# Tagged Questions

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

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### Kolmogorov backwards equation / stationary distribution

One can in the case of the Fokker-Planck / forward Kolmogorov equation, set the time derivative term to zero, and solve the remaining ODE to obtain the "forward-time" stationary distribution. Does ...
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### PDEs - traffic light, allocation of green to red light time

first time poster, so forgive me if I haven't done well with mathjax. doing some undergrad PDE traffic flow problems and we have a nasty. The set up is as follows. If you are looking at the correct ...
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### Sobolev Extension on boundary of Lipschitz domain

If $\Omega$ is Lipschitz and $f\in C^{0,\alpha}(\partial\Omega)$, then there exists $g\in C^{0,\alpha}(\Omega')$ such that $f=g$ on $\partial\Omega$, where $\Omega\subset\subset\Omega'$. Is there a ...
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### Show that $\lim_{|y|\to\infty}|y|^{n-2}u(y)>0$

For $n\ge 3$,Let $u\in C^2(R^n)$, $\Delta u\le 0, u>0$ in $R^n$ ,show that $\lim_{|y|\to\infty}|y|^{n-2}u(y)>0$. I consider the maximum principle,but I don't know how to deal with.
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### Can we write $\text{tr}[Q(x)^*\nabla^2u(x)Q(x)]$ for $u∈C^2(ℝ^d)$ and $Q:ℝ^d→\text{HS}(H,ℝ^d)$ in terms of a differential operator?

Let $H$ be a separable $\mathbb R$-Hilbert space $u\in C^2(\mathbb R^d)$ and $\nabla^2u(x)$ denote the Hessian of $u$ at $x\in\mathbb R^d$ $\operatorname{HS}(H,\mathbb R^d)$ denote the space of ...
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### How to tell if you have specified sufficient initial data for a differential equation?

I recently learnt that the following 'wave equation' is not well-posed $$\begin{cases} \partial_{tt}u=\partial_{xx} u, & (0,1)\times\mathbb R\\ u(t,0)=u(t,1)=0,&t\in [0,1] \end{cases}$$ ...
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### PDE basic traffic flow problem

I am analyzing a basic example of traffic flow presented here http://people.uncw.edu/hermanr/pde1/PDEbook/PDE_Main.pdf and have a question to the last transition in the traffic flow equation ...
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### Singularities in a PDE

This is more of a general question rather than anything specific but I was just wondering if someone could point me toward resources which discuss singularities in a PDE rather than in an ODE (by ...
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### Why is there no classification scheme for linear pde of third order or above

I am well aware of the classification of 2nd Order PDE into Elliptic, Parabolic or Hyperbolic type depending on the analogy with conic sections from analytic geometry. I have seen third order PDE ...
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### Please help: My MATLAB code for solving a 2D Schrödinger equation keep giving me weird output. [on hold]

I've been trying to solve the following Schrödinger equation numerically, -(\frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2})\Psi + \frac{\sinh^2(y) + \sinh^2(z)}{(\...
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### PDE: Classical Solution?

Part (a) is a standard problem which can be solved either using separation of variables or Fourier transform.I haven't seen a proper definition for 'classical solution' to answer (b). Any help is much ...
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### Heat Equation : Commutation of partial derivatives and summation

I'am having a problem when checking the validity of the solution i found for the heat equation: \begin{cases} U_{t}(x,t)=U_{xx}(x,t),\ {(x,t)\in (0,1)\times(0,+\infty)} \\ U(x,0) = x^2 - x\\U(0,t)=0\...
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### $-\Delta u = u^p$ in bounded domain

In my PDE lecture we had the following theorem and I am wondering how strong it is: Theorem Let $\Omega \subset \mathbb{R}^n$ be a bounded domain (with $\partial \Omega$ sufficiently smooth, lets ...
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### Why is uniqueness important for PDEs?

Every text on PDEs I come across will spend alot of time on showing the existence and uniqueness of solutions to a particular PDE. The importance of the existence of a solution to a PDE is obvious, ...
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### application of c*algebras to PDEs

I am preparing an introductory talk about c* algebras and I'd like to motivate C*algebras or show an application of them and I'd prefer an application in the field of partial differential equations. ...
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### How to derive a logarithmic potential from Newtonian?

Suppose we believe that the formula for Newtonian potential in $R^3$ is correct: $\varphi(\bar{x}) = \frac{1}{|x|} = \frac{1}{r}$, disregarding the constant. What is the justification of the fact ...
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### The set of all maximal ideals (Wiener Algebra)

I am trying to prove a proposition and in my proof I somehow need to find the set of all maximal ideals of a Banach Algebra. This is my working environment: Let $A(\mathbb{R}^2)$ be the (Wiener ...
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### Why this abuse of notation correctly solves the heat equation

Here's a stupid method I observed to solve the heat equation in $\mathbb R^d$, \begin{align*} \partial_tu=\Delta u,\quad u|_{t=0}=u_0. \end{align*} Pretend that $\Delta$ is a constant so this just ...
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### How to decide whether PDE is Homogeneous or non-homogeneous.

I am studying second order PDE. And I have seen homogeneous and non-homogeneous PDE. But I cannot decide which one is homogeneous or non-homogeneous. For examples; (1) $(D^3-3D^2D'+4D'^3)u=0$ ...
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### Separation of Variables and Linear PDEs

Separation of variables is a powerful method which comes to our help for finding a closed form solution for a linear partial differential equation (PDE). For example, we all know that how the method ...
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### PDE argument about passage to limit in Bochner space and weak derivative

I have a function $u \in L^2(0,T;H^1_0)$ with $u' \in L^2(0,T;H^{-1})$, all on some smooth bounded domain $\Omega$. In addition I have a sequence $u_n$ such that $u_n \to u$ in $L^2(0,T;H^1_0)$ and ...
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### Exercise involving sequence Palais-Smale and operator compact.

Let H be a Hilbert space and let K: H $\rightarrow{R}$ be $C^1$ and such that $\nabla K: H\rightarrow H$ sends bounded sets into precompact sets. Consider the functional $J: H\rightarrow R$ defined by ...
Given the Fokker-Planck equation on 1D with drift term $D^{(1)}(x)$, and diffusion term $D^{(2)}(x)$, and the governing equation of probability density function $$\frac{\partial f(x,t)}{\partial t} = ... 1answer 91 views ### Is the naive solution of this PDE/BVP unique? Problem statement Suppose I have a 2D or 3D equation of the form: \vec{\nabla} \cdot \left[ \vec{\vec{a}}\left(\vec{x}\right) \cdot \vec{\nabla} f\left(\vec{x}\right) \right] = \vec{\nabla} \cdot \... 0answers 25 views ### Estimates on the integral of an inner product Let X be an inner product space. For vector-valued functions F = (f_1,f_2), G = (g_1,g_2): [0,1] \to X^2, we define the inner product$$(F, G) = \int_0^1 f_1g_1 + f_2g_2.$$In particular,$$ ||F||...
Applying the method of eigenfunction expansion to the PDE $$u_t -c^2u_{xx}=F(t)$$ $$0<x<L, t>0$$ $$u(x,0)=f(x)$$ $$u_x(0,t)u_x(L,t)=0$$ for the homogenous part of this equation ($L[v(x,t)]=0$...