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

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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 $Lu=-D_{i}(a_{ij}(x)D_{j}u)$ ...
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12 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 ...
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7 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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6 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 ...
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14 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 ...
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20 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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2answers
44 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} ...
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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 ...
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2answers
31 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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55 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, ...
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1answer
17 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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1answer
31 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 ...
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1answer
44 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 ...
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19 views

Eigenvalues and Eigenvectors of an hyperbolic partial differential equations $\partial_t W + A \partial_x W = 0$

I read in a article dealing with a hyperbolic partial differential equations this statement : For any system of hyperbolic partial differential equations (pde), expressed as (1) ...
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18 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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10 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 ...
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13 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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2answers
33 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) ...
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1answer
15 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 ...
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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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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: ...
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8 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 ...
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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 ...
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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 ...
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29 views

Reduce the PDE to Canonical form. [on hold]

$u_{xx} + 5u_{xy} + 6u_{yy} = 0 $ Find the fundamental solution if possible. I think what needs to happen is you need to find dy/dx using the quadratic formula, then simplify the equation using a ...
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1answer
8 views

How to calculate the transition density for a multivariate jump process

I have the following stochastic process: $dX = (A-I)XdN$, where $X$ is a $2\times1$ vector of random variables, $A$ is a constant, real, symmetric, $2\times2$ matrix, $I$ is the identity matrix and ...
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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) = ...
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39 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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22 views

Solve IVP with the method of characteristics (quasilinear PDE) (+shockwaves)

The question i just can't figure out one bit is: Solve for the continious function $u(x,t)$: $$u_t+(1-2u)u_x=0 $$ $$-\infty < x < \infty, t>0$$ $$u(x,0)= \left\{ \begin{matrix} \frac{1}{4} ...
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1answer
20 views

Prove Continuity of a multivarible function.

I'm trying to prove the following: Let $f:\mathbb{R^n}\times\mathbb{R} \to \mathbb{R}$ be a continuous function. We define $$F(x,t) = \int_{0}^{t}f(x,s)ds $$ Prove that F is also continuous. I ...
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1answer
12 views

Non autonomous system?

Let us consider the wave equation $u_{tt}-u_{xx}=0$. I have the two following questions: a) If we have the boundary condition $u(0,t) = u(\pi+t,t) = 0$, for all $0 < t < \infty $ the given ...
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12 views

Potential theory solution for Variable coefficient Poisson with Dirichlet Boundary conditions

I am looking for a potential theory representation for the following equation in $2$D: $$\vec{\nabla} \cdot \left(a(x) \vec{\nabla}u\right) = 0 \,\, \forall x \in \Omega \,\, (\spadesuit)$$ $$u = g ...
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10 views

Poisson integral formula the following Harnack inequality and Liouville theorem

Suppose $u$ is a nonnegative harmonic function in $B_R(x_0)\subset \mathbb{R}^n$. Prove by the Poisson integral formula the following Harnack inequality: $$ ...
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15 views

Find the Green's function for the Laplace operator in the upper half space

Find the Green's function for the Laplace operator in the upper half space ($x_n>0$) and then derive a formal integral representation for a solution of the Dirichlet problem $$ \Delta u=0 \text{ ...
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1answer
36 views

Inequality for proof of Density Theorem

Someone could help me white this question or indicate some reference? Lemma:For any $\epsilon>0$ There exits a $C=C(n,\epsilon)$ such that for $u \in H^{1}(B_{1})$ with $|\{ x \in B_{1} ; ...
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1answer
17 views

What does $u_0(x)$ represent?

I am looking at the heat equation and in my notes it says the initial temperature distribution $u(0,x)=u_0(x)$. what does this mean? What does $u_0(x)$ represent?
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1answer
9 views

Where does $u(t,x) \to u(t,x)-a-(b-a)x$ come from?

I know the heat equation is $$\frac{\partial}{\partial t} u(t,x)=\frac{\partial^2}{\partial x^2} u(t,x)$$ I know that $u(t,x)$ is the temperature distribution at time $t$ at the point $x$. We assume ...
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0answers
11 views

Which numerical method gives the most accurate solutions of Helmholtz equation for arbitrary domains?

There are many numerical methods for the solutions of PDE's such as FDM, FEM, SEM, Meshfree methods etc. I'm wondering which method gives the most accurate Dirichlet eigenvalues (and corresponding ...
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1answer
20 views

Reducing a PDE to a dimensionless form with change of variables

I am working through the following example to refresh my memory on how to use the chain rule when changing variables: Change of variables (PDE) \begin{equation} \begin{split} ...
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1answer
34 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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24 views

heat equation on a surface

Probably I have not well understood the heat equation: please, can you confirm or correct the followings ? (The question raised in this post is similar to Heat Equation on Manifold but they don't ...
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1answer
13 views

PDE Method of characteristics with initial condition

I wanted to solve the following PDE with initial condition $$ u_t+tu_x=0, $$ $$ u(x,1)=f(x),$$ where $f(x)$ is a given function, using the method of characteristics. I explain what I have done. First ...
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1answer
19 views

How to show contradiction in the Hardy inequality when the singularity power is greater that 2.

Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have $$ \Lambda \int_{\Omega} ...
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0answers
21 views

fourier transform for pde equation

I was solving the pde using fourier transform: $u_{tt}-u_{xx}+m^2u=0$ with initial values $u(0,x)=f(x)$ and $u_t(0,x)=g(x)$. I have received the answer $$U(t,k)=Ae^{-it \sqrt {k^2+m^2}}+Be^{it \sqrt ...
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30 views

Is this partial differential equation solvable?

Ok so I am asked to set up a partial differential equation and then motivate why it is solvable. I'm only 2 weeks into my course so we are not asked to solve anything yet. However, if someone would ...
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15 views

About PDE solution: H^1 norm bounded = bounded in L^2?

Let $\Omega \subset \mathbb{R}^d~(d=2,3)$ be an open bounded set with Lipschitz continuous boundary $\Gamma$. We assume that $\Gamma$ consists of two disjointed parts, i.e, $\Gamma = \Gamma_{c} \cup ...
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2answers
42 views

how come $C^{1}_{0}$ is not complete under norm $||.||_{1,2}$

why the space $C^{1}_{0}$ is not complete under the norm $||.||_{1,2}$? by some counter example $||u||1,2=(\int_{\Omega} (|\nabla u |^2+|u|^2))^{1/2}$
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Are the set of probability functions with compact support in a fixed closed ball complete under the Wasserstein norm?

Let $B_R$ be a closed ball of radius $R$ in the space $\mathbb{R}^d$. As the title suggests I have this feeling that the set of functions $$S:= \left\lbrace f:\mathbb{R}^d \to \mathbb{R} ...
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1answer
32 views

Fourier cosine series giving nonsense answer

I'm currently trying to find the cosine Fourier series of $f(x) = \left | \sin \frac{\pi n }{L} x\right |$ on the interval $0 < x < L$. I first started by calculating the first term of the ...
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15 views

Sketching partial differential equation

I have found a solution to my pde however I want to try and sketch it however I don't know where to start. pde