Tagged Questions

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

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1
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0answers
10 views

Heat equation, boundary gradient singularity

Consider the Cauchy-Dirichlet problem for the heat equation in a non-cylindrical region $\Omega \subset \mathbf{R}^+ \times \mathbf{R}$: $\Omega = \{ (t,x): \; 0 \leq t \leq 1, \; x \leq ...
0
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1answer
11 views

Simple problem about Laplace Equation in a domain

Suppose that "$u$", is solution of the problem $$\triangle u=0, r<R $$ $$u_{r}(R,\phi)=f(\phi), 0<\phi\ < 2 \pi$$ Show that $$\int_{0}^{2 \pi}{f(\phi)d\phi}=0$$ I know what this question ...
4
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0answers
15 views

Unbounded Entropy Solution to Burger's Equation

I need to deduce that $ u(x) = \left\{ \begin{array}{lr} \frac{-2}{3}(t+\sqrt{3x+t^2}) & t^2+4x>0\\ 0 & t^2+4x<0 \end{array} \right. $ is an unbounded ...
0
votes
0answers
13 views

Helmholtz equation in 3D

I understand solving the Helmholtz equation, $\nabla^2u + \lambda^2u = 0$, when $\nabla^2 = \partial_{xx} + \partial_{yy}$. However, I am not sure of the form I should obtain when $\nabla^2 = ...
1
vote
1answer
18 views

Separation of variables in the PDE $u_{tt}=c^2 u_{xx}$.

I'm stuck trying to work with the constants to the solution to the SOV problem Given the following equation: $u_{tt}=c^2 u_{xx}$ and the following conditions: $u(0,t)=0=u(\pi,t)$, $u(x,0)=0$, ...
2
votes
1answer
14 views

How to use D'Alembert formula for Neumann boundary conditions on a finite interval?

I have a PDE to solve that I am not sure how to do. I know how to solve this using D'Alembert's formula for Dirichlet boundary conditions but I do not know how to solve it for Neumann boundary ...
0
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0answers
19 views

Traces of $W^{1,\infty}$ functions

Let $\Omega$ be a Lipschitz domain and $p \in (1,\infty)$. It's known that if $u \in W^{1,p}(\Omega)$ then $u_{|\partial \Omega} \in W^{1-\frac{1}{p},p}(\partial \Omega)$. I'm wondering if the ...
0
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1answer
29 views

Voltage Distribution Inside a Cylinder [on hold]

I was assigned this problem, and quite honestly I do not know where to begin. If I could get some help and an explanation of the Bessel function, also? Thank you. I know my conditions are: ...
1
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0answers
31 views

Solving a PDE equation [on hold]

Could you please help me to solve this equation: $$\frac{\partial^2}{\partial x^2}E(x,t)-LC_1 \frac{\partial^2}{\partial t^2}E(x,t)+LC_2 \frac{\partial^4}{\partial x^2 \partial t^2}E(x,t)=0$$ ...
0
votes
0answers
10 views

Second order PDE with initial and boundary conditions

I'm trying to solve the following PDE: $u=u(x,y)$ $\left\{ \begin{array}{1 1} \partial_x ^2 u-6\cdot\partial_x \partial_yu+9\cdot\partial_x^2u=x^2+y^2\ \\ \ u(0,y)=0 \\ \partial_x u(0,y)=y ...
0
votes
0answers
5 views

Dirichlet problem with Specific Boundary Condition

How to solve the Dirichlet Equation $D = \{(x, y) \in \mathbb R^2 : 0 < x < a; 0 < y < b\}$ for Laplace Equation for following boundary condition? $$u(x, 0) = x; \,u(x, \pi) = x( \pi-􀀀 ...
1
vote
1answer
31 views

d'Alembert's Solution: Should anything more be added to it?

I have this initial value problem: $$ u_{tt}- c^2u_{xx} = 0; \,0 < x < 1; \,t > 0; $$ $$ u(x, 0) = 0; \,u_t(x, 0) = 1; 0 <= x <= 1; $$ $$ u(0, t) = u(1, t) = 0; t >= 0: $$ ...
1
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1answer
25 views

Charge distribution on an arbitrarily shaped conductor

From physics we know that given a charged conductor put in vacuum (no external electric fields), the charge distribution on its surface is approximately proportional to the curvature of the surface on ...
0
votes
1answer
14 views

Dirichlet problem on rectangle with nonhomogeneous boundary conditions

I want to solve the following problem in Dirichlet Problem: Let $D = \{(x, y) \in \mathbb R^2 : 0 < x < a; 0 < y < b\}$ be a rectangle with boundary $\partial D$. I want to solve ...
0
votes
0answers
14 views

How to prove this property?

I have a function $S(x,y)$ which satisfies the following PDE $$\frac{\partial S(x,y)}{\partial y}=-H\left(x,\frac{\partial S(x,y)}{\partial x}\right)$$ where the known function ...
0
votes
0answers
11 views

Physical interpretation of the integral formula for the solution of Laplace equation with Dirichlet/Neumann boundary condition

Suppose we have a bounded domain $D$ with smooth boundary, with $G(x,y)$ being the Green's function for the Poisson equation on $D$, i.e. $G(x,\cdot)=0$ on $\partial D$ and $\Delta_y ...
1
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1answer
12 views

Positivity of Poisson kernel

Let $K(x,y)$ be the Poisson kernel for the Dirichlet problem of Laplace equation on a bounded domain $D$ with smooth boundary, i.e. for a harmonic function $u$ on $\bar D$ with $u|_{\partial D}=g$, we ...
0
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0answers
11 views

PDE: Green's function and the method of images

I am stuck on a problem with the method of images. The formulation is rather simple; Solve the for green's function given by $\nabla^2 G = \delta( \underline{x} - \underline{x}_0)$ in the wedge ...
2
votes
1answer
24 views

Semi-infinite plate problem when 2 edges are equal temperature and one edge is a function of position

Here is the problem posted: Now here is my solution for a) I knew to discard the $e^{ky}$ and $\sin(kx)$ due to $T\to 20$ when $y\to\infty$ and $T= 20$ at $x = 0$. Which leaves me with ...
0
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1answer
27 views

D'Alembert Formula where PDE has only one boundary condition

Solve the initial boundary value problem $$u_{tt}=4u_{xx}, \ x>0,t>0$$ $$u(x,0)=\frac{x^2}{8}, \ u_t(x,0)=x, \ x\ge0$$ $$u(0,t)=t^2, \ t\ge0.$$ I used D'Alembert forula and got ...
1
vote
2answers
30 views

Characteristics method applied to the PDE $u_x^2 + u_y^2=u$

I am trying to solve: $u_x^2 + u_y^2=u$ with boundary conditions: $u(x,0)=x^2$. Unfortunately it leads to equations that makes no sense (sum of squares is $0$ and all constants are $0$). I would be ...
0
votes
0answers
14 views

What conditions must an operator meet, to have only real eigenvalues?

Given the problem $Lu = \lambda u$, what properties must $L$ have, for all its eigenvalues to be real? An answer in the context of (partial) differential equations would be appreciated.
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0answers
11 views

Notational question on vector field/differential operator

I am looking at something related to Hormander condition. The paper is 'On local smoothness of generalised and hypotellipticity of second order differential equations' by Oleinik and Radkevich. It ...
3
votes
2answers
32 views

EigenFunction for $\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial x} =\frac{2f^2}{x}$

When studying a computer vision problem I end up with a function $f(x,t)$ that satisfying $\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial x} =\frac{2f^2}{x}$. My question includes two ...
1
vote
1answer
34 views

Change of variables - PDE

I was just wondering how do I use change of variables to obtain a more suitable equation to solve for the following PDE? If I know how to do that then I am sure I can solve the rest. ...
3
votes
1answer
22 views

Laplace's Equation in Polar Coordinates - PDE

Find the bounded solution of Laplace's equation in the region $\Omega=\{(r,\theta):r>1,0<\theta<\pi\}$ subject to the boundary conditions $u(r,\pi)=u(r,0)=0$ for $r>1$ and ...
0
votes
1answer
21 views

Holder continuity using Sobolev imbedding

We assume for any $V\subset \subset U$ and $1<p<\infty$ $||u||_{W^{2,p}(V)}\le C(||\Delta u||_{L^p(U)}+||u||_{L^1(U)})$ for some $C=C(V,U,p)$. Given, $B=\{x∈R^3,|x|<1/2\}$ and we suppose ...
1
vote
1answer
13 views

Laplace Equation Boundary Problem

Solve the boundary value problem $$u_{xx}+u_{yy}=0, \ 0<x,\ y<1$$ $$u_x(0,y)=0, \ u(1,y)=0, \ 0<y<1$$ $$u(x,0)=1,u_y(x,1)=0, \ 0<x<1.$$ I have, ...
2
votes
1answer
19 views

Fourier Series Coefficient

I am trying to review the basics. Find the Fourier series for the function $$f(x) =\left\{ \begin{array}{l l} 2x & \quad -\frac{\pi}{2}<x<\frac{\pi}{2}\\ 0 & \quad ...
0
votes
0answers
29 views

Can you help me with this Non-homogeneous PDE Problem

The Problem: Solve the wave equation with time-independent sources if an equilibrium exists. Analyze the limit as t approaches $\infty$. If no equilibrium exists, explain why and reduce the problem to ...
0
votes
1answer
19 views

MATLAB, 1st order 2d hyperbolic equation, problem with convergence.

Follow up to my previous question: MATLAB: solving 1st order hyperbolic equation in 2 spacial dimensions The equation I'm solving has the form: $$f_t + A y f_x - B x f_y =0$$ I wrote the following ...
1
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0answers
11 views

Continuum Approach to Modeling Cell Proliferation and Differentiation (PDE)

I'm new here so I hope this post is appropriate. I recently read in a bioengineering textbook about an approach to model cell proliferation and differentiation. They proposed the following partial ...
2
votes
0answers
33 views
+50

General form of a connection with zero curvature

I am looking for proofs of the following two theorems: Theorem 1. On a connected and simply-connected open set $\Omega\subset\mathbb{R}^3$, functions $L^p_{ij}\in C^1(\Omega)$ are given that satisfy ...
0
votes
1answer
16 views

D'Alembert Solution Formula

I have a test tomorrow and the only thing holding me back from getting a good grade is D'Alembert formula for boundary conditions. I have this example that I am trying to figure out. Find ...
0
votes
0answers
17 views

Suppose $\Omega\subset R^n$ is strictly convex. There exists a barrier function for each $\xi\in\partial\Omega$.

Suppose $\Omega\subset R^n$ is strictly convex. Then $\partial\Omega$ is regular in the sense there exists a barrier function for each $\xi\in\partial\Omega$. Basically, barrier function means there ...
1
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0answers
32 views

Laplace equation on infinite strip

I'm trying to solve the following problem using the Fourier transform: $$u_{xx}+u_{yy}=0$$ on the domain $\;0\lt y\lt b$ , $-\infty\lt x \lt \infty \;$ with the following conditions: $$ u(x,0)= ...
0
votes
1answer
16 views

Question about properties of harmonic functions

I am reading the book "Harmonic Function Theory-Second edition" by Axler, and on page 232 of the book, it says that: Theorem: If $\Omega$ is connected,$u$ is real valued and harmonic on $\Omega$ and ...
1
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0answers
21 views

l'application de la méthode du groupe de renormalisation à l'equation de schrodinguer non lineaire avec non linearité quadratique [on hold]

bonjour tous le monde , svp j'ai beson d'aide sur un projet d'etude de texte à presenter , voila le lien du fichier pdf http://ev03023.math.uregina.ca/SK_Analysis_Day/Talks/Abou-Salem.pdf ma ...
0
votes
0answers
30 views

Solving the PDE $\frac{\partial u}{\partial t}=a\frac{\partial^2 u}{\partial x^2}+b\frac{\partial u}{\partial x}$

I am trying to solve the PDE $\frac{\partial u}{\partial t}=a\frac{\partial^2 u}{\partial x^2}+b\frac{\partial u}{\partial x}$ for constants $a$ and $b$ with conditions $\frac{\partial u}{\partial ...
0
votes
1answer
19 views

Neumanns problem [on hold]

I'm trying to understand problem problem 3 in chapter 6 of http://www.columbia.edu/~la2462/Problems%20from%20Evans.pdf In the second part of the proof Im not really sure how to use that $\int_\Omega ...
1
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0answers
10 views

Two questions regrading to Laplace equation, the Green's Reconstruction Formula

All the following we use Evans notation. By Green's reconstruction formula, we could represent $u$ by $$ u(x)=\int_\Omega-\triangle u(y)G(x,y)dy-\int_{\partial \Omega}u(y)\partial_\nu ...
1
vote
1answer
18 views

Laplace equation in polar coordinates

Solve the Laplace equation in polar coordinates $u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0$ within the domain $0<\theta<\pi, 1<r<2$ subject to boundary conditions ...
1
vote
0answers
34 views

Measurability of solution of diffusion equation in sub-sigma algebra

I want to solve the following problem: Get $\omega \in \Omega \subset \mathbb{R}$, $x \in D \subset \mathbb{R}^2$ and $0<a_i\leq a(.,.)\leq a_x<\infty$. Let $a( x;. )$ and $f(x;.)$ be ...
1
vote
0answers
12 views

Green function for gradient

Does a Green function for the gradient exist? Specifically, consider the equation $$\vec{\nabla}_x\, G(\vec{x},\vec{x}') = \vec{\delta}(\vec{x}-\vec{x}'),$$ where $\vec{\delta}(\vec{x}-\vec{x}')$ is ...
1
vote
1answer
37 views

Energy functional in Sobolev Space

Let $E[u]:=\int_W |\nabla u|^2 +V(x)u^2 dx$ be the energy functional on functions $u\in H^1_0(W)$ and $V\in L^\infty(W)$. Can someone help me show that $E$ is bounded below for all $u\in H^1_0(W)$ ...
0
votes
2answers
27 views

Range conditions on a linear operator

While reading though some engineering literature, I came across some logic that I found a bit strange. Mathematically, the statement might look something like this: I have a linear operator ...
1
vote
1answer
17 views

Intermediate Forms Between Parabolic and Hyperbolic PDE (numerically)

Greetings MSE community, I have recently conducted some rudimentary experiments in matlab coding of PDE's. I have explicit and implicit numerical solutions to both the heat and the wave equation, for ...
0
votes
2answers
37 views

What is an alternative book to oksendal's stochastic differential equation: An introduction?

What is an alternative book to oksendal's stochastic differential equation: An introduction? But also An alternative that is over 300 pages and at the same level? Some professor refer that book as a ...
0
votes
1answer
19 views

Analytical solution to a second order PDE

Hey all here is my equation in a 2D system. $$\nabla^2u(x,y) = -\sin(\pi x)\sin(\pi y)$$ I haven't done anything like this in a while so could use a bit of guidance, how do I go about solving this ...
0
votes
0answers
19 views

Sobolev spaces and Cauchy sequences with respect to $L^2$-norm.

Let $z^n=(u^n,w^n,\phi^n)$ be a sequence in $H=H_*^1(0,\ell)\times H_0^1(0,\ell)\times H_*^1(0,\ell)$, where $H^1(0,\ell)$ and $H_0^1(0,\ell)$ are the usual Sobolev spaces and $H_*^1(0,\ell)=\{f\in ...