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

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pde problem in cartesian system with 2 neumann BCs

I need to solve Laplace's equation ($\DeltaΦ=0$) in the cartesian coordinates with 2 functions as BC's and 2 Neumann BC's (solving for Φ): http://i.stack.imgur.com/AuBtR.png the Neumann BC's are on ...
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2answers
25 views

Yamabe's equation

This is PDE Evans, 2nd edition: Chapter 9, Exericse 8(a). (a) Assume $n \ge 3$. Find a constant $c$ such that $$u(x) := (1+|x|^2)^{\frac{2-n}2}$$ solves Yamabe's equation $$-\Delta u = ...
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21 views

Solution to differential equation of function of two variables

Very simple question about differential equations, but I couldn't find anything online. Let $f(x,z)$ be a function of two variables that satisfies: $af+bf_x+cf_z+df_{xx}+ef_{zz}+gf_{xz}=q(x,z)$ ...
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0answers
27 views

Apply Banach's Fixed Point Theorem to a nonlinear boundary-value problem

I am attempting Exercise 5 in Chapter 9 of PDE Evans, 2nd edition: Consider the nonlinear boundary-value problem $$\begin{cases}-\Delta u + b(Du)=f & \text{in }U \\ \qquad \qquad \quad \, ...
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1answer
17 views

Heat Equation on $[0,l]$ with Neumann boundary conditions

I was reading the following pdf about the heat equation on an interval $[0,l]$ with Neumann conditions, http://texas.math.ttu.edu/~gilliam/fall03/m4354_f03/heat_N_web/heat_ex_homo_neum.pdf i.e. ...
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4answers
28 views

Proof of the construction of Dirac Delta

The Dirac Delta function pops up in a wide variety of applications, especially in applications that require Laplace and Fourier transforms. But my question is: what's the proof that the distribution ...
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0answers
18 views

Gradient w.r.t. boundary conditions in PDE

I am trying to solve the following problem. Suppose I have a field $\Phi(r)$, which is the solution to a partial differential equation: $\mathcal{L}\Phi(r) = s(r)$, as long as $r \neq r_0$ Here ...
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0answers
37 views

How to prove $H_0^1(\Omega)=H_0(div;\Omega)\cap H_0(curl;\Omega)$

It was shown in the book "Finite Elements Methods for Navier-Stokes Equations" by Girault and Raviart that $$H_0^1(\Omega)=H_0(\operatorname{div};\Omega)\cap H_0(\operatorname{curl};\Omega).$$ The ...
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1answer
21 views

a question on the extension of an operator?.

It is known that $C_0^{\infty}(\Omega)$ is dense in $W_0^{1,p}(\Omega)$ where $\Omega$ is a bounded domain of $\mathbb{R}^n$. Let $T:C_0^{\infty}(\Omega)\rightarrow\mathbb{R}$ be a continuous linear ...
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0answers
23 views

Harmonic function in $\mathbb{R}^n$ is not one-to-one, for $n\geq 2.$

Le $u:\mathbb{R}^n\to\mathbb{R}$ a harmonic function. Prove that if $n\geq2$ then every $y\in Im\{u\}$ is attained infinite times, but it's not true for $n=1$. I no have idea to start, someone has a ...
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PDE: Laplace equation

Any idea to start this problem I got stuck with: $$-\Delta \phi +\beta^2\phi=0,\quad \beta>0 \quad (\beta \,constant)$$ in $\quad D=\{(x,y)\in \Bbb R^2:x\in \Bbb R, \quad y\in [-h,0]\}\quad$ with ...
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2answers
47 views

PDE: Heat equation problem

I'm trying this PDE: $$u_t = u_{xx} + g(x);\quad x\in[0,\pi]$$ With boundary conditions: $$u_x (0,t)=u_x(\pi,t)=0$$ And initial condition: $$u(x,0)=f(x)$$ I think variable separation proposing a ...
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1answer
21 views

Simplifying PDE by change of variables

How does one use the substitution $\hat{x}=e^{-r(T-t)}x$ to get from, $$ \frac{\partial \hat{F} }{\partial t} + \frac{1}{2}\sigma(x, t)^2x^2\frac{\partial^2 \hat{F} }{\partial x^2}+rx\frac{\partial ...
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1answer
21 views

Can we change the Laplace equation to the wave equation with a linear substitution

I would like to know, is it possible to make a linear change of dependent and independent variables such that Laplace's Equation $u_{xx}+u_{yy}$ transforms to the Wave Equation ...
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1answer
16 views

Deriving the Helmholtz equation in polar form

The two dimensional helmholtz equation is $$\frac{\partial ^2 \phi}{\partial x^2}+\frac{\partial^2 \phi}{\partial y^2}+k^2 \phi=0$$ and I have that $$\nabla^2 u(r,\theta)=\frac{\partial^2 u}{\partial ...
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1answer
25 views

Remark below the proof of Rellich-Kondrachov compactness theorem

The following if a remark left at the end of the proof of the RK compactness theorem p.274 Evans. Here I think I got through everything more or less, but have trouble proving the final claim $$ ...
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0answers
9 views

2D wave equation: decay estimate

Let $u$ be a solution to $$\begin{cases}\Box u =0,\; \; (t,x)\in \mathbb{R}_+ \times \mathbb{R}^2\\(u,u_t)\restriction_{t=0} = (f,g),\end{cases}$$ where $f,g$ are smooth functions with compact ...
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1answer
18 views

Uniqueness for the wave equation on an interval

We search to prove that the following problem admits a unique solution. $$ \begin{cases} \dfrac{\partial^2 u}{\partial t^2}= a^2 \dfrac{\partial^2 u}{\partial x^2}\\ u(0,t)=u(l,t)=0\\ u(x,0)=f(x)\\ ...
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0answers
9 views

Harmonic functions locally null on connected open set

Let $u$ be a harmonic function on $U$ connected open set of $\mathbb{R}^n$ and suppose there is a open set $V\subset U$, such that $u(x)=0$ for every $x\in V.$ Show that $u=0$ in $U$. So, I tried to ...
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1answer
21 views

Solution to the 1-D heat equation

In solutions to the heat equation $u_t(x,t)=cu_{xx}(x,t)$ I've seen they've used the set of boundary conditions $$u(0,t)=u(L,t)=0$$ $$u(x,0)=u_0(x)$$ These set of boundary conditions is set to model ...
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2answers
16 views

A general solution of a partial differential equation with $f(x,y)$

I need to find a general solution to such a PDE: $$u_x-u_y=f(x,y)$$ I am able to find a solution if $f(x,y)=0$ or $f(x,y)=u$. But I have no idea how to get the general solution. Has anybody got any ...
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0answers
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Notation for vector valued integration

Suppose we have a vector field $\mathbb v=(v^1,v^2)$ on the two dimensional torus $\mathbb T^2$ and we wish to compute $\int_{\mathbb T^2} \mathbb v \cdot \Delta \mathbb v $, where the Laplacian acts ...
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2answers
30 views

Finding the particular solution of a pde.

I have solved a PDE up to the point of finding the particular solution. I am trying to find the constant $$C_n$$ I have the expression $$3x-x^2=\sum_{n=1}^{\infty} C_{n} \, \sin\left(\frac{\pi n ...
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1answer
18 views

Is $C^\infty_0(\Omega)$ complete with the norm $\|u\|_\Delta:=(\|u\|_{L^2(\Omega)}^2+\|\Delta u\|_{L^2(\Omega)}^2)^{1/2}?$

Let $\Omega$ be an open subset of $\mathbb R^n$. Is it true that $C^\infty_0(\Omega)$ is complete with the norm $$\|u\|_\Delta:=(\|u\|_{L^2(\Omega)}^2+\|\Delta u\|_{L^2(\Omega)}^2)^{1/2}?$$ Above ...
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2answers
25 views

Numerically solving a transport equation

I would like to solve this transport PDE numerically : $$ \partial_t f + v(f) \partial_x f = 0 $$ What I would want to do is "freeze" the velocity $v$ and solve a classical transport equation by ...
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0answers
22 views

wave equation on a circular domain

Consider the wave equation for the displacement $$\text{u(r,$\theta $,t)}$$ in a circular domain $$\text{0 $<$ r $<$ a, -$\pi $ $<$ $\theta $ $<$ $\pi $}$$ How do I use the separation ...
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25 views

What is the motivation for solving the Bessel equation.

My course is highly theoretical. For most part, we're taught to solve equations. But as a Physics student, I would very much like to know the motivation behind seeking the solution to the Bessel's ...
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1answer
16 views

Ordinary point of a Bessel DE

The Bessel DE: $$z^2\frac{\text d^2f}{\text{d}z^2}+z\frac{\text{d}f}{\text{d}z}+\left(z^2-m^2\right)f = 0.$$ The Bessel DE can be rewritten as: $$\frac{d^2f}{\text{dz}^2} + a(z)\frac{df}{ dz } + ...
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1answer
14 views

Reducing a Bessel's differential equation to a more 'useable' form

Suppose the given equation is: $$r^2\frac{\text d^2f}{\text{d}r^2}+r\frac{\text{d}f}{\text{d}r}+(\lambda r^2-m^2)f = 0$$ My text demonstrates the following: Let $$\text{z = }\sqrt{\lambda }r$$ So ...
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1answer
29 views

A domain in $\mathbb{R}^n$ with $C^2$-boundary satisfies an “outer spherical condition”

Let $\Omega\subseteq\mathbb{R}^n$ be a domain and $\partial\Omega\in C^2$, i.e. $\Omega=\overline{\Omega}^\circ$ For all $x_0\in\partial\Omega$, there exists a neighbourhood $U\subseteq\mathbb{R}^n$ ...
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1answer
13 views

reducing a pde to a canonical form

I'm really struggling with this one and I can't seem to find what's wrong with my approach. I am given a PDE in the form $$U_{xx} + x y U_{yy} = 0,$$ and I am supposed to bring it to its canonical ...
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0answers
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Characteristics and additional conditions for differential equation

I need to solve such a DE: $$(1+x^2)u_x+u_y=0$$ And then I need to draw its characteristics. The second part of the task says: Write three additional conditions such that this equation: Has one ...
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1answer
14 views

PDE of the form $x \partial_x T - y \partial_y T = F(x,y)$ where $F$ is a given function.

Is there a known solution, or technique, for solving the following PDE? $x \partial_x T - y \partial_y T = F(x,y)$ Here, $F$ is a given smooth function $\mathbb R^2 \to \mathbb R$, and $T: \mathbb ...
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1answer
29 views

What is the definition of ``2nd-order quasilinear parabolic'' ? for partial differential systems?

I have to know why the mean curvature flows are 2nd-order quasilinear parabolic. Let $\Omega\subset\mathbb{R}^n$ be a bonded domain (or a smooth manifold of $n$ dimensional) and $N\geq 2$. When the ...
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1answer
16 views

Neumann boundary conditions for Laplace equation with Raviart-Thomas elements

I am working on creating a finite element model for the Darcy equation using Raviart-Thomas elements and the mixed hybrid formulation. The problem in mixed form is this: $\mathbb{K}\nabla p = ...
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1answer
27 views

Integrating a Poisson kernel in $n$ dimensional unit sphere

Let \begin{equation*} P(x,y)=\frac{1}{\omega_n R} \frac{R^2-|x|^2}{|x-y|^n} \end{equation*} be a Poisson kernel where $x$, $y$ are in $R^n$, $|x|<R$, $|y|=R$, $\omega_n$ is area of n dimensional ...
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2answers
67 views

Why does $D \det(I)B=\frac{d}{d t}\det(I+t B)|_{t=0}$ hold?

The following problem is from Taylor's PDE I. I do not get why $D \det(I)B=\frac{d}{d t}\det(I+t B)|_{t=0}$ hold. Since $\det(I)=1$ and $D$ is $n\times n$ matrix, the left side seems to be a matrix, ...
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2answers
47 views

If $\frac{\partial F^i}{\partial x^j}=0$ on a connected open set, is $F$ constant?

Let $U$ be open in $\mathbb{R}^n$ and let $$F:U\to \mathbb{R}^m$$ be a smooth map, i.e. $F\in C^\infty(U)$. It is easy to prove that if $U$ is convex and $$\frac{\partial F^i}{\partial x^j}=0\tag{1}$$ ...
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0answers
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How to construct the explicit solution of this boundary-value problem of the PDE of the first order

I want to find the explicit solutions of this boundary value problem of the first order PDE \begin{cases}\tag{1} \text{PDE: } \frac{1}{4}(u_x)^2+u u_y=u(x,y),\quad y\neq \frac{x^2}{2},\\[2ex] ...
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1answer
13 views

Numerical method with a time derivative boundary condition

I'm trying to reproduce a result from a paper I'm reading using a numerical scheme that I'm coding myself. The equation is a reaction diffusion PDE. $$\frac{\partial M}{\partial t}=\frac{\partial^2 ...
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1answer
21 views

If $u_k$ converges uniformly on $\partial \Omega$, does it converge uniformly on $\Omega$?

Let $u_k$ be continuous on $\overline \Omega$ and harmonic in $\Omega$. Suppose $u_k$ converges uniformly on $\partial \Omega$. Can we conclude that $u_k$ converges uniformly on $\Omega$?
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1answer
21 views

Why $\int_{\partial(\Omega \backslash B_p(y))}v(x) \Delta u(x)dx=\int_{\Omega}v(x)\Delta u(x)dx$ as $p \to 0$?

Suppose $v(x)= \Gamma(x-y)=\frac{1}{n(2-n)w_n}x^{2-n}$ when $n>2$. Then $\int_{\partial(\Omega \backslash B_p(y))}u\frac{\partial v}{\partial n}=\int_{\partial\Omega}u\frac{\partial v}{\partial ...
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1answer
58 views

Caccipoli Inequality

I am reading the book Elliptic Partial Differential Equation by Fanghua Lin and I got stuck at the lemma 1.36 ( the Caccipoli inequality). The conditions of this lemma are: $u\in C^1(B_1)$ ($B_1$ is ...
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0answers
16 views

Heat Transfer FEM 2D PDE Matlab [on hold]

I'd love to know if this looks right in any way since I'm unfamiliar with Heattransfer. The Domain is correct. The heat transfer coefficient is 1. The Dirichlet BC is u(0,x) = 1 and u(y,1) = 0 ...
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32 views

Nonnegative harmonic functions

Suppose $U \in \mathbb{R}^n$ is an open domain, and $u\in C^2(U) \cap C(\bar{U})$ such that $\Delta u = 0$ in $U$. I'm working on a couple of problems pertaining to the mean value formula/harmonic ...
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1answer
41 views

Radial symmetry

This is a full theorem and proof copied from PDE Evans, 2nd edition, pages 558-559. My two questions about two parts of the proof are on the bottom of this post. THEOREM 2 (Radial symmetery). Let ...
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0answers
24 views

Help in finding a paper on nonlinear Schrodinger equations [closed]

I'm looking for the following paper: Authors: Baillon, Jean-Bernard; Cazenave, Thierry; Figueira, Mario Title : Equation de Schrodinger non linéaire. (C. R. Acad. Sci. Paris Ser. A-B 284 ...
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1answer
26 views

Semilinear Poisson PDE - proving a (hopefully) simple inequality

This is from page 557 of PDE Evans, 2nd edition. My question is at the bottom of this post, but for now, here is some context for my question: LEMMA 2 (Boundary estimates). Let $u \in ...
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1answer
24 views

Is ODE essentially different from PDE or actually PDE is the generalization of ODE? [closed]

Is ODE essentially different from PDE or actually PDE is the generalization of ODE? If so, how are they essentially different from each other?
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10 views

Applying different boundary conditions in a quasi Helmholtz problem.

I have tried to solve this exercise from Applied Partial Differential Equations-Richard Haberman, Consider the two-dimensional eigenvalue problem $$ \nabla^2\phi+\lambda\sigma(x,y)\phi=0 $$ ...