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

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1answer
121 views

RObin problem (Laplace equation)

Let $\Delta u = 0 $, $ \frac{\partial u}{\partial v}(x) + \alpha u(x) = 0 $ be the Laplace equation with Robin conditions. How do I prove it has at most one solution. If I could prove that any two ...
2
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0answers
51 views

Energy Equality: how to derive the energy equalities?

Sorry! I'm wondering if there is some way of deriving the energy equality of a given equation. If possible, I would like to see a specific example, which is $ u_ {tt}- \Delta u + u^3 = 0 $ ($ u $ is ...
2
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1answer
566 views

weak solution of biharmonic equation

Consider $U$ a open and bounded subset of $R^n$, with smooth boundary. A weak solution for the problem : $$ \Delta^2 u = f \ \in \Omega \ and \ u=\frac{\partial u}{\partial\nu} = 0 \text{ in } ...
4
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1answer
563 views

Laplace equation with periodic boundary conditions

Suppose I had the problem $$\nabla^2 u(x,y) = 0 \text{ in } \Omega=[0,1]^2$$ with the periodic boundary condition: $u(0,y)=u(1,y)$ and $u(x,0)=u(x,1)$ Note that I'm purposefully omitting periodic ...
3
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1answer
864 views

Heat equation in polar co-ordinates

I was studying the heat equation, when i saw a new variant of it. Here's the statement: "the edge $r=a$ of a circular plate is kept at temperature $f(\theta)$. The plate is insulted so that there is ...
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0answers
98 views

Approximate the solutions to the following parabolic partial-differential equations.

Use the: a)forward difference method b)backward difference method c)Cranck-Nicolson algorithm to approximate the solution of the parabolic partial differential equation: $$\frac{\partial u}{\partial ...
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0answers
58 views

Showing transition density for Brownian motion satisfies the heat equation

Let $B$ be a Brownian motion in $\mathbb R^n$. Let $U$ be a (sufficiently regular) open subset of $\mathbb R^n$. Let $\tau_U$ be the time $B$ exits $U$. Let $D_\epsilon(y)$ denote the ball of radius ...
3
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1answer
167 views

Advices on learning SDE/PDE for junior undergrad

Everyone. I am about to take an ODE Course in the summer. I wonder if it will help my understanding in stochastic differential equation and partial differential differential equation. My future plan ...
3
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0answers
70 views

Confused by a (standard?) gradient estimate.

This sort of inequality appears frequently in a paper I'm reading, but never with any justification. Take $u(x) \in H^{1}(B(x_{0}, r))$ and a cutoff $\phi(x) \in C_{0}^{\infty}(B(x_{0}, r))$ where ...
5
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1answer
164 views

Self adjointness of an elliptic differential operator

Let $A:D(a)\to L^2(\mathbb{R}^n)$ be an elliptic partial differential operator $$ A(f)=\sum_{i,j=1}^{\infty}\partial_{x_j}(a_{ij}(x)\partial_{x_i}f) $$ where $a_{ij}\in C^{\infty}_b(\mathbb{R}^n)$, ...
3
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1answer
125 views

variational question

Let $\Omega$ a bounded domain, connexe and regular, and let $f \in L^2(\Omega).$ Let the variational problem: Find $u \in H^1(\Omega)$ such $$\int_{\Omega} \nabla u \nabla v dx + (\int_{\Omega} u ...
2
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2answers
121 views

Solve system of Charpit equations

I have a partial differential equation $$u_xu_y = xy \mbox{ with } u(0,y)=y+1$$ Calling $u_x = p, u_y = q$ gives the following Charpit equations $$\frac{dx}{dt} = q, \frac{dy}{dt} = p, \frac{dp}{dt} ...
5
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1answer
138 views

Calculus on surfaces and chain rule

Define the surface gradient operator on any surface $S$ as $$\nabla_S f = \nabla f - \nabla f \cdot \nu_S \nu_S$$ where $\nu_S$ is the outward unit normal on $S$. Let $T:S_1 \to S_2$ be a $C^2$ ...
6
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1answer
300 views

solving the PDE of a beam under a moving load using Laplace transform

Solve this PDE using Laplace transform : $$ EI {\partial^4 y(x,t)\over\partial x^4}+\mu {\partial^2y(x,t)\over\partial t^2}= F(x,t) $$ $$F(x,t)= P\delta(x-u) / ...
4
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1answer
79 views

A problem about mollification

The problem is : Given $M > 0$ a constant, show that exists $\phi \in C^{\infty}(R)$ with the following properties: i) $\phi(x) = x , \forall x \in [-M,M] $ ii) $ 0 \leq\varphi^{'}(x) \leq 1, ...
4
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1answer
243 views

Proof of weak maximum principle.

I read a proof in my book on pde which I find a bit strange. Let $\Omega$ be some bounded domain. For $f\in\mathscr C^2(\Omega)\cap\mathscr C^0(\Omega)$ satisfying $\Delta f\geq0$ it holds that ...
2
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2answers
571 views

Solve Laplace equation in the upper half plane

I need to solve \begin{eqnarray} u_{xx} + u_{yy} = 0 \quad \quad y>0 \quad -\infty < x< \infty \end{eqnarray} With boundary condition \begin{eqnarray} \frac{\partial u(x,0)}{\partial y} = ...
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2answers
156 views

Definition of $C^k$ boundary

Can someone give me a resonable definition of $C^k$ boundary, e.g., to define and after give a brief explain about the definition. I need this 'cause I'm not understanding what the Evan's book said. ...
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2answers
2k views

Conceptual difference between strong and weak formulations

What are the conceptual differences in presenting a problem in strong or weak form? For example for a 2D Poisson problem the strong form is: \begin{split}- \nabla^2 u(\pmb{x}) &= ...
7
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2answers
119 views

When does a PDE solve a variational problem?

I understand that for a functional $J[f]$ on the space of differentiable functions $f$ on some domain, setting $\delta J[f]|_{f=f_0} = 0$ yields a (possibly nonlinear) partial differential equation in ...
2
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1answer
105 views

a gentle introduction to well posedness issues in fluid dynamics ( reference request)

I am looking for an introduction, either books, survey papers, to well posedness issues in fluid dynamic systems like the Navier Stokes, Euler equations or Stokes' equation and / or even for other ...
3
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3answers
223 views

Introduction to Pseudodifferential operators

I'm interested in elementary introduction to pseduodifferential operators and its application to hyperbolic PDE's. I know measure theory, Fourier analysis and some elementary(linear) hyperbolic PDE's ...
4
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1answer
160 views

Bounded vector field has globally defined flow

Let $X$ be a vector field on $\mathbb R^n$, and suppose that $\|X\|$ is bounded, where the norm is taken with respect to the Euclidean inner product. I am trying to show that $X$ has globally defined ...
2
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1answer
120 views

How to show $C_p^k([-\ell, \ell])$ is not a Banach space?

I need to show the space $$C_p^k([-\ell, \ell])=\{f\in C^k(\mathbb R; \mathbb C); f(x+2\ell)=f(x), \forall x\in\mathbb R\},$$ is not a banach space with the norms ...
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1answer
837 views

Solving Laplace's equation using finite differences

I having coding in MatLab to approximate solutions to Laplace's equation in 2D using finite differences. I was able to do it without much problem. I learnt about how to implement this using this: ...
2
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1answer
473 views

Use Lax-Milgram theorem to prove the existence of weak solution for an elliptic equation

Let $\Omega$ an open bounded and regular domain to $\mathbb{R^n}$ and let $\{\overline{\Omega_1},\overline{\Omega_2}\}$ a partition of $\Omega.$ $\bar{\Omega} = \bar{\Omega_1} \cup \bar{\Omega_2}.$ ...
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0answers
37 views

Regarding traffic flow and position

I need help starting this problem I don't understand how to utilize the equation for q, traffic flow. I understand that this problem wishes to have one utilize the method of characteristics but how ...
1
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1answer
144 views

Mean Value Property of Harmonic Functions

I can't prove this theorem: "Let $\Omega$ is a bounded domain, $u\in C^2(\Omega)$ satisfy $\Delta u=0(\geq0,\leq0)$, then for any ball $B=B_R(y)\subset \subset \Omega$, we have ...
1
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1answer
295 views

Determine Greens function inside a semi circle

I need to determine Greens function $G(x,x_0), x \mbox{ and }x_0 \in \mathbb{R}^2$ inside a semi-circle $(0<r<a, 0<\theta<\pi)$ with $\nabla G = \delta(x - x_0)$ and $G = 0$ on the ...
3
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0answers
131 views

Reference: Hölder regularity for linear elliptic equation, Neumann problem

I checked the Book of Gilbarg Trudinger on elliptic equations for (Hölder)-regularity results on weak solutions for elliptic pde. In particular, my equation is of the form $$\nabla \cdot ...
0
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1answer
26 views

Convergence of a series of a given metric..

I'm trouble with a metric defined over a given set: Consider $\mathcal{P}=C_p^\infty([-\pi, \pi])$, that is, $\mathcal{P}$ is the set of all infinitely differentiable functions $f:\mathbb R\rightarrow ...
2
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0answers
56 views

What is the correct canonical form of this elliptic PDE?

$u_{xx}+2u_{xy}+(1+x^2)u_{yy}=1$ where $\eta = y-x$ and $\psi = \frac{1}{2}x^2$ such that the characteristic curves are given by $\eta\pm\psi i=\text{constant}$. I have tried and achieved ...
3
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1answer
318 views

1D Green's function: from interval to infinite line

Let's consider two problems for diffusion equation. The first one: $$ u_t = a^2u_{xx},\qquad 0<x<l,\quad 0<t\leq T $$ $$ u(x,0) = \phi(x), \qquad 0 \leq x \leq l $$ \begin{equation} ...
1
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2answers
72 views

About smooth approximation in a Sobolev space

I want to prove the following fact : Consider $\Omega \subset R^n$ a bounded and open set. Let $v \in H^{1}_{0}(\Omega)$ a nonnegative function. Then exists a sequence $v_m$ in ...
3
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1answer
115 views

How to solve following multidimensional equation?

Let's have an equation $$ u_{t} = \Delta u ,\quad \mathbf x є R^{n},\quad t > 0, \quad u(\mathbf r , 0) = e^{-(x_{1} + ... + x_{n})^{2}}. $$ How to solve it? I tried to reduce the equation to the ...
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0answers
98 views

Scattering of waves by an obstacle

I wish to study the paper by Melroe and Taylor: Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle. Adv in Math, Vol 55 (3), 1985, 242–315. ...
5
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2answers
149 views

Sobolev spaces - about smooth aproximation

Consider $\Omega $ a open and bounded set of $R^n$ . Let $u \in H^{1}(\Omega)$ a bounded function. I know that exists a sequence $u_m \in C^{\infty} (\Omega) $ where $u_m \rightarrow u$ in ...
2
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3answers
161 views

Poisson equation

Find the solution of the Poisson equation inside a ball $B(0,1)$, i.e.: $$\begin{cases} \Delta u=x^{2} \ \ \ in \ B(0,1)\\u=3 \ \ \ on \ \partial B(0,1)\end{cases}$$ ($x^2=x\cdot x$, $x\in ...
1
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1answer
133 views

Centered-Difference Scheme to approximate BVP

I have the following boundary value problem: $\frac{\partial ^2u}{\partial x^2}+\frac{\partial ^2 u}{\partial y^2}=0, \; 1<x<2, \;0<y<1;$ $u(x,0)=2\ln x, \; u(x,1)=\ln(x^2+1), \; 1 \leq ...
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0answers
34 views

Difference Schemes, Accuracy and Consistency

I have the following question: Say $u_j^{n+1}=(1-2\alpha-2\beta)u^n_j+\alpha(u_{j+1}^n+u^n_{j-1})+\beta(u^n_{j+2}+u^n_{j-2})$ is a scheme for $u_t=u_{xx}$. When $\Delta t/(\Delta x)^2$ and ...
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0answers
76 views

$u\in W^{1,p}(B)$ implies $u\in W^{1,p}(\partial B_t)$ for almost $t\in (0,1]$?

Suppose that $u\in W^{1,p}(B)$ where $B=B(0,1)\subset\mathbb{R}^N$ and $p\geq 1$. It was showed on this post (as an application of Fubini's theorem) that for almost $t\in (0,1]$ $$u_{|\partial ...
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1answer
93 views

Unicity of solution of pde

Let the pde $$\dfrac{\partial^2 u}{\partial t^2} - \dfrac{\partial^2 u}{\partial x^2}=f(x)$$ The question is: Find the limit condition such that this pde admit a unique solution in $[a,b] \times ...
4
votes
1answer
156 views

Divergence theorem in volume integral

We have a partial differential equation \begin{equation} \nabla \cdot (p_1^2\nabla\alpha)=0\,. \end{equation} Question: from this equation how can I write the following condition? \begin{equation} ...
1
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1answer
301 views

Constants in Laplace's equation for a cube

I'm working a Laplace's equation $\Delta F=0$ for a cube in Cartesian coordinates $((0,0,0),(a,a,a))$ and after separation I have $$\frac{X''(x)}{X(x)} +\frac{Y''(y)}{Y(y)}+\frac{Z''(z)}{Z(z)} = ...
4
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2answers
68 views

curl of what yields $(0,s^{-1},0)$ in cylindrical coordinates?

In cylindrical coordinates $(s,\theta,z)$, what function $\mathbf{A}$ has the property $$\nabla\times \mathbf{A} = (0, \frac{1}{s} , 0) $$ I know generally that $$\nabla\times \mathbf{A} = ...
2
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1answer
264 views

Change of variables in a convolution..

I'm in trouble with change of variables in a convolution: Definition: The convolution of two $2\pi$-periodic functions $f$ and $g$ is defined as $$(f*g)(x)=\frac{1}{2\pi}\int_{-\pi}^{\pi} ...
2
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2answers
56 views

How do you find the second constant in a parabolic pde solution?

Okay, so I am given the following Parabolic PDE: $y^2u_{xx}-2xyu_{xy}+x^2u_{yy}=x^{-1}y^2u_x+y^{-1}x^2u_y$ I find the characteristics to be: $\frac{dy}{dx} = \frac{-x}{y}$ and therefore $\psi(x,y) ...
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1answer
375 views

Relationship between Turing bifurcation, saddle-node bifurcation, and Hopf bifurcation?

Quoting from http://jxshix.people.wm.edu/2009-harbin-course/mississippi-bifurcation-2.pdf a Turing bifurcation occurs when for an ODE and related PDE $u' = f(u,v), v' = g(u,v)$ $u_t = d_1 \nabla ...
3
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0answers
109 views

Prove the equivalence of the solution existence of Stokes equation's variational problem to the original problem

I have this exercise. We put $X = L^2(\Omega)^4 \times L^1_0(\Omega), M = H^1_0(\Omega)^2.$ $$a : X \times X \rightarrow \mathbb{R}; ((\sigma,p),(\tau,\alpha)) \rightarrow (\sigma \otimes \tau)$$ ...
9
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1answer
259 views

How do we know that PDE solutions obtained via separation of variables are the only ones?

You can find solutions to, for example, the 1D Schrödinger equation $-\frac{\hbar^2}{2m}\Psi_{xx}(x,t) + V(x, t)\Psi(x, t) = i\hbar\Psi_{t}(x,t)$ by assuming solutions of the form $\Psi(x,t) = ...