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

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Heat equation for x approaching infinity

I'm trying to solve a PDE problem given to me, but I'm stuck. It's all about heat equation applied to the groud temperature. We assume that the temp in the ground is function of time $t$ and depth ...
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22 views

D'Alembert's solution and Heaviside function

Starting with the wave equation $$u_{tt} = c^2 u_{xx}$$ and given the initial conditions $$u(x,0) = \phi(x)\\ u_t(x, t) = \psi(x)\\ -\infty<x<\infty $$ We can derive D'Alembert's solution ...
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36 views

What is spectrum for Laplacian in $\mathbb{R}^n$?

I know very well that Laplacian in bounded domain has a discrete spectrum. How about Laplacian in $\mathbb{R}^n$?(not in some fancy-shaped unbounded domain, but the whole domain) Where can I find ...
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2answers
44 views

Ladyzhenskaya's inequality

I know a version of Ladyzhenskaya's inequality, that is Let $\Omega$ be bounded domain in $\mathbb R^2$, we have $$\|u\|^2_{L^4} \leq C\|u\|_{L^2}\|\nabla u\|_{L^2}, \quad \forall u\in H^1(\Omega).$$ ...
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15 views

Properties of the distribution function.

Let $\Omega\subset\mathbb{R}^n$ be open. Let $f:\Omega\rightarrow\mathbb{R}$ be a measurable function and $g\in L^p(\Omega)$, for some $p\geq 1$. For a measurable function $f: ...
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11 views

(2+1) dimensional PDE and numerical analysis

How to write boundary conditions in the following MATLAB code? How to get the following integral $$\int_0^{\infty} m(u_T,w,t)P(u_T,w,t)dw $$ where $m=u(1-u(1-au)-w+I)$ Is there any other boundary ...
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1answer
15 views

Linear Operator for Heat Equation

Show that if $u(x,t)$ satisfies $u_t = k \Delta u$ in a bounded region G, then for any $L>0$, $u(Lx, L^2t)$ solves the same equation for $x \in L^{-1}G$, where $L^{-1}G$ is the set of points ...
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57 views

Question about separation of variables

This is for the heat equation, where $$\frac{\partial U}{\partial t}-k \frac{\partial^2 U}{\partial x^2}=1$$ with the conditions $$U(0,t)=0, \; U(x,0)=0 \text{ and } \frac{\partial U}{\partial t} ...
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12 views

Traffic Flow via Characteristics

I can do the maths with this question its just the bit at the bottom of the solution that I have highlighted in green. How can i predict that the position where $\rho=\rho_{max}/2$ after 2 hours is ...
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22 views

Domain decomposition

In order to solve the $1$-D problem \begin{align*} u^{''}=f \in (0,1),\\ u(0)=u(1)=0, \end{align*} $f\in L^{2}(0,1)$, if we choose our solution space to be $H^{1}(0,1/2)\times H^{1}(1/2,1)$ and then ...
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Method of PDE solution by Fourier transform

In Rudin's Functional Analysis (Chapter 7, exercise 17), Rudin claims that for $n=1$ or $2$, if $u$ is a distribution on $R^n$ with compact support $K$, whose Fourier transform $\hat{u}$ is a bounded ...
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1answer
21 views

Coercivity for functional and complete orthonormal system

Consider with $\rho \in W^{1,2}([0,\pi])$ the following functional $$J(\rho)=\frac{1}{2}\int_{0}^{\pi}{\rho^2\,dx}$$ I know that in the $L^{2}([0,\pi])$ the coercivity condition is satisfied, but i'm ...
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1answer
149 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 ...
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2answers
121 views

Prove there exist a unique $u\in H^1$ such that $\int_{\Omega}(\kappa\nabla u\cdot\nabla v + \frac{1}{\kappa}uv) =\ \int_{\Omega}fv$ for $f\in L^2$

I need help with this problem from my homework Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ and let $\kappa : \Omega\rightarrow\mathbb{R}$ be a continuous function, such that there's ...
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2answers
76 views

Hint for integral

Could someone provide a hint as to why $$\nabla \cdot \vec a(\vec x) = -i\,\,\,b\,\,\,c(\vec x)$$ where $b$ is a constant, $i$ is $\sqrt {-1}$, implies that $$2\int d^3x \,\,x_ia_j(\vec ...
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1answer
24 views

Application of Sobolev inequality

I saw this inequality without knowing how to justify it. This is probably an application of Sobolev inequality (or not). Here it is: $u,v\in H^1(\mathbb{R}^d)$, then ...
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1answer
202 views

A question about Fisher's Equation and the Traveling Wave Equation

I am dealing with the Fisher's Equation: Consider the PDE: $u_t=u_{xx}+u(1-u)$, where $-\infty<x<\infty$ and $t>0$.Prove that there exists $c^*>0$ such that for each $c>c^*$ there ...
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35 views

the dual space of $L^p$

I am reading some preliminary material to develop a good background in order to study PDE and I came across the following fact The dual space of $L^p$ is $L^q$ where $q$ is the Holder's Conjugate of ...
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20 views

Elliptic PDE - max principle

The maximum principle for elliptic PDEs is established for the nondivergence form as in http://www.ann.jussieu.fr/~frey/cours/UdC/ma691/ma691_ch3.pdf. But what if we are dealing with the divergence ...
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1answer
14 views

Obtaining characteristic v on Cauchy Problem

$(x-y)p+(y-x-z)q=z$ Find the integral surface which the curves it passes are $z=1$ and $x^2+y^2=1$ Here is my try. $$\frac{dx}{x-y}=\frac{dy}{y-x-z}=\frac{dz}{z}$$ So we have ...
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9 views

Wave Characteristic diagram question

Say $F(x)=v(x)=0$ when $|x| \geq a$ whilst $F(x)=h(x)$ and $v(x)=c \ h'(x)$, both when $|x|<a$. As we know, the wave equation in D'Alembert's form is: $$ ...
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28 views

Decoupling system of two partial differential equations

If I have the following systems of PDE $$ u_t+x^2u_{xx}-\dfrac{h_1(t)}{h_0(t)}e^{-(v-u)}-\dfrac{h_0'(t)}{h_0(t)} = 0,\\ v_t-\dfrac{h_0(t)}{h_1(t)}e^{-(u-v)}-\dfrac{h_1'(t)}{h_1(t)} = 0, $$ where ...
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1answer
30 views

Show the equality holds for any $x \in [0, \pi]$

We are considering a $2\pi$ periodic function defined on $x\in(-\pi,\pi)$ by $$f(x) = \pi - x, 0<x<\pi $$ and 0 otherwise. I already computed the full Fourier series is equal to: $$f(x) = ...
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1answer
435 views

How to find an orthonormal basis for $L^2(\mathbb{R},\mathbb{C})$?

Consider the Hilbert space $X:=L^2(\mathbb{R},\mathbb{C})$ Now consider the operator that takes the second derivative, i.e. $A := \partial_{x}^2$, i.e. $A: H^2(\mathbb{R},\mathbb{C}) ...
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18 views

Stochastic PDE representation

I am trying to find a pde which $u$ satisfies when $u(x) = E^{x}[\cos(X_1)]$ where $dX_t = \sin(nX_t)\,dt + dW_t$ and $X_0 = x$. I have tried using Feynman-Kac but I can't seem to get it into the ...
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1answer
34 views

Changing variables for a partial differential equation

If I have the following systems of PDE \begin{align} u_t+x^2u_{xx}-\dfrac{h_1(t)}{h_0(t)}e^{-(v-u)}-\dfrac{h_0'(t)}{h_0(t)}=0\\ v_t-\dfrac{h_0(t)}{h_1(t)}e^{-(u-v)}-\dfrac{h_1'(t)}{h_1(t)}=0, ...
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29 views

Caccioppoli inequality

Assume we have established the following version of Caccioppoli inequality $$\int |\nabla u|^2 \psi^2 dA\leq C \int u^2 |\nabla \psi| ^2 dA$$ for $C^2(\mathbb C)$- smooth functions $u\geq 0$ with ...
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1answer
27 views

Traffic flow vs Density

This is a pretty simple question but I can't seem to understand it conceptually. The question is: If the traffic flow is increasing as $x$ increases ($\frac{\partial q}{\partial x}>0$), explain ...
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1answer
63 views

Proving $u\mapsto |u|^2u$ is Lipschitz on bounded subsets of $H^2(\Omega)\cap H_0^1(\Omega).$

I'm reading a paper and am stumped verifying two details. Let $\Omega$ be a bounded region in $\mathbb{R}^2$ with smooth boundary. I'd like to show that the map $u\mapsto |u|^2u$ is a map from ...
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1answer
19 views

Steps in the solution of Korteweg-deVries PDE

In the following solution of the Korteweg-deVries PDE $$ u_t + 6uu_x + u_{xxx} = 0 \qquad (3.1) $$ I do not understand the second integration step and how they arrive at the expression for the ...
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46 views

Extension of function from $W^{1,p}(\Omega - \{x\})$ to $W^{1,p}(\Omega)$

Suppose $\Omega\subset \mathbb{R}^n$ is open, $p\geq 1$, $n\geq 2$ and $u \in W^{1,p}(\Omega-\{x\})$. Show that $u$ extends to a function in $W^{1,p}(\Omega)$. So far I have; clearly $u$ and each ...
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1answer
19 views

von Neumann stability analysis for irregular meshes

All the litterature I have come across about the von Neumann stability analysis is performned on regular grids. Can the analysis be performed analytically on irregular grids, or does it have to be ...
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30 views

Show $u(x,t)$ is analytic in time

$$u_t + u_x + u u_x - u_{xxt} = 0$$ {know: $u$ can be differentiated $\infty$ times with respect to $t$. this fact may or may not be helpful in the proof} how would one approach such problem? i ...
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A question about the boundary values of Dirichlet functions.

Let $u_1$ and $u_2$ be two Dirichlet functions; hence both attain their maximum and minimum values on the boundary of the domain $D$ (let us call the boundary $B$). My book says the following: ...
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1answer
63 views

How do you pronounce Korteweg–de Vries

As stated in the title, how do you pronounce Korteweg-de Vries? I've always just heard it referred to as "KdV" but I have to give a talk on it so I'd like to know how to pronounce it properly.
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Characteristic curves of 2nd-order PDEs under invertible coordinate transformations

First off, I'm not very experienced with the subject and English is also not my first language, so if there are any inaccuracies in the following text, let me know. Given a linear, scalar, ...
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1answer
45 views

Heat Equation Steady state question

Say you have a slab of material occupying the region $0\leq x\leq a$. Heat is supplied at a constant unit rate so the temperature $T(x,t)$ satisfies $$\frac{\partial T}{\partial t}= k ...
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11 views

Inhomogenous partial differential equation

I am stuck getting the particular solution to the following partial differential equation: $$ \Big(\frac{\partial ^2}{\partial t^2}-\nabla^2+\alpha\mu^2\Big)\,f(\mathbf{x},t) = g(\mathbf{x},t) $$ ...
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How to prove comparison principle for parabolic PDE (nonlinear)

Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for $x > 0$ and $F:(0,\infty) \to (0,\infty)$ continuous and increasing with $F(0) = 0$. Consider the PDE $$u_t = \Delta F(u) ...
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what does well posdeness results tells us concerning non linear evolution equations?

Consider a nonlinear Shr\"odinger equation, $$iu_{t}+\bigtriangleup u + f(u)= 0, u(0)= u_{0}$$ where $u(t, x)$ is complex valued function of $(t,x) \in \mathbb R \times \mathbb R^{n}$, $i=\sqrt{-1}, ...
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99 views

How exactly does the constant $C$ in the Sobolev inequality depend on the domain?

The Sobolev inequality theorem -as stated here- says Let $U$ be a bounded open subset of $\mathbb{R}^N$, with a $C^1$ boundary. Assume $u \in W^{k,p}(U)$. If $k<n/p$ then $u \in L^q(U)$, where ...
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183 views

How find this $\frac{yf_{y}-z}{f_{x}}+\frac{xf_{x}-z}{f_{y}}-xf_{x}-yf_{y}+x+y+z=C$ solution

In plane $R^3$,Find $z=f(x,y)$, such the length of the portion of any tangent line to the astroid $$z=f(x,y)$$ cut off by the coordinate axes is constant $C$, This problem is from this post (when I ...
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PDE Heat Equation Question: Finding T(x,t) with limited information.

Say our equation for temperature at position x and time t is shown by: $$ T(x)=T_0(1-x/a) $$ This equation holds for a rod of length a from x=0 to x=a. Initially T(0,t)=$T_0$ and T(a,t)=0. ...
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66 views

Variational methods : Why i can't apply this theorem?

Consider the following problem: Find a weak solution for $$u'' + u = \sin t ,\,\, 0 < t < \pi$$ $$u(0) = u(\pi)=0 $$ the corresponding functional for the problem is $\varphi(u) = ...
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1answer
90 views

Second Derivative of Eigenvalue

Consider the potential $V_t=tx$ and the corresponding Schrödinger operator $H_t=- \frac{\partial ^2}{\partial x^2}+V_t$ on $L^2([0,R])$ with Neumann or Dirichlet boundary conditions. Let ...
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1answer
52 views

can the first eigenfunction of the Dirichlet Laplacian have any saddle points

Let $\Omega$ be a connected, bounded region of $\mathbb{R}^2$. The Laplacian $\Delta$ has a discrete spectrum of functions satisfying $$\Delta f = \lambda f$$ on $\Omega$ with $f=0$ on the boundary ...
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31 views

Why can we assume the separation constant?

I am starting a project using the wave equation and I don't understand why when doing separation of variables, we can assume the following, where the equation has already been separated. ...
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1answer
30 views

Fundamental solution of the Laplacian on the surface of a cylinder

Does the Laplace operator have a fundamental solution on the surface of a cylinder in $\mathbb{R}^3$? Intuitively, I can visualize a function that diverges to $\infty$ at a point, decreases to a ...
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2answers
191 views

Positivity of principal eigenvalue for $L\phi=-\triangle \phi + \nabla \cdot( u \phi )$

EDIT: This question is still unresolved as of April 18. The two answers provide useful work in the right direction, but neither resolves the question. A counterexample should have $u \in ...
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FEM PDE's How do I find the best solution using abstract methods for FE.

I have a homework problem I am working on, and I need a little help or maybe a nudge in the right direction.... Question: (Purpose: Review abstract finite element methods.) Consider the ...