Questions tagged [partial-differential-equations]
Questions on partial (as opposed to ordinary) differential equations - equations involving partial derivatives of one or more dependent variables with respect to more than one independent variables.
22,072
questions
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How to perform integration by parts on a term containing the gradient of a gradient?
I have $u\in\mathbb{R}^3$ and the term $\epsilon = (\nabla u)^\top + \nabla u$.
Since $\nabla u = \begin{pmatrix}
\frac{\partial u_1}{\partial x} & \frac{\partial u_1}{\partial y} &...
- 101
2
votes
1
answer
31
views
Burgers' PDE with a given solution (Traffic flow)
Burgers' equation:
$u_t+u u_x=0$
This solution $u(t,x)=c(1-\frac{2 \rho(t,x)}{\rho_0})$ should be verified. $c$ is the maximal velocity and $\rho_0$ is the maximal vehicle density, by which the ...
- 64k
6
votes
1
answer
79
views
How to show the continuity of the Laplacian of the heat kernel (in t)
Let $K_t(x):=(4\pi t)^{-\frac{d}{2}}e^{-\frac{x^2}{4t}}$ be the heat kernel for $t > 0$. How can I show that the function $t \mapsto (\Delta K_t)*f$ is continuous in $L^p(\mathbb{R}^d)$ (...
- 4,654
-2
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0
answers
18
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convolution with mollifier
It is said that $u_{\epsilon\epsilon'}=u_{\epsilon'\epsilon}$. I can't figure out how this is true.
$u_{\epsilon\epsilon'}=\int_{U_\epsilon}\eta_{\epsilon'}(x-y)u_\epsilon(y)dy$, where $\eta_\epsilon$ ...
- 1
2
votes
0
answers
14
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ODE Solver Implementation for f(t, x, u)
Recently I've been learning about solving DEs numerically. In order to understand various methods, I've implemented some of them in Julia. So far I have implementations to solve a system of DEs of the ...
- 21
0
votes
0
answers
20
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Maximum Principle for Poisson’s Equation
I understand how to prove the maximum principle for $u_{xx}+u_{yy}=0$, but how does this extend to a maximum principle for the equation $u_{xx}+u_{yy}=f$?
I believe this is called Poisson’s equation. ...
- 415
1
vote
1
answer
39
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Why does the foliation $\mathcal{F}$ of this Lorentzian manifold also solve the backwards heat equation?
Consider a linear parabolic partial differential equation:
$$t \partial_{tt}\varphi_t(x)=\pm x\partial_x \varphi_t(x)$$
which (essentially) takes the form of the backwards heat equation (minus sign) ...
- 350
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votes
0
answers
50
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Existence of a continuous function whose Fourier series diverges at a point
Given the following function
\begin{align*}
g: \mathbb{R}&\rightarrow\mathbb{C}\\
x&\mapsto \begin{cases}
x &\mbox{, $-\pi<x<\pi$}\\
0&\mbox{, $x=-\pi$}\\
g(...
2
votes
1
answer
41
views
If $f$ is a eigenfunction of $-\Delta$ in $L^2[0,1]$, is it necessarily $C^\infty$?
I am a little bit confused about the properties of the Laplacian $-\Delta$ on $L^2[0,1]$ with the periodic boundary conditions.
At least I know that $-\Delta$ is an unbounded self-adjoint operator on $...
- 2,062
1
vote
1
answer
27
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Understanding estimation for $\Vert (1-\Pi)F\Vert$
I am referring to this paper, page 6.
Can somebody explain to me, why line (16),
$\Vert A F\Vert\leq\frac{1}{2}\Vert (1-\Pi)F\Vert$
implies that
$$
\vert\langle A(F-F_\infty),F-F_\infty\rangle\vert\...
- 10.3k
0
votes
0
answers
17
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PDE with additive seperation
I have a problem with the following PDE:
$x^2u_x+\frac{1}{y}u_y+\alpha u=0$, for $x,y>0$ and $\alpha \in \mathbb{R}$
With additive seperation:
$u(x,y)=X(x)+Y(y)$
Substituting in the PDE yields:
$x^...
- 25
-2
votes
0
answers
29
views
How i can prove this set is closed?
be $\Omega\subset\mathbb{R}^{N}$, $u\in C^{1}(\overline{\Omega})$, the set is the lower contact set of $u$:
$S(u)$ = {$x\in\Omega;\hbox{ }u(y)\ge u(x)+\nabla u(x)(y-x)\hbox{ }\forall y\in\Omega$}
2
votes
1
answer
36
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Ways to show that the solution to the heat equation is $C^{\infty}((0,\infty) \times \mathbb{R})$
We say that $u: [0,T) \times [-\pi, \pi] \rightarrow \mathbb{C}$ solves the heat equation with initial condition $u_0: [-\pi, \pi] \rightarrow \mathbb{C} $ if:
$$
\begin{cases}
\partial_t u(t,x) = \...
- 45.6k
0
votes
1
answer
22
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Coefficients of the solution to the 2D heat equation with Neumann boundary conditions
I've solved the heat equation with Neumann boundary conditions on the rectangle $0 < x < a$ and $0 < y < b$, with initial condition $u(x,y,0)=f(x,y)$. The equation to solve was $$u_t = D^2 ...
- 29.9k
1
vote
1
answer
44
views
Weak solution is strong solution
I came across the following statement: If $V=\{v \in C^1[0,1])~ |~ v(0)=0\}, ~~f \in C([0,1])$ and $u \in C^2([0,1])$ then any solution to
\begin{equation}
u(0)=0 ~~\text{and}~~ \text{for all}~~ v \...
- 13.3k
1
vote
0
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28
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Why does the limit of the average of $L^\infty(\Omega\times(0,T))$ function over $\Omega\times [t,t+\epsilon]$ when $\epsilon \rightarrow 0$ exist?
In these lecture notes on PDEs
http://www.sbai.uniroma1.it/pubblicazioni/doc/phd_quaderni/02-1-and.pdf (p. 26, sec 4.1)
it is claimed that
$$\lim\limits_{\epsilon \rightarrow 0} \frac{1}{\epsilon} \...
- 11
2
votes
0
answers
56
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The Leibniz integral rule
I've had problems applying the Leibniz Rule.
The professor's slides say:
Consider the following value function
\begin{equation}
\nu(t) = \int_{t}^{\infty} e^{- \int_{t}^{s}r(z)dz} \pi(s) ds
\end{...
- 30.6k
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0
answers
26
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How to find the eigenfunctions of this PDE?
Consider the differential operator $$ \mathcal{L} = -A \partial_x f_1(x,y) - B \partial_y f_2(x,y) + \frac{A}{2} \partial_x^2 f_1(x,y) + \frac{B}{2} \partial_y^2 f_2(x,y)$$ where $f_1 = (x+1)(M-x-y)$, ...
- 123
0
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0
answers
46
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A lower bound for total variation in multivariable functions
Let $v$ be a function of bounded variation with the total variation defined as
$$
|Dv|(\Omega) =\sup\left\{- \int_\Omega v\, {\rm div}\, \phi\; dx:\, \phi \in C^\infty_0(\Omega, \mathbb{R}^n),\, |\phi|...
2
votes
0
answers
32
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Computer algebra system for applying Cartan's test to systems of PDEs
It is my understanding that if one has a (possibly overdetermined) system of PDEs, one can check for compatibility by applying Cartan's test (see for example [1], Chapter 7). It involves first writing ...
3
votes
1
answer
155
views
+100
Weak equality equation with a negative fractional Sobolev space.
Sobolev Preamble
For $N\geqslant 2$, let $\Omega \subset \mathbb{R}^N$ be a connected, bounded, open set with Lipschitz continuous boundary $\partial \Omega$.
Let the space $H^{1/2}(\partial \Omega)$ ...
- 19k
-1
votes
0
answers
29
views
Checking implicit solution for a PDE of order 1
Let a quasi-linear PDE of order 1 be:
$u_{t}+uu_{x}=0$
while $u=u(x,t)$.
Find an implicit solution and a solution as a parametrical surface for the condition:
$u(s,0)=f(s)$
I got confused because one ...
- 30.6k
3
votes
1
answer
1k
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Eigenvalue problem with Robin boundary conditions at both ends
From Walter A. Strauss's Partial Differential Equations:
Consider the eigenvalue problem with Robin boundary conditions at both ends:
$$ -X'' = \lambda X, \qquad X'(0) - a_0 X(0) = 0, \qquad X'(l) - ...
- 20.7k
0
votes
0
answers
28
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Notation Question: Function spaces in Lie groups
I'm trying to define a functional space of curves in the special Euclidean group SE(3), but the function depends on both space and time.
Given $h(x,t)$ where $x \in [0,L]$ and $t \in \mathbb{R}_{\geq ...
-4
votes
0
answers
30
views
Burgers' Equation with initial condition [closed]
Define the characteristics (or characteristic traces) of a first order PDE.
Draw the characteristics for the problem
Uy+ UUx = 0, |x| < ∞, y > 0 satisfying the initial condition
u(x, 0) = (
0, ...
- 1
0
votes
1
answer
36
views
Some Algebra In the Exponent of Inequalities arising in the Proof of Young's Inequality
I'm getting lost with a simple computation -- these are exponent manipulations in a holder-type situation:
First define convolution in this setting: let convolution be defined for real- or complex-...
- 3,369
7
votes
1
answer
134
views
Doubly periodic solutions to $\frac{|\partial w|^2}{(1+|w|^2)^2} = C^2$?
I am interested in solutions to $$\frac{|\partial_z w|^2}{(1+|w|^2)^2} = C^2$$ for $C>0$ a constant and $w = w(z,\bar z)$ is allowed to have poles. For example $\tan(|z|)$ is a solution. But I am ...
- 3,435
1
vote
0
answers
31
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Existence of weak solution to Poisson equation with nonlinear Neumann boundary conditions
This is problem 8.11 in Evans.
Let $U\subset\mathbb{R}^n$ be open and bounded, with smooth boundary. Assume $\beta:\mathbb{R}\to\mathbb{R}$ is smooth with $0<a\leq\beta'\leq b$ for constants $a,b$....
- 11
0
votes
0
answers
21
views
How to prove the uniqueness and stability of solution to Poisson equation using the maximum principle?
I came across this problem in an exam. Given a Poisson equation on a bounded region $\Omega$,
$$\begin{cases}
-\Delta u=f& u\in\Omega\\
\alpha u+\beta\dfrac{\partial u}{\partial n}=g&u\in\...
- 319
-3
votes
0
answers
23
views
Simulación de EDPs mediante diferencias finitas [closed]
Enunciado: Encontrar soluciones numéricas de una EDP utilizando este procedimiento. Partís de una EDP cualquiera y obtenéis la ecuación en diferencias asociada. Con ésta, podéis calcular distintas ...
3
votes
1
answer
100
views
Feynman–Kac formula: conditional expectation vs. Wiener integral
The Feynman–Kac formula for the solution $u(t,x)$ of the one-dimensional heat equation
\begin{align*}
\partial_t u &= \frac{1}{2}\Delta_x u,\\
u(0,x) &= f(x)
\end{align*}
is given by
\...
0
votes
1
answer
56
views
On the existence of Green's functions by Peter Lax
I'm reading Peter Lax's paper On the existence of Green's functions. He showed that the regular part of Green's functions is continuous on the boundary. My question is : to have the normal derivative ...
- 925
0
votes
0
answers
65
views
Subleading terms in Weyl's Law
The two term Weyl's Law states that
$$N(\lambda)\sim\frac{area(\Omega)}{4\pi}\lambda-\frac{perimeter(\partial\Omega)}{4\pi}\sqrt\lambda$$
where $\Omega$ is a bounded domain in $R^2$, and $N(\lambda)$ ...
- 83
3
votes
2
answers
296
views
Characteristics of equations of the form $u_{xy}=f(u_x,u_y,u)$
In the usual treatment of hyperbolic differential equations, it is always assumed that there are two families of characteristics. That is, if the equation $L[u]-f(u_x,u_y,u)=au_{xx}+2bu_{xy}+cu_{yy}-f(...
- 8,903
0
votes
0
answers
32
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Stability of time stepping in PDE
I am currently trying to perform a stability assessment on solving a parabolic PDE of the form
$$
\frac{\partial u}{\partial t} = g(x,t) \frac{\partial^2 u}{\partial x^2} + u
$$
I am using Crank-...
- 1
1
vote
1
answer
441
views
Using a change of variable to reduce a linear 2nd-order PDE to a 1st-order PDE
I created the following example, but am unsure if my final answer is correct or in the nicest possible form. (The examples in my textbook don't involve the mixed partial, and none have the integral ...
- 36k
0
votes
0
answers
31
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Has it been proven that the Cheeger constant is attainable on surface?
Recently, I have been studying the monotonicity of the Cheeger constant under Ricci flow on surfaces. In fact, I want to use the monotonicity to prove the convergence of Ricci flow on $S^2$, which ...
- 5,495
2
votes
0
answers
27
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Using Crank-Nicolson to solve the diffusion equation with variable diffusivity and flux boundary condition
I'm trying to solve the diffusion equation with variable diffusivity and a zero-flux boundary condition at one boundary, and a flux that is proportional to concentration at the other boundary. The ...
- 173
0
votes
1
answer
47
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Comparison principle for p-harmonic functions.
The proof begins by considering the open set $D_{\varepsilon} = \{ x \in \Omega : u(x) > v(x) + \varepsilon \}$ for some $\varepsilon > 0. $ Then it says that $D_{\varepsilon} = \emptyset$ or $...
- 101k
-1
votes
2
answers
760
views
Complete integral of PDE $x^2p^2+y^2q^2-4=0$
I have the following non-linear first order PDE before me :
$x^2p^2+y^2q^2-4=0$
I have to find two complete integrals for this PDE.
I wrote the Charpit equations as below:
$\dfrac{dp}{2p^2x}=\dfrac{...
- 1
-4
votes
0
answers
48
views
PDE Parallelogram rule proving that u(x,t) is a solution for a homogenous wave function [closed]
I need to prove that if u(x,t) holds the parallelogram rule in all of R², then it is a solution to the homogeneous wave function. I have tried several ways to prove this but have encountered problems ...
- 33.8k
0
votes
1
answer
47
views
Are eigenvalues of elliptic PDE real when $u$ is real valued function and all coefficients are real?
I asked this to my professor and am more confused than before.
Let's say we have $(L-\lambda)u=0$. In order for $u$ to be nontrivial real valued function, shouldn't $\lambda$ necessarily be real value?...
- 3,142
0
votes
0
answers
25
views
Reversal of Variable Transformation in the Solution of Inhomogeneous Dirichlet PDE
Initially, we are dealing with a non-homogeneous Dirichlet problem:
\begin{equation}
\begin{cases}
q_t(z,t)-D(z,t)q_{zz}(z,t) = 0 & \text{for $0<z<L,t>0$} \\
q(0,t)=a &...
- 1
0
votes
0
answers
24
views
Analytical solution of nonlinear Advection Diffusion equation
Is there any analytical solution to the nonlinear Advection Diffusion equation of the following type?
$$\frac{\partial f}{\partial t}= \frac{\partial}{\partial x}\left(f^{\alpha}f+f^{\beta}\frac{\...
- 121
-2
votes
0
answers
34
views
How would you solve this PDE problem, I am running into all sorts of integration issues. I am aware y=e^s, but thats as far as I got. [closed]
Image of the problem I wish to solve, any help massively appreciated!
-1
votes
0
answers
20
views
Suppose $0\leq A\leq B$ and $B$ has constant density $\rho_B<\infty$. Then $0\leq \rho_A(x)\leq\rho_B$ for almost every $x$. [closed]
Let $A,B$ on $L^2(\mathbb R^d,\mathbb C^d)$ such that $0\leq A\leq B$. Suppose $\rho_B(x)=K_B(x,x)$ where $K_B=K_B(x,y)$ is the integral kernel of $B$ exists and is constant, $\rho_B(x)=\rho_B>0$ ...
- 235
2
votes
0
answers
40
views
Does the Laplacian commute with the Heat semigroup?
Let $f \in L^p(D)$, with $1 \leq p\leq \infty$. We say that $f$ admit a weak $L^p$-Laplacian if there exists a function $g\in L^p(D)$ such that for all functions $\phi \in C_c^{\infty}(D)$ we have $$\...
- 81
0
votes
1
answer
25
views
Parameterized ODE: If the data is $C^k$, then so is the solution.
I want to prove the following "regularity".
Assume $f \in C^k(\mathbb R^3)$ and $g \in C^k(\mathbb R)$. Consider the parameterized ODE
$$\begin{cases}\displaystyle
\frac \partial {\partial ...
- 120k
0
votes
1
answer
43
views
Pure Birth Process. Finding the probability we are state n after time t, in general (David Kendall 1949)
Hi, I'm stuck understanding a paper by David G Kendall (Stochastic Processes and Population Growth, 1949). The paper demonstrates how to derive $p_n(t) = \mathbb{P}(X_t = n)$ for a pure birth process ...
- 1,589
0
votes
0
answers
32
views
Bounding solution to elliptic problem with inhomogeneous Neumann boundary condition
I am having a hard time understanding the passage at the end of page 478 and the beginning of page 479 from Blowing up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity....
- 437