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

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2
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0answers
57 views

Separation of variables with harmonic, time-dependent boundary condition

I am studying the vibrating dynamics of beams. In its resting state, the beam is flat, on what I call the $x$ axis. The variable $w(x,t)$ measures the vertical deflection from that axis for any point ...
0
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1answer
95 views

Radial solution in Wave equation

Let $u$ be a solution of the initial wave equation in $\mathbb{R}^n$: \begin{equation} u_{tt}-\Delta u = 0 \mbox{ in } \mathbb{R}^n\times\mathbb{R}_+ \quad ; \quad u(x,0) = g(x) \quad ; \quad u_t(x,0) ...
3
votes
1answer
54 views

Initial value problem and characteristic equations

I have this initial value problem: $u_t + xu_x = -u^2$, with $u(x,0)=1$. So from this we have $ \dfrac{dt}{1} = \dfrac{dx}{u} = \dfrac{du}{u^2}$ so $\dfrac{du}{dt} = -u^2 \implies ...
1
vote
0answers
34 views

maximum principle on compact manifolds with boundary

Let us consider the equation $Lu + f(u) = 0$ on a compact manifold $\overline{M} = M \cup \partial M$ with boundary, with Dirichlet boundary conditions. $L$ is a linear elliptic operator, and $f$ ...
3
votes
1answer
80 views

Solving Laplace $\nabla^2 \phi=0$ in $x,y \geqslant 0$

I'm trying to solve $\nabla^2 \phi=0$ in $x,y \geqslant 0$ $\phi(x,y)=0 $ as $x^2 +y^2 \rightarrow \infty$ $\phi_x(0,y)=0$ and $\phi(x,0)= \frac{1}{1+x^2}$ I know the solution is ...
0
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1answer
279 views

Zeros of a harmonic function

Prove that the zeroes of a Harmonic function is never isolated. All I can think of is a very rough idea of a proof by contradiction.
4
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2answers
99 views

How could we show that $u=0$?

In my notes there is the following example about the energy method. We want to show that the problem $$w_{tt}(x, t)-w_{xxtt}(x, t)-w_{xx}(x, t)=f(x, t), 0<x<1, t>0$$ $$w(x, 0)=\phi(x) \\ ...
3
votes
1answer
52 views

Generalising Liouville's Theorem for Real Numbers

I've been tasked with the following; Let $u : \mathbb{R}^n \to \mathbb{R}$ be harmonic and $R \ge 0$, $\beta \in \mathbb{N}$. Suppose that; $$\limsup_{|x| \to ...
2
votes
1answer
130 views

distributions whose derivative is zero?

I just learned about the notion of tempered distributions $\mathcal{S}'(\mathbb{R})$. But it is unclear that if such a distribution has a 0 derivative (of course in the distribution sense) then it ...
1
vote
1answer
39 views

Fourier series method

I have the following Boundary value problem $U_t-U_{xx}=0$ from zero to one; $t>0$ $ U(x,0)=x$ from zero to one $U(0,t)=U(1,t)=0$ I need to solve it using the Fourier series method, But I ...
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vote
0answers
44 views

Explicit solution for conductivity equation with discontinuity

Let $B$ the unit disk in $\mathbb{R}^{2}.$ Let $B_{\epsilon}$ be the disk centered at the origin at of radius $\epsilon.$ Let me define $\displaystyle ...
0
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1answer
84 views

Heat Equation Solution Dependence on Diffusion Coefficient

Given a heat equation of $u$ with spatial dependent diffusion coefficient $\alpha(x)\ge 0$ $$\frac{\partial u_\alpha}{\partial t}=\alpha(x) \frac{\partial^2 u_\alpha}{\partial x^2}$$ where $(t,x)\in ...
1
vote
1answer
43 views

Orthogonal Level Sets and a generalization of harmonic functions

Forgive my ignorance of differential equations and analysis. I was playing around with orthogonal level curves of real valued functions in the plane, and realized this is one way a person could be ...
0
votes
0answers
29 views

Wave equation in three space dimensions

Suppose that $u$ satisfies the Wave Equation in $R^{1+3}$ with initial conditions $u(0, x) = 0$, $u_t(0, x) = g(x)$ for $x \in R^3$ and $g \in L^p(R^3)$. Show that $u(t, ·) \in L^p(R^3)$ for all $t$ ...
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vote
0answers
52 views

Spectral convergence for collocation methods

Spectral methods work (simplified) as follows. Consider the problem \begin{align} \partial_t u(t,x) = \mathcal{L} u(t,x) \end{align} where $\mathcal{L}$ is some differential operator. We then try to ...
1
vote
1answer
57 views

Parameters in the Hamilton-Jacobi Equation

I'm reading through Gelfand and Fomin's 'Calculus of Variations', and they've just derived the Hamilton-Jacobi Equation: $$\frac{\partial S}{\partial x} + H \left(x, y_1, \ldots, y_n , \frac{\partial ...
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0answers
50 views

Home work of PDE

$h(x)\in C^2(R^n)$, and $\omega_n$ is the area of unit sphere.Let: $$ M_h(x,r)=\frac{1}{\omega_nr^{n-1}}\int_{|y-x|=r}h(y)dS_y $$ proof: $$ \left\{ \begin{aligned} \frac{\partial^2M_h}{\partial ...
1
vote
3answers
66 views

Solutions of the Laplace equation

How do I find solutions $u=f(r)$ of the two-dimensional Laplace equation $u_{xx}+u_{yy}=0$ that depend only on the radial coordinate $r= \sqrt{x^2+y^2}$
2
votes
2answers
115 views

Characteristic curves

I have to solve the initial value problem: $$2u_{xx}(x, t)-u_{tt}(x, t)+u_{xt}(x, t)=f(x, t), x \in \mathbb{R}, t>0 \\ u(x, 0)=0, x \in \mathbb{R} \\ u_t(x, 0)=0, x \in \mathbb{R}$$ using ...
2
votes
1answer
36 views

Choosing which variable to take the Fourier Transform with respect to

Why here do we take the fourier transform with respect to $x$ as opposed to $t$?
1
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1answer
32 views

pde initial condition problem

$$\frac{dt}{d\tau }=1$$ $$t=\tau $$ $$t(\tau=0)=0$$ $$\frac{dx}{d\tau}=u$$ $$x=g(\xi)\tau+\xi$$ $$x(\tau=0)=\xi$$ $$\frac{du}{d\tau}=0$$ $$u=g{\xi}$$ $$u(\tau=0)=g(\xi)$$ How should I take my ...
2
votes
0answers
29 views

Is one dimesional cubic NLS is globally wellposed in $H^{s}(\mathbb R), (0<s<1)$?

We consider the one dimensional cubic nonlinear Shr\"odinger equation (NLS): $$i\partial_{t}\phi (x,t) +\Delta \phi (x,t)= \pm |\phi (x,t)|^{2} \phi(x,t), \ (x, t\in \mathbb R),$$ $$\phi (x,0) = ...
1
vote
0answers
136 views

Wave equation with discontinuous initial condition

Use the method of images to find the solution $u(x, t)$ to the initial-value problem for the wave equation $u_{tt} = u_{xx}$ on the half-line $x ≥ 0$ with Dirichlet boundary condition $u(0, t) = 0$ ...
2
votes
1answer
76 views

Solution to a particular Wave Equation

Consider the partial differential equation \begin{align} \frac{1}{c^{2}} \, \frac{ \partial^{2} U}{\partial t^{2}} &= \frac{\partial^{2} U}{\partial x^{2}} + x \, \frac{\partial U}{\partial x} + ...
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vote
0answers
40 views

Differentiating both sides of a DE

In general if you have a differential equation with two variables such that: $$L(x,y)=h_1[f(x),f'(x),f^{(2)}(x),...,f^{(n)}(x),g(y),g'(y),g^{(2)}(y),...,g^{(n)}(y)]\\ ...
1
vote
1answer
118 views

Strong solution of inviscid Burgers' equation with initial data $u(x,0)=x^2$

I'm studying for an exam and having trouble solving the following: Find the strong solution to the inviscid Burgers' equation $u_t+uu_x=0$ with initial data $u(x,0)=x^2$ Using the initial ...
1
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0answers
46 views

Maximum Principle for the Derivatives of Parabolic PDE Solution

Is there a maximum principle for the spatial derivatives of the solution of a parabolic PDE with coefficients in front of the spatial partial derivatives depending on spatial variables only, similar ...
2
votes
1answer
24 views

Periodic solutions to the wav equation with seperated solutions

Consider the wave equation $ u_{tt}=a^2 u_{xx} $ and a separated solution $u(x,t)=T(t) \varphi (x) $, with boundary condition $u(0,t)=c, \ u(1,t)=d$. Then I want to show that all separated solutions ...
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0answers
41 views

a question in analysis of Sobolev spaces.

I just wanted to know wthether the following is OK or not - The space $Y=\{u \in H_0^1(\Omega): ||u||_{\infty}\leq M, ||\nabla u||_{2}\leq r\}$ is closed in $X=H_0^1(\Omega)\bigcap L^{\infty}(\Omega)$ ...
1
vote
1answer
77 views

a priori estimates involving Sobolev norms

Let $\sigma,$ $f$ and $g$ be $C^{2}(\overline{\Omega})$ functions, with $0<\frac{1}{M} < \sigma < M.$ We have the Dirichlet problem: $\text{div}\sigma \nabla u=f, \hspace{3mm} \text{in} ...
1
vote
1answer
159 views

wave equation with neumann boundary and initial condition hat function

I have the following boundary and initial value problem on $0\leq x\leq 1$: $u_{xx}=u_{tt}$, $ f(x)=u(x,0) = \begin{cases} 0& \textrm{ if $0\leq x\leq 1/4$} \\ x-1/4& \textrm{ ...
0
votes
1answer
17 views

Inequation in Sobolev

Let $u\in W^{1,2}(\Omega)$. I need to prove that: $\|(\nabla u)u\|_{W^{0,2}(\Omega)}\leq \|\nabla u\|_{W^{0,4}(\Omega)}\;\|u\|_{W^{0,4}(\Omega)}$ I'm using the usual Sobolev notation (see for ...
2
votes
2answers
79 views

How to prove this limit

Let $g:[0,\infty)\to\mathbb{R}$ with $g(0)=0$. For $x>0$ and $t>0$, define: \begin{equation} u(x,t) = \frac{x}{\sqrt{4\pi}} \int_0^t \frac{1}{(t-s)^{3/2}} e^{-\frac{x^2}{4(t-s)}} g(s) ...
2
votes
0answers
39 views

Linear PDE of second order

I have the following problem: $ \rho_{tt} +a\rho_{xt}-c^2\rho_{xx}=bv_{xx}$, where $\rho=\rho(x,t)$, $v=v(x,t)$ and $a,b,c$ are constant. My attempt to solve such an equation is to treat $v$ as any ...
1
vote
1answer
67 views

Extensions and reflections of wave equations - general question

I understand the concept of making even and odd extensions of the initial data to satisfy the boundary conditions - using an even extension in the Neumann case and odd extension in the Dirichlet case. ...
3
votes
1answer
468 views

Energy Method to show uniqueness of solution of PDE

In my notes there is the following example about the energy method. $$u_{tt}(x, t)-u_{xxtt}(x, t)-u_{xx}(x, t)=0, 0<x<1, t>0 \\ u(x, 0)=0 \\ u_t(x, 0)=0 \\ u_x(0, t)=0 \\ u_x(1, t)=0$$ ...
2
votes
1answer
176 views

Why does the Hamilton Jacobi Bellman Equation imply Pontryagin's Minimum Principle

I'm having difficulty understanding the proof that allows us to go from the Hamilton-Jacobi-Bellman equation to to the Pontryagin Min(Max) Principle. Lets consider $x(t)$ and $u(t)$ as real valued ...
2
votes
2answers
63 views

Solution of this non-linear PDE

What is the general solution $h: \mathbb{R}^2 \rightarrow \mathbb{C}$ (or maybe $h: \mathbb{C}^2 \rightarrow \mathbb{C}$ if necessary), $(x, y) \mapsto h(x, y)$ to the non-linear partial differential ...
1
vote
1answer
98 views

Poisson's eqution and radial symmetry

I don't understand why $\Phi$ does not depend on $\theta$ or $\phi$. On the boundary it makes sense because $\Phi$ on the boundary does not depend on any of the variables. I get that the sphere ...
6
votes
1answer
178 views

Differential equation - Green's Theorem

I want to find the solution of the following initial value problem: $$u_{tt}(x, t)-u_{xt}(x, t)=f(x, t), x \in \mathbb{R}, t>0 \\ u(x, 0)=0, x \in \mathbb{R} \\ u_t(x, 0)=0, x \in \mathbb{R}$$ ...
0
votes
0answers
30 views

Extension of a self adjoint Operator

Suppose we have a open (bounded) domain $\Omega$ in $\mathbb R^d$. And let a plane $\mathcal P$ in $\mathbb R^d$ divides the domain in two (disjoint) open sets. (say $\Omega_1$ and $\Omega_2$) Hence ...
0
votes
1answer
36 views

Elliptic PDE Integral inequality

Suppose $\zeta\in C_0^1(\Omega)$ and $u>0$ in $\overline{\Omega}$. Suppose $\lambda|\xi|^2\leq a_{ij}(x)\xi_i\xi_j\leq\Lambda|\xi|^2$ for $x\in \Omega$ and $\xi\in\mathbb{R}^n$. Apparently $$ ...
2
votes
0answers
61 views

Kahler-Einstein Metrics in Physics - Topic Suggestions

I am hoping to get some topic suggestions for a presentation I have to give in a couple of weeks. The course the presentation is for is called Kahler-Einstein metrics. I would really like the ...
4
votes
2answers
152 views

Solve integral (convolution) equation

Given a function: $u(t) = \exp\left( -\frac{At^2}{1+t}\right),$ $A>0, t>0,$ and an equation: $\frac{d u(t)}{dt} = \int^{t}_0 \phi(t-\tau) u(\tau) d \tau .$ How to find a closed expression for ...
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2answers
54 views

$\frac{\partial T}{\partial t} = \alpha \nabla ^2r$ for spherically symmetric problems

The standard one dimensional partial differential diffusion equation in Cartesian coordinates has the form; $$\frac{\partial T}{\partial t} = \alpha \frac{\partial ^2 T}{\partial x^2} \tag{1}$$ For ...
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0answers
85 views

What does a well-posed problem imply?

A well-posed problem in the sense of Hadamard states that: A solution exists The solution is unique The solution's behavior changes continuously with the initial conditions. Now in order to prove ...
2
votes
0answers
48 views

Reformulate this PDE in different notation

I would like to rewrite this general PDE \begin{equation} \alpha\partial_tu+\beta\partial_xu+\gamma\partial_{xx}u+\delta u=\varepsilon \end{equation} in this form $$c\left(x,t,u,\frac{\partial ...
0
votes
1answer
51 views

The phrase “periodic boundary conditions” for a two-variable PDE

I'm currently working on trying to solve a system of PDE's of the form $c_t=D_x(c_{xx} + c_{yy})+K_1 c + K_2 d$ $d_t= D_y(d_{xx}+d_{yy})+K_3 c + K_4 d$ that has "periodic boundary conditions" on a ...
0
votes
1answer
20 views

What condition does this impose on the coefficients $ B_n$ in the vibrating string problem

Let: $$y(s; t) =\sum\limits_{n=1}^{+\infty} B_n \cos\left(\dfrac{n\pi ct}{ L}\right) \sin\left(\dfrac{n\pi s}{L}\right),$$ be a solution of the vibrating string problem. Suppose that the string is ...
0
votes
1answer
41 views

Fourier series solution of the heat equation on $-2<x<2$

I have to solve the following boundary value problem: $u_t=u_{xx}$, $u(t,-2)=u(t,2)=0$ and $u(0,x)=f(x)$. I tried to solve the problem using the method of separation of variables. So assume ...