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

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1answer
12 views

Compare analytic model with numerical, mass spring system.

So I'm trying to solve a problem here and I have been working on it all day, clearly i'm in need of some guidance. I have a rod of length $L$ and cross section area $A$, Young's modulus $E$ and ...
0
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0answers
21 views

The eigenfunction of $-\Delta$.

Let $u_k$ be eigenfunctions for operator $-\Delta$ over domain $\Omega\subset \mathbb R^2$, open bounded, smooth boundary. Then, we know that $u_k$ forms a basis for $L^2$. Let $u\in H_0^1(\Omega)$ be ...
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0answers
10 views

method of characteristics for non-linear PDE

I'm trying to solve the PDE $u_x^2-u_y^2=8u$ with initial conditions $u(x,x)=f(x)$. I have that $F(x,y,u,p,q)=p^2-q^2-8u$, with $p=u_x, q=u_y$, and then \begin{equation*} \begin{array}{ll} ...
1
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1answer
28 views

Combining two results from partial integration

I have a set of two PDEs: $$\partial_{\tau}\theta+\partial_{\eta}\psi=0$$ $$\partial_{\tau}\psi=-\partial_{\eta}\theta+\alpha\partial_{\eta}^{2}\psi$$ These can be combined into a wave equation of ...
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1answer
26 views

System of differential equation (Matrix form)

I'm trying to solve this system $$ M\ddot{X}(t) = KX(t) $$ where M is a known diagonal matrix and K is a symmetrical known matrix. I'm asked to do the ansatz $Y(t) = M^{1/2}X(t)$ where $M^{1/2} = ...
2
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0answers
13 views

Step 2 of Strichartz's Estimate Proof.

I am stuck with Step 2 of the Strichartz's estimate. My question is actually a continuation of a topic which has been raised some time ago and it could be seen here Technical question about Strichartz ...
6
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1answer
122 views

Construction of Monotone function which is differentiable on the given set

Given a set $A \subset \mathbb{R}$ of measure $0$, is it possible to construct a monotone function whose set of non differentiable points is $A$ ?
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0answers
13 views

Differentiation with integration region depending on $x$ to solve for decreasing energy of wave equation

I want to show that for the general wave equation $u_{tt} - \nabla \cdot (c^2\nabla u) + qu = 0, \quad u(x, 0) = \phi(x), \quad u_t(x, 0) = \phi(x)$ we have $$ E(t) = \int_{|x-x_0| < R_0 - c_2t} ...
1
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1answer
17 views

Apply Periodic Boundary to PDE (Fourier Transform)

Use Fourier Transform to solve the BVP: \begin{cases} u_t + a u_x - b u_{xx} = 0, & \mbox{for } x \in [-1,1] \\ u(x,0) = f(x) \\ u(x+2,t) = u(x,t) \end{cases} I solved the problem (attached); ...
4
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1answer
118 views

Technical question about Strichartz estimate's proof.

I was studying the proof of Strichartz estimates from the book "Semilinear Schrödinger equation" of T. Cazenave. The proof is divided in several steps. Here we can assume ...
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2answers
2k views

Question from Evans' PDE book

How do you do the second part of question 8, chapter 5, of Evans' PDE book (first edition)? I have proven the inequality for smooth, compactly supported functions using integration by parts, and I ...
2
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0answers
15 views

Heat equation with mixed boundary conditions

I am trying to solve the following problem $$\left\{\begin{matrix}\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},& 0<x<1,t>0,\\ u(0,t)=\frac{\partial u}{\partial ...
1
vote
1answer
454 views

Vanishing at the infinity of a function in the Sobolev space.

If $f \in H^s (\Bbb R^n)$ for $s > 1 + \frac{n}{2} $ then the Sobolev inequality implies that $f$ and $\nabla^\alpha f$ ($|\alpha| =1$) vanishes at the infinity. ($\alpha$ : multi-index). But in ...
0
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1answer
11 views

Taking derivative of energy of wave equation

Consider the variable coefficient, real valued wave equation $$ u_{tt} - \nabla \cdot (c^2 \nabla u) + qu = 0, \quad u(x,0) = \phi(x), \quad u_t(x, 0) = \phi(x), $$ where $c, q \geq 0$ depend only on ...
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0answers
5 views

Equivalence of first order quasilinear PDE to linear PDE

Given a system of nonlinear PDE of the special form: $\sum_{i=1}^n A_i(x, \phi) - \frac{\partial \phi_j}{\partial x_i} = B_j(x,\phi) $ $(1)$ with $(j=1,...,m)$ and $x \in R^n,\phi \in R^m$. If we ...
1
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1answer
40 views

Solve $A \partial_t w + B \partial_t\partial_x^4 w + C \partial_x^4 w + \partial_t^2 w = 0$

a non-mathematician wants me to solve a PDE. The problem is that I don't know a lot of theory to solve PDE's except the fouriertransform. This is the PDE $$A \partial_t w + B \partial_t\partial_x^4 w ...
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0answers
21 views

Nondimensionalization of PDE with non constants coefficients

I need to try to solve, by some numerical method, a variant of the Navier-Cauchy equation of motion for an elastic, linear, isotrophic, not homogeneus body: $$ \rho_0 (\ddot{\textbf{u}} - ...
1
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1answer
14 views

uniqueness of heat equations and the squared integrable assumption

I am looking at the classical proof of uniqueness for the heat equation in Evans. Clearly, we differentiate under the integral sign of the square of $w$. A very basic question is, why are the ...
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1answer
29 views

Functions of the form $u(x,y) = f(x-y)$ are weak solutions of $u_x + u_y = 0$

This is a problem out of Logan's Applied Math book. Section 6.7, problem 2. Show that for any locally integrable function f on $\mathbb{R}$ the function $u(x,y) = f(x-y)$ is a weak solution to ...
2
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0answers
24 views

Poincare Inequality in n-dimensions

I am trying to prove the Poincare Inequality on a n-dimensional box. That is a domain $ \Omega = (0,1)^n$ for $f(x) \epsilon H^{1}_{0}(\Omega) $, show there exists a constant $C$ such that ...
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0answers
36 views

Solving PDE's: Laplace equation on semi-infinite cylinder

Solve Laplace’s equation $\nabla^2$u = 0 inside a semi-infinite cylinder 0 < z, 0 < r < 1 with boundary conditions u(r = 1, $\theta$, z) = $e^{−z}$ and u(r, $\theta$, z = 0) = 0, where (r, ...
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0answers
52 views
+100

Proving that if $f\in\mathcal{F}C^{1}_{b}(X)$ then $f\in W^{1,p}(X,\gamma$) for $p>1$

Let $X$ be a separable Banach space endowed with a centered nondegenerate Gaussian measure $\gamma$ and $H$ the Cameron-Martin space. Then consider $f\in\mathcal{F}C^{1}_{b}(X)$. I want to prove that ...
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0answers
17 views

Inequality in the proof of Weak Harnack Inequality

Let $\Omega \subset \mathbb{R}^{n}$ a bounded domain s.t $B_{1} \subset \Omega$ , $u \in H^{1}(\Omega)$ a nonnegative supersolution in the weak sense of the equation $Lu=-D_{i}(a_{ij}(x)D_{j}u)$ ...
1
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1answer
16 views
+50

Where can i find references to proofs of 1D,2D (partially 3D) Navier Stokes Equation?

I'm currently trying to get into PDE's and as part of a course i'm focusing on proofs on existence of solutions to the Navier-Stokes Equations. Although existence of solutions has been proved for 1D ...
4
votes
3answers
165 views

Initial-value problem for non-linear partial differential equation $y_x^2=k/y_t^2-1$

For this problem, $y$ is a function of two variables: one space variable $x$ and one time variable $t$. $k > 0$ is some constant. And $x$ takes is value in the interval $[0, 1]$ and $t \ge 0$. At ...
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0answers
16 views

Show that the function is identically Zero in certain subset

We are given a open ball D (radius = 1) in $\mathbb R^2$. and let $\{x_n\}$ be the dense sequence in the set D. Around each point $x_n$ we make a hole of radius $r_n$. The sequence $r_n$ satisfy the ...
0
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0answers
23 views

long time behavior of heat equation

Given the heat equation \begin{align} {{u}_{t}}-{{u}_{xx}}&=0,\quad x\in \mathbb R,\,t>0 \\ u\left( x,0 \right)&=f\left( x \right),\quad x\in \mathbb R. \end{align} If ...
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0answers
11 views

PDE: Traffic problem with initial density a piecewise function

I was studying a bit of PDE and I found very interesting traffic problems, however, I have some troubles to deal with them. I wanted to solve the following: Consider the traffic problem ...
0
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2answers
45 views

Method of characteristics - finding the particular solution using initial conditions

I am trying to use the method from my previous question to solve this PDE: $$ 3u_x + 2u_t = \cos x $$ with initial condition $u(x,0) = x^2$. So I need to solve these: \begin{align} \frac{dx}{ds} ...
3
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1answer
177 views
+50

Biharmonic Equation in a Rectangle with Some Uncommon Boundary Conditions

Consider the following boundary value problem (BVP) $$\matrix{ {{\Delta ^2}H = 0,} \hfill & {} \hfill & {{\rm{in}}\,} \hfill & \Omega \hfill \cr {\partial _y^2H = 0} \hfill & ...
2
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0answers
22 views

PDE - three restrictions, wave equation (1 dimension)

I'm not very good at PDEs but this particular problem seems... Strange. It requires that the answer be "continuous (!!)" all in bold. \begin{align} u_{tt}&=9u_{xx},\quad x>0,\, t>0, \\ ...
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0answers
17 views

Weak derivative of a piecewise defined function

I am currently looking at these online notes on PDEs, page 59. How does it follow that if $f^R = \phi(x/R) f(x)$ $ \phi(x) = \left\{\def\arraystretch{1.2}% \begin{array}{@{}c@{\quad}l@{}} 1 ...
1
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1answer
18 views

checking that an initial condition holds for the heat equation

I'm trying to follow a video lecture on solving the heat equation. $I) \space u_t = ku_{xx}, x \in \mathbb{R}, t > 0$ $II) \space u(x,0)=\phi (x), $ $k$ is const, $\phi (x) $ is a ...
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0answers
10 views

capillary surface problem [on hold]

Consider the capillary surface problem (⋆) )   Du  div 1 + |Du|2 = κu in Ω on∂Ω,  Dηu  1+|Du|2 =β where κ > 0, η is the outward pointing unit normal to ∂Ω and β ∈ C1(Ω) satisfies |β| ≤ 1 ...
1
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2answers
33 views

Method of characteristics - eliminating variables

I am trying to follow a guide for the method of characteristics; quoting the first example: We use the method of characteristics to solve the problem $ 2u_x - u_y = 0, \;\; u(x, 0) = f(x) $ ...
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0answers
60 views

About the gradient of a function in $H^{1}(\Omega)$

let $\Omega \in \mathbb{R}^{n}$ a bounded domain and $u \in H^{1}(\Omega)$ a real function. In the Leoni's Book - A First Course in Sobolev Spaces, the author define $\nabla u = (D_{1} u,\dots, ...
0
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1answer
33 views

More equations than unknowns for maxwell equations?

I had one curiosity regarding maxwell equations in 3-D From the curl equations, you get 6 unknowns, with 6 equations. The divergence equations add 2 additional equations. When these are combined, we ...
2
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1answer
53 views

How do I determine if the equation is a conservation law?

We have the PDE $\frac{\partial u}{\partial t}+a(x,y)\frac{\partial u}{\partial x}+b(x,y)\frac{\partial u}{\partial y}=0$. What would be conditions on $a$ and $b$ for the equation to constitute a ...
0
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1answer
12 views

Solving the heat equation with piecewise IC

I have the solution to the heat equation, with the BC's and everything but the IC applied. So I am just trying to solve for the coefficients, the solution without the coefficients is $$u(x,t) = ...
2
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0answers
14 views

Where does the name “tracking type problem” come from?

In PDE-constrained optimization problems, the distributed constrol problem $$ \begin{array}{ll} \displaystyle \min_{y,u} & J(y,u) = \frac{1}{2}\|y-y_d\|_{L^2(\Omega)}^2 + ...
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0answers
11 views

Question with D'Alembert formular

We have the solution of the wave equation $u_{tt}-u_{xx} = 0$ with boundary condition $u(0,t) = u(L,t) =0$ is $u(x,t) = \int_{t-x}^{t+x}u_x(0,s)ds$. My question is that can we replace the formular as ...
1
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1answer
75 views

Laplace Equation in a Cylinder with Some Uncommon Boundary Conditions

While I was working on some theorems in PDEs, I encountered the following axisymmetric boundary value problem $$\matrix{ {{\nabla ^2}H = 0} \hfill & {{\rm{in}}} \hfill & \Omega \hfill ...
0
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2answers
35 views

Separation of Variables for second order PDE

I have a PDE that I have attempted to solve using the method of 'separation of variables' $$u_t = (1+2t)u_{xx} \,\,\,\, 0 \leq x < \pi, t \geq 0 $$ With initial and boundary conditions: $$u(0,t) ...
12
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1answer
156 views

Interpretation of an integral transform from the wave equation to the heat equation

I'm having troubles with understanding the physical meaning of a certain transform. If $u$ is a solution to the wave equation $$\partial_t^2u-\Delta u=0\ \mathrm{in}\ ...
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0answers
27 views

Solve this recurrence relation via a first order partial differential equation?

Find a general formula for $a_{n,k}$ , for $n,k\geq1$. We have initial values $a_{1,1}=1$, and $a_{1,k}=0$ for $k>1$. The recurrence relation is: $a_{n+1,1}=-a_{n,1}$ , for $n\geq1$ and ...
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0answers
36 views

solution of linear elliptic equation

Can you please help me to show that if $\Omega\subset\mathbb R^n$ is a $C^2$ domain and $f$ is an application which belongs to $L^2(Ω)$ and $u$ is a weak solution of the linear elliptic equation: ...
2
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1answer
36 views

Analyze : $u_t-u^2u_x +cu =0, u(x,0)=g(x)$

Analyze : $u_t-u^2u_x +cu =0 $, $ u(x,0)=g(x)$. From This we have following $$\begin{align} \frac{dt}{ds} &=1 \\ \frac{du}{ds} &=c \\ \frac{dx}{ds} &=-u^2 \end{align}$$ then how to ...
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0answers
9 views

transform to autonomous linear equation

I would like to ask that which are the equations could be write as the form of autonomous linear equation $u_t = Au_{xx}$ I just known the heat equation, (we take $A = Lapacian$) or the wave ...
1
vote
0answers
29 views

rayleigh quotient of eigenvalue problem (sturm liouville theory and partial differential equations)

I am reading "A First Course in Partial Differential Equations with Complex Variables and Transform Methods" (Weinberger, p. 168). if we have the eigenvalue problem $$ (pu')'- qu + \lambda \rho u = 0 ...
1
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0answers
10 views

PDE Von Neumann Problem- Physical Interpretation

The Von Neumann Problem is as such: $\Delta u = f(x,y,z)$ in $\ D$ $\frac {\partial u} {\partial n} = 0$ on bdy $\ D$. Using this you can prove that for there to be a solution to this Von Neumann ...