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

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Show that the boundary layers diffuse out from the plate with speed $\sqrt{\frac{\nu}{t}}$

I was wondering if someone could help me with this problem. I think it's quite simple, but I'm not sure how to do it! An infinite horizontal plate moves with speed $U$ in its on plane relative to ...
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1answer
17 views

Separation of variables to get two separate ODEs

[ so far I have: u(x,t) = f(t) h(x). h(x) df(t)/dt+ f(t) d^h(x)/dx^2=- ρ f(t) h(x) 1/f(t) df(t)/dt + 1/h(x) d^2 h(x)/dx^2 =- ρ is my solution so far correct? I'm confused from here on, what do I ...
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Boundary conditions for a radiative heat transfer problem

Consider the heat equation $$ \frac{\partial T}{\partial t} - a\Delta T + \mathbf v \cdot \nabla T = S $$ where $S$ is a source term dependent of the radiation intensity $I$ and the temperature $T$. ...
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19 views

Verification of some conditions and facts of the Laplacian on a Riemannian manifold

I came across the following introduction to a paper I was reading: "Let $L$ be a second-order, elliptic, differential operator, with smooth coefficients, on a Riemannian manifold $M$. Assume $-L$ is ...
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Choosing between SOV/Green's functions/Laplace transform for solving PDE - Guideline for choosing the most appropriate method?

Forgive me if this questions seems silly, but I have a question which is keeping me busy. I'm not really looking for a mathematical proof (but it is welcome), however I'm more looking for guided ...
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2answers
36 views

Solving Simple Partial Differential Equation

I don't remember how i can solve this simple partial differential equation. Can someone help me? $$x\frac{\partial \phi}{\partial x}+y\frac{\partial \phi}{\partial y}+ (\alpha+1-x)\phi =0$$ Update ...
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2answers
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uses of Riemannian geometry for questions not related to Riemannian geometry

Poincaré conjecture (whose formulation does not make use of notions from Riemannian geometry: "Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere") was eventually proved using ...
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1answer
27 views

Solution of $xu_x + yu_y = 0$

I have the first oder PDE $$xu_x + yu_y = 0 \; \text{on} \; \mathbb{R}^2$$ and I found the solution of that PDE is $$u(x,y) = f\left(\frac{y}{x}\right) = e^C = K$$ which is a constant solution. So, ...
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19 views

Differential Operator simplifying

I read in chapter 2 "Weisner Method" in the book "Obtaining Generating Functions" by Elna Browning McBride In Sec 5" The extended form of the group generated by B and C " I did not understand ...
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1answer
22 views

Duals of Sobolev Spaces vanishing on parts of the boundary

I am revising for a Finite Elements course and have the following question about the definition of $H^{-1}$. Let $D\subseteq \mathbb{R}^2$ be bounded Lipschitz domain and let $\Gamma_0 \subseteq ...
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19 views

How to scale a problem involving heat and the flow of a viscous fluid?

I recently got set this problem and I was wondering if anyone would be able to give me some help on the later parts. An incompressible thermal conducting fluid is contained between two infinite ...
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2answers
223 views

Can I follow a graduate course in PDE without having studied ODE

Hi I am considering taking the first course on Partial differential equations at my university next semester. I have already taken a first course on functional analysis . I haven't taken a proof based ...
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31 views

Regularity of elliptic PDE in terms of weighted Sobolev space

There seems to be a whole industry of weighted ver. of the Poincaré's inequality (Weighted Poincare Inequality) I wonder if there are results like, weighted $L^2$ equivalent of (interior/boundary) ...
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1answer
20 views

Poisson functional on bounded domain

I was wondering if it is actually clear that on bounded domains the Poisson integral is bounded from below: $$I[u]=\int_{\Omega} \left( \frac{1}{2}\lvert \nabla u \rvert^2 - u\rho \right)\, dx,$$ I ...
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13 views

PDE Singularity and time of shock

Given the following PDE: $$∂p/∂t+p*∂p/∂x=0$$ $$p=p(x,t)$$ $$p(x,0)=e^{-x^2}$$ It is requested to find its solution, show that it is not well behaved by verifying what happens to $∂p/∂t$ and $∂p/∂x$ ...
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1answer
20 views

Damped wave equation on $\mathbb{R}^{2}/2\pi\mathbb{Z}^{2}$

Let $a \in (0, 1)$ and let $u$ satisfy \begin{align*} u_{tt} - \Delta_{x}u + au_{t} &= 0\\ u(x,0) &= 0\\ u_{t}(x, 0) &= f(x) \end{align*} with $t \geq 0$, $x \in ...
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1answer
23 views

Domain monotonicity of eigenvalues

Let $\Omega_{1}$, $\Omega_{2}$ be subsets of $\mathbb{R}^{2}$ with smooth boundary and $\Omega_{1} \subsetneq \Omega_{2}$. Let $-\lambda_{1}$ and $-\lambda_{2}$ be the smallest (in magnitude) ...
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35 views

Coupled partial differential equation, with boundaries specification

Please, help me to find a books or samples to learn how to solve such coupled equations $$\begin{eqnarray} \frac{\partial T_1(x,t)}{\partial t}&=& \alpha_1 \frac{\partial^2 T_1(x,t)}{ ...
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14 views

PDE Variable transform (integral) Chain Rule

Can somebody help me with the variable transform where the new variable is integral variable? This is a moving boundary problem where the radius of particle, $R(t)$ changes with time. The equations ...
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2answers
60 views

How to simplify $ \int_{\Bbb{R}^2}\Delta\varphi(x)\log|x|^2\ dx $ using Green's indentity?

Let $\varphi\in C_c^\infty(\Bbb{R^2})$ (infinitely differentiable functions with compact support) and consider $$ I=\int_{\Bbb{R}^2}\Delta\varphi(x)\log|x|^2\ dx, $$ the existence of which is ...
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Existence and uniqueness of Dirichlet problem

Let $U= \{x \in \Bbb R^n: |x|>1 \}$. Suppose $u \in C^2 (U) \cap C(\bar U)$ is a bounded solution of the following Dirichlet problem: $\Delta u=0 \in U$ and $u=\phi$ on $\Gamma=\{x \in \Bbb R^n: ...
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15 views

How do I find out the Partial Differential Equation for the given expression? [on hold]

$$z=x^2f(y)+y^2g(x)$$ Well, I found out that $$ z_x=2xf(y)+y^2g'(x)\\ \ z_y=x^2f'(y)+2yg(x).$$ How do I relate them to get the partial differential equation?
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1answer
28 views

Help in understanding the notation

I am reading the paper in this link https://dl.dropboxusercontent.com/u/20327748/99-16.ps.pdf Please help me in the notation used in page 5, $(M \vee \phi_n)\wedge M$ it is in line 2 of page 5. ...
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1answer
44 views

solution of 1st order PDE

Find the solution of PDE, $$u_xu_y = u$$ with the initial condition $u(x,0) = 0$ in the domain $x \geq 0$ and $y \geq 0$. I have try the method of characteristic, but it seems like not working for ...
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1answer
14 views

Let $u(x)$ be harmonic for $x \in \Bbb R^3$. Suppose that $\nabla u \in L^2 (\Bbb R^3)$. Prove that $u$ is a constant.

Let $u(x)$ be harmonic for $x \in \Bbb R^3$. Suppose that $\nabla u \in L^2 (\Bbb R^3)$. Prove that $u$ is a constant. I tried with Green's equation but it didn't work. Could anyone tell me which ...
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1answer
20 views

Some clarification on nonlinear PDEs

In Strauss' book $\textit{Partial Differential Equations: An Introduction}$, one of the important PDEs listed on p. $2$ is the shock wave equation given by $$u_x + uu_y= 0 \text{.}$$ It is nonlinear, ...
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1answer
23 views

Reflection from the upper half ball to the whole ball is harmonic

I have a question about problem 9(b) in Chapter 2 of Evans' PDE book. It says if we have $u$ is harmonic in the open upper half ball $U^+$ and $u\in C^2(U^+)\cap C(\bar{U^+})$, $u=0$ for $x\in ...
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1answer
40 views

Use the Laplace Transform to solve the following PDE.

I need to use the Laplace Transform to solve the following PDE, but I don't think I'm doing it correctly. $u_{t}(y,t)=\nu\nabla^2 u(y,t)$ with $u(0,t)=u_{0}$ and $u(y,0)=0$. What I have so far: ...
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24 views

Show that boundary layers diffuse out from the plate with speed $\sqrt{\frac{\nu}{t}}$

I was wondering if somebody would be able to help me with this problem. I know how to solve it using dimension arguments but I'm unsure what is meant by 'transform techniques'. Any help would be ...
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2answers
32 views

Difference between two solution of inhomogeneous linear equation

Show that the difference between two solutions of an inhomogeneous linear equation $Lu =g$ with the same $g$ is the solution of the homogenous equation $Lu=0$ I know the definition of linearity, but ...
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1answer
37 views

Why does the limit $ \lim_{\varepsilon\to 0+}\int_{\varepsilon}^M \frac{\varphi(x)-\varphi(0)}{x}\ dx$ exist for smooth $\varphi$?

Let $D(\Bbb{R}):=C_c^\infty(\Bbb{R})$ and $$ p.v.(1/x)(\varphi):=\lim_{\varepsilon\to 0}\int_{|x|>\varepsilon}\frac{\varphi(x)}{x}\ dx. $$ for $\varphi\in D(\Bbb{R})$. I'm trying to understand ...
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1answer
42 views

1D Wave PDE with “strange” Boundary Conditions

I've just arrived home from an exam and cannot come to terms with the fact I couldn't solve the following question: Find a solution to $$ \left\{ \begin{array}{ll} u_{tt} - u_{xx}, & ...
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34 views

Solving Heat Equation with Laplace Transform

I am trying an alternative method to separation of variables to the following equation $$ \begin{cases} u_{xx} =4u_t , 0 < x < 2, t>0\\ u(0,t)=0, u(2,t)=0, t>0\\ u(x,0)=2\sin(\pi x), 0 ...
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1answer
40 views

Fundamental solution Laplace-Poisson equation

Let $\Phi:\mathbb{R}^n\setminus\{0\}\rightarrow\mathbb{R}$ be the fundamental solution of the Laplace equation (see e.g. in the book of Evans). For a function $f\in\mathcal{C}_c^2(\mathbb{R}^n)$ we ...
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1answer
29 views

Transport equations with constant coefficients

Let $X$ be the vector field given by $X = b \cdot \nabla_x + \partial_t$ where $b \in \mathbb{R}^n$ is fixed. Let $f \in C^1(\mathbb{R}^{n+1})$. Assume that $u \in C^1(\mathbb{R}^n \times ...
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1answer
21 views

Partial Differential Equation Formation (Arbitrary Functions) [closed]

Form the partial differential equation by eliminating the arbitrary functions from: $$z=f(x^2+y^2)+x+y$$
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1answer
30 views

System with arbitrary function of an unknown

How can I solve the following system $$ (u_x)^2 - (u_t)^2 = 1 \\ u_{xx} - u_{tt} = f(u) $$ where $f$ is an arbitrary function of $u$, $u$ and $f$ to be determined. I don't know any approach, ...
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1answer
19 views

Bounded linear functionals in solving PDE

Many theorems in functional analysis, Rietz, Hahn-Banach, for example, are used to find linear functionals in certain spaces. But why are bounded linear functionals useful in solving PDE? This ...
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Steklov eigenvalue on unit ball [closed]

Show that the eigenvalues of the Dirichlet-to-Neumann map of the unit ball $B^n$ of the n- dimensional euclidean space $R^n$ are 0, 1, 2, ... . Furthermore, the eigenspace of k is given by space of ...
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20 views

Existence of solution to a linear second order PDE

I want to show the existence and the uniqueness of a solution to the following partial differential equation: \begin{equation} ...
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27 views

Banach contraction mapping to find unique solution of $y(x) = \int_0^1 G(x,s) f(s,y(s)) ds$

Problem statement: Use the Banach contraction mapping theorem to show that $$y(x) = \int_0^1 G(x,s)\ f(s,y(s))\ ds$$ has a unique, continuous solution $y = y(x)$ if (1) $G(x,s) \ge 0$ for $0 \le x, ...
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1answer
30 views

Evans pde book: details on an bound for a Sobolev norm in the proof of the Meyers-Serrin theorem

Let $U$ be an open subset of $\mathbb{R}^n$ and $f\in W^{m,p}(U)$. Suppose that $$\|f\|_{W^{m,p}(V)}\leq\delta\tag{1}$$ for all $V\subset\subset U$ (that is, all $V$ such that $V\subset\overline{V} ...
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1answer
24 views

Prove why the shock equation is not linear.

I need a hint not an answer before answering this question: Two part question: The homogeneous shock equation is given by $u_x$ + $u$$u_y$ = 0 Part 1) Show why the shock equation is not linear. ...
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1answer
36 views

Locally flat manifold from Frobenius, differential forms approach

It is a known fact that a Manifold $M$ is locally flat if and only if the Riemann curvature tensor (Levi-Civita connection) vanishes. To show the if part is easy, but I am trying to follow the details ...
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4answers
64 views

Solving this 1st Order PDE [closed]

I am trying to solve the following PDE with an initial condition: $$u_x + u_y = x + y$$ with $$u(x, 0) = 0$$ I am not sure which method to use to solve this. Thanks
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1answer
32 views

Solving this 2nd Order non-homogeneous PDE

I am trying to solve the following equation: $$3u_{xx} - 10u_{xt} - 3u_{tt} = \sin(x + t)$$ I know that the left hand side is a quadratic equation which I have to factorise. Then I let one of the ...
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1answer
37 views

Gradient flow of Dirichlet energy

I have heard that the gradient flow of the Dirichlet energy gives a solution of the heat equation, i.e. if $u(t,x) \in C^1( [0,\infty) \times \mathbb R^d)$ solves $$ u_t(t,x) = - dE(u(t,x)), $$ where ...
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26 views

Derivative of mollification

This is in response to a claim made in the second line of the question here, namely: Given the standard mollifier $\eta$ and a locally integrable function $f:U \rightarrow \mathbb{R}^n$, by defining ...
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20 views

Uniqueness of the Green's function

Given a linear operator $L$, a Green's function $G(x,s)$ is any solution of $$\tag{1} LG(x,s) = \delta(x-s)$$ where $\delta(x-s)$ is the Dirac Delta function. The Green's function can also be used in ...
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1answer
32 views

Finding general solution to Partial Differential Equations

I am asked to find the general solution $f(x, y)$ of the partial differential equation: $\frac{\partial ^2 f}{\partial x \partial y}=e ^ {x+2y}$ I know these are relatively easy to solve, I haven't ...