar\Omega)$ solves $$ -\sum_{i,j=1}^n a_{ij}(x)\,u_{x_ix_j}(x)=f(x)\quad ...
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0answers
7 views

Spectral collocation

I'm new to spectral methods, and I'm trying to solve an advection-diffusion equation in 1d: $\frac{\partial u}{\partial t}=A \frac{\partial u}{\partial x} + B \frac{\partial ^2u}{\partial x^2} $ I ...
1
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1answer
17 views

How to solve a simple differential equation in the way of weak solutions?

I want to prove that the solution equation $y'=y$ is $y=Ce^x$, where $C$ is a constant. Here $y$ belongs to the space of linear operators on $C_0^\infty(\mathbb{R})$, and $y'$ is its weak ...