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

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1answer
60 views

Show that $\max_{\bar{U}} |u|\le C(\max_{\partial U} |g|+\max_{\bar{U}} |f|)$

Let $U$ be a bounded, open subset of $\mathbb{R}^n$. Prove that there exists a constant $C$, depending on only $U$, such that $$\max_{\bar{U}} |u|\le C(\max_{\partial U} |g|+\max_{\bar{U}} |f|)$$ ...
3
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1answer
160 views

Solving cauchy hyperbolic second order pde

I'm currently taking a course in partial differential equations. I'm trying to solve the following problem (which is, as far as I can tell, a bit above the level of the course): $$\begin{align} u_{...
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1answer
85 views

Solving a one dimensional wave equation

Consider the partial differential equation $$\frac{\partial^{2}u}{\partial t^{2}}=c^{2}\frac{\partial^{2}u}{\partial x^{2}}$$ For the region $0<x<\pi$ where $t>0$. With the boundary ...
6
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2answers
267 views

What physical information does the mean value property of heat equation convey?

I'm reading through the Evans' book on PDE, the chapter on heat equation. The definitions are the same as here. I see that mean value property of heat equation is useful for proving maximum principle ...
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0answers
54 views

Trace map $\gamma:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$ invertible on $\{ u \in H^1(\Omega) \mid \Delta u =0 \}.$

It is apparently true that the trace map $\gamma:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$, is invertible when restricted to the domain $$\{ u \in H^1(\Omega) \mid \Delta u =0 \}.$$ I've been ...
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1answer
38 views

Restriction estimates

What is the defining property of what someone in the harmonic analysis community would call a "restriction estimate?" I see sobolev norms, Fourier transforms, and inequalities relating these. The ...
2
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0answers
85 views

Duhamel's principle in constructing heat kernel

I want to construct the heat kernel $k_t(x,y)$, which is the fundamental solution to the heat equation $(\partial_t + \Delta_x)u(t,x) = 0$. There is something that I don't understand about using ...
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0answers
48 views

Basis of $H^1(\Omega)$ which is orthonormal wrt. $L^2(\partial\Omega)$ inner product?

Let $\Omega$ be a domain with $\partial\Omega$ bounded. Is it possible to find a smooth basis of $H^1(\Omega)$ and $L^2(\Omega)$ which is orthonormal wrt. the $L^2(\partial\Omega)$ inner product? ...
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1answer
347 views

The rate of convergence for finite difference methods for Poisson's equation with piecewise constant data

I am solving the following PDE; $$ \nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \rho, $$ where $\rho(0.5,0.5) = 2$ (zero elsewhere), $0\leq x,y\leq1$ and the ...
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1answer
44 views

Finding the fundamental mode of wave equation in a rectangle

Consider the two-dimensional wave equation: $$\frac{\partial^{2}u}{\partial t^{2}}=c^{2}\left[\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}\right]$$ for some $c>0$. I ...
0
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1answer
77 views

Solving 2D Laplace eigenfunction equation

I want to solve the equation $$\nabla^{2}\,{\rm P}(x, y) = \frac{k}{c^{2}}{\rm P}(x, y)$$ or $$\frac{\partial^{2}{\rm P}}{\partial x^{2}}+\frac{\partial^{2}{\rm P}}{\partial y^{2}}=\frac{k}{c^{2}}{\rm ...
1
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1answer
147 views

Equilibrium Temperature in insulated rod from BVP

this problem is giving me some trouble. It is a review problem given to us to study for our Final Exam this week. I would love some help in understanding how to solve it so that I can study it and ...
0
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1answer
151 views

What is the solution to poissons equation with a point source?

I am currently trying to solve the following PDE using various numerical methods - direct and iterative. $$ \nabla ^2 u = \rho $$ where $u = u(x,y)$ and $\rho(0.5,0.5) = 2$ (zero elsewhere). The ...
0
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2answers
55 views

Help with first order linear PDE with initial condition

I would like to solve the following pde: $2y\cdot \partial_x u(x,y)-3x\cdot\partial_yu(x,y)=0$ and $u(x,x)=e^{x^2}$ Without the initial condition I got the following result: $\frac{1}{4}(2y^2+3x^2)...
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0answers
38 views

nonlinear coupled partial differential equations

Is there any order for solving a set of nonlinear coupled partial differential equations analytically i.e. without a numerical algorithm. I cant solve the following set of equations $$ \partial_xf_1(...
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2answers
101 views

How would I compute this sum?

So I would to compute this integral which is coupled by a sum: $$ \int_{x = 0}^{x = \lambda} \sum_{k=-\infty}^\infty e^{-( \frac{x-k \lambda}{\sigma} )^2} dx$$ I was thinking about using parseval's ...
0
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1answer
32 views

first order equation problem

I have a first order PDE with the initial condition: (a) $$\displaystyle\frac{\partial f(x,t)}{\partial t}-\displaystyle(xt)\frac{\partial f(x,t)}{\partial x}=0$$ $$f(x,0)=\frac{1}{1+x^2}, \hspace{...
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0answers
30 views

Second order equation and regularity

Let $U$ be an bounded open set in $\mathbb R^3$ and ${V}\subset \subset{W} \subset \subset {U}$. Let $u \in {H^1}(U)$, and $f\in L^{2}(U)$ satisfies $$\int_{\mathbb R^3} \nabla u(x). \nabla \varphi (...
2
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1answer
79 views

Proof of an inequality in Sobolev space.

I want to show the next inequality: $$\| D^a u D^b v \|_{L^2(\mathbb{R}^n)} \leq C \| u \|_{H^s(R^n)} \| v \|_{H^s(R^n)}$$ (for $s>n/2$) Where $D$ is a differential operator. What I did so far is ...
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1answer
49 views

Fundamental solution to the bi-harmonic operator?

I am not sure about what the hint means. If $\Delta u =\frac{1}{2 \pi}(1+\log|x|)$. Since $\log|x|$ is a fundamental solution of $\Delta u =0$. Does that mean $\frac{1}{2 \pi}(1+\log|x|)$ is a ...
3
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1answer
128 views

Prove the energy is constant in a PDE?

I calculated the $$ \begin{align} \frac{dE(t)}{2\,dt} & = \int_\Omega u_tu_{tt}+DuDu_t+u^3u_t\,dx \\ & =\int_\Omega [u_t(u_{tt}-\Delta u)+u^3u_t] \, dx+\int_{\partial \Omega} u_t \frac{\...
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0answers
52 views

Confusion about domains in PDE and “every $C^k$ manifold can be thought of as a smooth manifold”

I have read that every $C^k$ domain can be described instead by $C^\infty$ chart maps. So in this sense every $C^k$ domain is a $C^\infty$ domain. But consider standard PDE where people say thing like ...
2
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1answer
53 views

How do I accurately find a point within a grid of 4 based on varying values?

I am specifically trying to find a source of heat, with heat sensors giving readings from 4 different points eg below: The diagram above represents a 4 x 4 grid where the numbers are heat sensors ...
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0answers
72 views

Interior boundary value problem for the Helmholtz equation

Let $D \subset \mathbb R^d$ be a $C^2$ bounded domain. I consider the following boundary value problem for the Helmholtz equation $$ (\Delta+k^2)u = 0 \quad \text{in $D$}, \\ u|_{\partial D} = u_0,...
0
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1answer
31 views

Finding (exactly) the electric potential, in presence of non-constant dielectric

In a medium with homogenous dielectric, the electric field can be solved as an instance of Poisson's equation, but this is not the case in general. I can find the variational form and solve with ...
2
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1answer
58 views

Partial Derivatives versus Proper Derivatives

I'm having some difficulty understanding exactly what a partial derivative is. I had been content with the definition $$\frac{\partial F}{\partial x_i } = \lim_{\Delta x \rightarrow 0} \frac{F(x_0, ...
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0answers
34 views

Inverse Fourier Transform of $1/k^2$ in $\mathbb{R}^N $

This comes up in the context of finding the Green's function of Poisson's equation for $\mathbf{x} \in \mathbb{R}^n $ $$ \nabla^2 G(\mathbf{x}) = \delta(\mathbf{x})$$ Attempt by using Fourier ...
3
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3answers
117 views

A problem on infinite domain diffusion equation

Consider the following problem $$u_t-u_{xx}=p(x,t), -\infty<x<\infty,t>0$$ $$u(x,0)=0$$ $$u\rightarrow0 \text{ as } x\rightarrow \pm \infty$$ This can be solved using many sub problems as ...
2
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0answers
68 views

use of Strichartz estimates

I have been learning(understanding) the Strichartz estimates. My naive Questions: (1) Can you give some ideas, how Strichartz estimates plays role in solving nonlinear dispersive equations(e.g. ...
0
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1answer
39 views

Calculate $\int_{B_R(0)}u$ for a very weak solution $u$ of a Dirichlet boundary problem

Let $B_R:=B_R(0)\subseteq\mathbb{R}^n$ be the open ball with radius $R>0$ around $0$, $f\in\mathcal{L}^\infty(B_R)$ be radial and $u$ be a very weak solution² of $$\left\{\begin{matrix}-\Delta u&...
0
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1answer
52 views

Uniqueness of the fundamental solution of a 2nd order linear parabolic PDE

I'm reading Avner Friedman, Partial Differential Equations of Parabolic Type. Let \begin{eqnarray} L:=a_{ij}\dfrac{\partial}{\partial x_i\partial x_j}+b_i\frac{\partial}{\partial x_i}+c-\frac{\partial}...
0
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1answer
48 views

Meanvalue formula of harmonic functions - translation invariance of surface measure

In almost every proof of the mean value formula for harmonic functions (e.g. Evans 2.Theorem 2) one finds the following calculation $$ \frac{1}{|{\partial B(0,1)}|}\int_{\partial B(0,1)} D u(x+rz)\...
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1answer
43 views

Basic question about weak solution of p-Laplace equation

Let $p \geq 2$ and $B(x_0, d)$ an open ball in $R^n$ ($n \geq 2$). Let $u \in W^{1,p}(B(x_0,d))$ a weak solution of $\Delta_p u = f$ , $u = 0 , \ on \ \partial B(x_0,d)$, that is, $\int_{B(x_0,d)} \...
0
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0answers
87 views

Uniqueness of the damped wave equation

I want to prove the uniqueness of the following. \begin{cases} u_{tt} + u_t - u_{xx} = 0 & 0<x<b, t>0 \\ u(0,t) = u_x(b,t) = 0 & t\geq 0 \\ u(x,0) = f(x), u_t(x,0) = g(x) & 0\leq ...
0
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1answer
68 views

Bump function on set

Given an arbitrary compact set $K \subset \mathbb{R}^n$ and an open set $V \subset \mathbb{R}^n$, I want to construct a bump function $\zeta$ that is $1$ on $K$ and has its support in $V$, where I ...
3
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1answer
50 views

Finite differences and conservation law

I am using a Finite Difference scheme to solve a simple PDE in conserved form: $$\partial_t u = \partial_x (\partial_x u +au\partial_x u) = (1+a)\partial_x^2u +a(\partial_x u)^2 $$ $$\frac{u_{n+1,j} ...
5
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1answer
249 views

Solving a PDE with Feynman-Kac Formula

I'm trying to solve this PDE using Feynman-Kac formula Now i follow the regular steps Here is where I don't know how to proceed. How do I calculate this expectation?
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0answers
23 views

method of characteristics in a nutshell

Good morning everybody, I need a quick reference for the following inhomogeneous first-order pde... namely $$f(x,y,z)=A\partial_x\varphi+B\partial_y\varphi+C\partial_z\varphi,$$ where $\varphi\in C^...
3
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1answer
79 views

solution of multidimensional PDE

I'm looking for a way to find a solution 'f' to the following PDE. $$ y \frac{\partial f}{\partial r} + g_1(r)\left(z\frac{\partial f}{\partial y} - y\frac{\partial f}{\partial z}\right) + g_2(r)\...
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1answer
141 views

Solving characteristic base curves initial value PDE

I'm trying to solve the characteristic base curves of an initial value problem. $$ \left\{ \begin{matrix}\ xy\frac{\partial u}{\partial x} + (2y^2 - x^6)\frac{\partial u}{\partial y} = 0 ; x>0,y\...
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1answer
111 views

Diffusion-Reaction PDE - radial coordinate

I am trying to obtain an expression for the concentration $C$ based on this stationary equation : $\frac{\partial C}{\partial t} = \frac{1}{r} \frac{d}{dr} \left(r \frac{\partial C}{\partial r}\right)...
0
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1answer
89 views

Solve laplace equation inside a rectangular

My answer is $U = Acos(nπx/L)e^-nπy/L$ I really have no idea how to solve the particular solution. Please advise me.
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0answers
52 views

Transforming 1D Burger's Equation into infinitely many coupled ODE's

I've been working on the following problem but I can't justify my steps, would a savvy mathematician kindly tell me what, if any, violations I've made. Problem: Show Burger's equation can be written ...
3
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0answers
57 views

when does a partial differential equation have unique solution? [duplicate]

The differential equation $ xu_x + yu_y = 2u$ satisfying the initial conditions $y = xg(x), u=f(x)$ with $f(x) = 2x, g(x) = 1$, has no solution $f(x) = 2x^2, g(x) =1$, has infinite number of ...
3
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1answer
145 views

Is fractional order Sobolev spaces reflexive?

Let $0<s<1$, we define $$ W^{s,p}(\Omega):=\left\{u\in L^p(\Omega),\,\frac{|u(x)-u(y)|}{|x-y|^{\frac{N}{p}+s}}\in L^p(\Omega\times\Omega)\right\} $$ with norm $$ \|u\|:=\left(\int_{\Omega} |...
1
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1answer
157 views

2D linear inhomogeneous wave equation with inhomogeneous time-independent initial conditions

I'm looking for any insight into solving the following PDE: $$u_{tt}=c^2 (u_{xx}+u_{yy})-\sin(y)$$ $$u=0, y\in {0,\pi} $$ $$u_x=0, x\in {0,1}$$ $$u(x,y,0)=\cos(\pi x)\sin(3y) $$ $$u_t(x,y,0)=0$$ ...
4
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1answer
76 views

Convolution integral problem

In the process of solving a certain PDE, I've arrived at a convolution integral: $$\int_{\mathbb{R}^3} G(x-y) \nabla p(y) dy$$ where $x \in \mathbb{R}^3$, $G(z)=\frac{1}{\| z \|}$ and $p(z) = \frac{...
2
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0answers
52 views

Solving a system of two linear PDE: $u_x+v_x +u_y=0$ and $v_x+u_y-{1\over 2} v_y=0$

trying to solve the following cauchy problem: $$u_x+v_x +u_y=0\\v_x+u_y-{1\over 2} v_y=0\\u(x,0)=1-x,v(x,0)=x$$ my solution is: 1. multiply each equation by $t_1,t_2$ and sum the two equations like ...
3
votes
1answer
107 views

Solving a first order nonlinear PDE

I have to solve this equation : $$ \frac{\partial u}{\partial x} \cdot \frac{\partial u}{\partial y} = xy $$ with initial condition $u(x,0) = x$. I know it is easy with separation of variables, but ...
3
votes
1answer
225 views

Construction bump function with positive Fourier transform

I am looking for the construction of a smooth bump function, $f$, mapping the real line to itself which has two special properties: (1) $f$ is constant on some interval in its support (for instance $...