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

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14 views

Conformal mapping

I know nothing about conformal mapping. I want to find analytical solution of Laplace equation in a hexagon with Dirichlet boundary condition at each wall. I already know the analytical solution of a ...
2
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1answer
42 views

Rayleigh-Bénard convection

I have a nondimensionalized linear perturbation system relevant to the appropriate pure conduction solution for Rayleigh-Bénard convection in upper planetary atmospheres under the compressible gas ...
1
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2answers
44 views

PDE Initial-Boundary value problem question

The problem: The ends of a stretched string are fixed at the origin and at the point, $ x=\pi$ on the horizontal x-axis. The string is initially at rest along the x-axis, and then drops under it's ...
1
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1answer
32 views

Mixed boundary condition for the wave equation, using reflection method

Solve for $u(.5,13)$: $$u_{tt}=4u_{xx}\quad 0<x<1$$ $$u(x,0)=x, \ u_t(x,0)=1$$ $$u(0,t)=0, \ u_x(1,t)=0. $$ Using D'Alembert formula I got that I must take the odd extension and then apply ...
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19 views

Two dimensional non homogenous pde

Would you please help me with some examples like this: I need 3 examples of 2D nonhomogenous PDE with complete solution (Sturm liouville) thank a lot.
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18 views

Estimates of partial derivatives

Suppose $f\in C^{\infty}(\mathbb{R}^n)$ is real analytic and $\Delta f(o)\not=0$. Are there pointwise estimates for $\frac{\partial^{\alpha}f}{\partial x^{\alpha}}$ in terms of ...
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1answer
35 views

Verifying Distribution Equivalence for Fourier Series Expansion

In my lecture notes, given a periodic distribution $T \in (C_{per}^\infty([-\pi,\pi]^n))'$, the Fourier coefficients are defined by $$\hat T(m) = T({1 \over (2\pi)^n}e^{-i m \cdot x}),$$ for $m \in ...
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0answers
32 views

Extend concept of weak derivatives?

Let $f \in H^m(\mathbb{R}^n)$, then we have for $|\alpha| \le m$ that $$\langle \partial^{\alpha}f, \phi \rangle = (-1)^{|\alpha|} \langle f , \partial^{\alpha} \phi \rangle$$ for all test ...
0
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1answer
39 views

Inverse fourier transform for the heat equation

I'm trying to find the Inverse Fourier Transform for the following heat equation. Question: Solve the problem $$u_t=ku_{xx}+\frac{1}{\sqrt{2kt}}e^{\frac{-x^2}{4kt}}, -\infty < x< \infty, t ...
1
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1answer
41 views

What concepts do I need to solve this nonlinear problem?

Problem Details: Let $L$ be a linear operator and $N(u)$ be a nonlinear function of $u$. Consider the IVP for the following nonlinear PDE: $$\partial_{t}u + L(u) = N(u)$$ defined on $(x,t) \in ...
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1answer
111 views

solving a PDE with with coordinates

First i need to find eigenfunctions in spherical coordinates with source term. I am having trouble with this . Can someone help guide me i also have 2 b.c.'s.
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0answers
19 views

Radial nondecreasing function

I would appreciate if someone can please answer these questions: Suppose $u(x)\in C_{loc}^2(\Re ^3) satisfies -\Delta u+(x+b(x)).\nabla u\le 0, |u(x)|\le C(1+|x|)^{2014}$ where $b(x)$ is a smooth ...
1
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1answer
20 views

A polynomial can be written as the difference of sub-harmonic functions

Let $\Omega\subset \mathbb R^N$ open bounded be given, I am trying to prove that first any Polynomial can be written as difference of two sub-harmonic functions, and then for any continuous function ...
2
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0answers
37 views

Stuck trying to solve wave equation in $n$-dimensions.

Solving the wave equation $u_{tt} = c^{2} \Delta{u}$ subject to $u(0,x) = f(x)$ and $u_{t}(0,x) = g(x)$ gives us d'Alembert's formula. I'm looking to solve the wave equation, subject to these same ...
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1answer
22 views

Harmonic function on an open set is infinite dimensional

I want to prove that the space of harmonic function on an open set $\Omega\subset R^N$ $(N\geq 2)$ is uncountablely infinite dimensional. That is, I want to prove that $$A:=\{u\in ...
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0answers
41 views

Using Fourier Analysis to determine Green's Function of Laplace's equation

I have previously seen the Green's function for Laplace's equation in two spatial dimensions determined using the method of images. Since then, I have learned some more Fourier analysis and have ...
3
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1answer
35 views

An elliptic PDE problem of Laplace function

Let $u$ be a harmonic function and we define $$ q(r)=\int_{\partial B(0,r)}u^2(x)\,dx $$ I want to prove that $q$ is monotone and convex. I tried the usual trick, by changing of variable, we have $$ ...
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2answers
38 views

First order partial differential equation

How to find L if the form is: $$\left(\frac{\partial L}{\partial x}\right)^2-\left(\frac{\partial L}{\partial y}\right)^2=-1$$ The author wrote, $$L=y+ax^2+...$$ but I didn't get how? Edit: where L ...
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1answer
52 views

Is this derivative somehow bounded?

I have a function $\phi: \mathbb{R} \rightarrow \mathbb{R}$ that is a test function and $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined by $f(x) = \phi(\frac{\|x\|}{n})$. Now if I take any ...
0
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0answers
71 views

Solving differential equation by weak formulation and minimizing a functional

I want to give a weak formulation of the boundary value problem \begin{align*} -(c(x)(u'(x)-1))' & = 0 \textrm{ on } \Omega = (-1,1) \\ u(-1) = u(1) & = 0 \end{align*} where $c(x)$ is ...
2
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1answer
41 views

Approximate $C^{\infty}$ functions by test functions in the Sobolev space norm

I am looking for a way to approximate a function $f \in \mathbb{C}^{\infty} \cap H^m(\mathbb{R}^n)$ by test functions such that I approximate $f$ and all of $f's$ $m-$ derivatives in the canonical ...
2
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1answer
38 views

Solve Basic partial differential equation question [on hold]

Help me solve the partial differential equation. $$\frac{\partial^2z}{\partial x^2} + 2 \frac{\partial^2z}{\partial y^2} - 3\frac{\partial^2 z }{\partial x \partial y} = e^{2x-y} + e^{x+y} + \cos(x ...
2
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1answer
18 views

Solution Operator for inhomogenous Dirichlet Problem

Writing up a report, I want to understand the following solution operator better. Suppose $\Omega$ is an open bounded domain in $\mathbb{R}^d$ with boundary $\partial\Omega$ of class ...
2
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0answers
26 views

notation question: What is $\nabla^{l}$ where $l\in \mathbb{N}$

Does $\nabla^{l}$ where $l\in \mathbb{N}$ mean anything? Could it mean dimension? We do have $\nabla^{2}=\Delta$, but I've never seen $\nabla^{3}$ . Found it in page 17 ...
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0answers
7 views

Can you give some information for rothe method

I want to learn a numerical method for PDEs other than finite difference method. After some research on internet i have found Rothe method and it looks interesting to me. Unfortunately, i couldn't ...
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2answers
53 views

Separation of Variables (PDEs): What about $0$?

Question: [See the context given below.] $\rm\color{#c00}{(a)}$ When we divide by the functions $T$ and $X$ to obtain $(1)$, aren't we assuming that the functions will be non-vanishing on ...
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1answer
77 views

Proving $\frac{200}{\pi}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)\cosh\left(\frac{\pi}{2}(2n+1)\right)}=25$

I solve a partial differential equation (Laplace equation) with specific boundary conditions and I finally found the answer: $$U(x,y)=\frac{400}{\pi}\sum_{n=0}^{\infty}\frac{\sin\left((2n+1)\pi ...
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1answer
10 views

How can I calculate the argument of amplification factor?

For example, I have an amplification factor of upwind scheme for hyperbolic conservation law, $$\lambda(k)=1-\nu(1-e^{-ik{\triangle}x})$$magnitude of which is ...
2
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0answers
20 views

Compactness of a sequence

Let $\theta_n(x,t)$ a sequence such that $$\theta_n\rightarrow\theta\;\;\mbox{in}\;\;C((0,T],H^s)\;\;\mbox{where}\;\;s>1.$$ Consider $\phi \in C^{\infty}$ and ...
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1answer
34 views

Two exercises on Evans PDE book.

Those two problems bothers me for a while. I think I got most of it but I do want to have a nice and clean solution, so I post it here for discussion. All below I will use Einstein summation. The ...
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1answer
28 views

Heat equation fundamental solution

The following is from a book of PDEs and I have cannot seem to figure out a particular step in it with regard to the derivation of the fundamental solution of the heat equation. I have highlighted it ...
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1answer
29 views

D'Alembert Formula nonhomogenous bounday condition

I just need help converting the nonhomogeneous boundary condition into a homogenous boundary condition using change of variables $u=v+w$. Since D'Alembert formula is for the infinite domain ...
5
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1answer
94 views

Transforming a PDE given basis vectors

I have a non-orthogonal coordinate system defined by $\mathbf x=\mathbf x(r,\beta,z)$, and so I can find the basis vectors as $$ \mathbf g_r=\frac{\partial \mathbf x}{\partial r},\quad\mathbf ...
4
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1answer
44 views

Green's operator for elliptic differential operator

Let $$ P:\Gamma(E)\rightarrow\Gamma(F) $$ be an elliptic partial differential operator, with index zero and closed image of codimension 1, between spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections ...
0
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1answer
20 views

Fastest numerical way to solve steady-state reaction-diffusion equation

I have a reaction-diffusion equation in 2 dimensions of the typical form: $\frac{\partial u}{\partial t} = D\nabla^2u - \Phi(u(x))$ I want to stress that $\Phi(u(x))$, is not a constant, but depends ...
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0answers
41 views

Property of a vector field.

Let us consider a regular vector field $A:\mathbb{R}^3\rightarrow \mathbb{R}^3$, such that $\mathrm{div} A=0$ in $\Omega$, where $\Omega\subset \mathbb{R}^3$ is a bounded, open, and smooth domain. Is ...
7
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1answer
132 views

Nonlinear partial differential equations with applications

Has anyone studied the book 'Nonlinear Partial differential equations with applications' by Tomas Roubicek? I am interested in discussing a point of interest in this book. Specifically, on page 52, ...
3
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0answers
17 views

Compactness of solution space of semi-linear parabolic PDE

Under what conditions a closed and bounded subset of solution space of following parabolic PDE is compact? $$x_{t}=x_{zz}+f(x,z)$$ Thank you!
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1answer
28 views

Minimal surfaces maximum principle

This is homework so no answers please. The problem is The domain is unit disk in $\mathbb{R}^{2}$ Suppose u,w satisfy the minimal-surface equation $div(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}})=0$, ...
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0answers
44 views

I want to proof that the lebesgue measure of the set below is positiva. Help me!

Let $\Omega$ be a domain limited with smooth boundary in $\mathbb{R}^{n}$ and consider the Sobolev space $H^{1}_{0}(\Omega)$ equiped with the norm $||u||=\int_{\Omega}|\nabla u|^{2}dx$. Let ...
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1answer
30 views

Bounded Laplacian Equation

Find the bounded solution of $\triangledown^2u(x,y)=0$ in $D=\{(x,y):x^2+y^2\le 1\}$ assuming that on the boundary of $D, \ \frac{\partial u}{\partial n}$ is equal to ...
1
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1answer
27 views

Does this equation has a non-trivial solution

Suppose $u:[0,1]^2\to [0,1]$. The equation $$\int_{[0,1]}\frac{\partial u}{\partial u_1} (y,x)dx=\int_{[0,1]}\frac{\partial u}{\partial u_2} (x,y)dx, \phantom{00}\forall y\in[0,1]$$ has trivial ...
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0answers
38 views

Solving PDE on manifold via Hodge theory

Let $(M, g)$ be a Riemannian manifold, where $M$ is compact without boundary. The Hodge decomposition tells us that $$\Omega^k = \ker (\Delta) + \text{Im} \ d + \text{Im}\ d^* . $$ Note that we can ...
2
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1answer
29 views

Energy associated with PDE

Let $a \in \mathbb{R}$ and consider a solution (sufficiently regular) of the equation $$\left\{ \begin{aligned} u_{tt} +a u_t - u_{xx} & = 0, \quad t > 0, \quad x \in ]0,1[, \\ u(0,t) & = ...
3
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1answer
37 views

Density of smooth compactly supported functions in Sobolev space over unbounded domain.

Prove that $C^{\infty}_ {c}(\mathbb{R}^n)$ is dense in $W^{k,p}(U)$ for any open $U\subset \mathbb{R}^n$ with $\partial U\in C^1.$ In which $p\in [1,\infty)$ Note: In Lawrence Evans's PDE text, the ...
2
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1answer
39 views

Prove that partial differential equation has no weak solution

For a Dirichlet problem $$\begin{array}[t]{rcl}Lu&=&-\frac{\text{d}}{\text{d}x}(x^2\cdot u'(x))=1=:f\,\,\,\text{ in }(0, 1)\\ u&=&0\,\,\,\text{ in }\{0,1\} \end{array}$$ there is a ...
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1answer
17 views

What is the “usual continuation argument”?

Let $u:[0,T]\to\mathcal{H}$ be the local solution of some initial value problem. Suppose we have proved that there exists a constant $C$ (that depends only on initial data) such that ...
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0answers
8 views

Variational form of boundary value problem

Given Dirichlet boundary value problem $$ \begin{array}[t]{rcl} -\Delta u&=f &\text{ in }\Omega\\ u&=0 &\text{ on }\partial\Omega, \end{array} $$ we can use Green's theorem and ...
2
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1answer
44 views

Is the on-diagonal heat kernel “local” with respect to the metric?

Question Let $X$ be a manifold, and $\mu_A$, $\mu_B$ two Riemannian metric on it which agree on an open subset $U\subset X$, i.e. $\mu_{A\,|U} = \mu_{B\,|U}$. Let $K_A(t;z,w)$ resp. $K_B(t;z,w)$ be ...
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2answers
47 views

Weak formulation of the Poisson equation with discontinuous source

We consider the equation $$ -\Delta u =f\chi_\Omega\quad \mbox{in }\mathbb{R}^3, $$ where $\Omega$ is a bounded and smooth domain, $\chi_\Omega$ is the charasteristic funtion of the set $\Omega$ and ...