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

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Deflection of String

I am trying to determine u(x,t) for a string of length L=1 and c^2=1 when the initial velocity is 0 and initial deflection with small k(.01) is as follows: ...
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27 views

Hyperbolic PDE with strange condition.

The 1D wave equation part is not tricky, but I am having trouble dealing with the max condition. I was thinking of using d'Alembert's formula somehow but I am not sure how to use it in this case. ...
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22 views

Elliptic regulartiy for nonlinear elliptic equations

Here is the question: Let us consider the Schrodinger type equation $$ \left\{ \begin{aligned} &-\Delta u + u = |u|^{p-2}u \quad \text{in } \mathbb{R}^N \\ &u\in H^1(\mathbb{R}^N) \qquad ...
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98 views

Elliptic regularity in an unbounded domain

Let $\Omega \subset \mathbb{R}^N$ be an unbounded domain with nonempty boundary. Let $$\mathcal{D}^{1,2}(\mathbb{R}^N) = \{ u \in L^{2^*}(\mathbb{R}^N) : \nabla u \in L^2(\mathbb{R}^N ; \mathbb{R}^N) ...
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1answer
13 views

Neumann boundary conditions for PDE

I have a question about Neumann boundary condition for PDE. Suppose $\Omega$ is an open bounded set in $R^n$ with a smooth boundary $\partial \Omega $. Then, a homoegenous Neumann boundary condition ...
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1answer
154 views

Does $\operatorname{div}\left(\nabla G +xG\right)=0\Longleftrightarrow \nabla G +xG=0$?

Let $G$ be a function of $\mathcal{C}^2(\textbf{R}^d$;$\textbf{R}^*_+)$ such that $G \in \operatorname{L}^1(\textbf{R}^d)$. I read on the Internet that one has the following equivalence ...
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1answer
35 views

Existence and uniqueness for the Cauchy problem for the Laplace equation

Given a Laplace equation $u_{xx}+u_{yy}=0$ in $\mathbb{R}^2$ with initial conditions $u(x,0)=u_0(x)$, $u_y(x,0)=u_1(x)$. Is this problem uniquely solvable in general? Does someone have a ...
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22 views

What are the main methods to solve an evolution PDE and how are they applied?

When one sees an evolution PDE, what are the reflexes that he should have in order to tackle it. What I mean is what are the main methods that have been developed so far and to which kind of PDEs ...
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1answer
29 views

Second order PDE

Kindly help me with this... $$U_{xy} + yU_{yy} + \sin(x+y)=0$$ Here $A =0$, so how to calculate the characteristic equations ? as $$ {dy\over dx} = {B^2 \pm \sqrt D\over2A} $$
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16 views

What does the Sturm-Liouville theory say about separation of variables?

In this question Completeness of solutions and the separation of variables method one of the comments says that the condition for the solutions formed by separation of variables to be a basis for all ...
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1answer
42 views

The mysterious $\dot{H}^{-1}$ notation.

I have encountered the $\dot{H}^{-1}$ notation in one of the SIAM Journal on Mathematical Analysis articles. It appears to be standard (or at least not uncommon) to use this one in the field, since ...
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1answer
24 views

Understanding of the Hopf-Lax formula

This is an exercise in the book Partial Differential Equations (2nd edition) by Evans: Here $L^*(q)=\max_{y\in {\Bbb R}^n}\{q\cdot y-L(y)\}$ and $L$ is assumed to be such that it is convex and ...
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7 views

Lower order perturbations of 2nd order differential operators

Consider the well-known Hormander's sum of squares $P = \sum_{j = 1}^m X_j^2$, where $X_j$ are vector fields on a compact manifold $M$ of dimension $n$. Also assume, as is usual to this theory, $m ...
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21 views

$u_{xx}+u_{yy}=0,1<r<3,0<\theta<\frac{\pi}{2}$

$u_{xx}+u_{yy}=0,1<r<3,0<\theta<\frac{\pi}{2}$ $u(1,\theta)=u(3,\theta)=0,0\leq\theta\leq\frac{\pi}{2}$ $u(r,0)=(r-1)(r-3),u(r,\frac{\pi}{2})=0,1\leq r\leq3$ I have no idea how to solve ...
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1answer
17 views

Numerically solving a steady state equation (diffusion reaction with monod kinetics)

I have a system that I'm interating in time via finite differences, but one of the equations is to be solved at steady state each iteration: $D\Delta S=\frac{S}{S+a}\rho$ I want to solve it via a ...
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1answer
22 views

Fourier transform of a scaled variable [duplicate]

If $f\hat(k)$ is the fourier transform of $f(x)$, what is the fourier transform of $f(x/c)$ where $c$ is a real number greater than $0$?
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1answer
30 views

Necessary and sufficient condition for separation of variables to give all solutions.

Lets say we have a partial differential with derivatives of $y$ with respect to $x$ and $t$ is there a necessary and sufficient condition that must be obeyed by such an equation for the superposition ...
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1answer
21 views

Are the solutions of a Sturm-Liouville equation entire in the spectral parameter?

In $[1]$ the following (paraphrased) claim is made: Let $q\in L^1_{loc}([0,\infty);\mathbb{R})$, and suppose $\varphi$ and $\theta$ solve the one-dimensional Schrödinger equation \begin{equation} ...
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1answer
64 views

Consider the differential equation

Consider the equation $$u\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = \frac{1}{2}u$$ a) Find a solution satisfying $u(x, 2x) = x^2$ b) Is the solution unique? I have no idea how to ...
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1answer
18 views

Maximum Principle for a Poisson Equation?

Let $u$ be a $C^2$-solution of $\Delta u=u^3-u$ on a bounded domain $\Omega$ with $u=0$ on $\partial\Omega$. How can one show that $-1\leq u\leq 1$ for every $x\in\Omega$? Also is it possible for $u$ ...
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30 views

When do the paths of two dispersive waves cross?

Given $u_t=u_{xxx}$, consider initial conditions consisting of two wave packets. The first wave starts at $x_1=0$ and has wave number $k_1=2$. The second wave starts at $x_2=10$ and has wave number ...
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16 views

Using Banach Contraction Principle to prove existence of PDE solution

I want to prove there exists a solution $u(x,t)$ satisfying the following equation and initial conditions for $x\in\mathbb{R}, \ t\in(0,T]$ for some small enough $T$: $$ u_t-u_{xx}=u^2 , \ ...
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1answer
27 views

Laplace transform nonlinear equation

How can I apply the Laplace transform on a the following nonlinear PDE $$ \frac{\partial y}{\partial t}=\frac{\partial y^n}{\partial x}$$ where $n$ is a natural number? When I say apply the Laplace ...
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1answer
20 views

Conservation of norms by the 2-d euler vorticity equation

In the book of Filho Lopes, Weak solutions for the equations of incompressible and inviscid Fuid dynamics. Page 59 They want to prove the following: Take $w^{\epsilon}_0$ a ...
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2answers
51 views

Finding a solution to a PDE

EDIT: This is for a production scheduling problem with quadratic production and linear inventory costs. The goal is to \begin{equation*} \max_{u} \int_{0}^{T} -(c_{1}u^{2} + c_{2}y) dt ...
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2answers
49 views

Laplace Equation on the Corners and Boundary of a Rectangle?

Consider for some rectangle $[a,b] \times [c,d] \in \mathbb{R}^2$, we have a generic boundary value problem: \begin{equation*} \begin{cases} \frac{\partial ^2 u}{\partial x ^2}+\frac{\partial ^2 ...
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23 views

Proving a fourier transform expression with green's formuls

Using Green's formula, show that: $${\cal F}\left[\frac{d^2f}{dx^2}\right]= -w^2F(w) + \frac{e^{iwx}}{2\pi}\left(\frac{df}{dx} - iwf\right) \\(evaluated\ from\ -\infty\ to\ \infty)$$ last part is ...
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1answer
37 views

Nonlinear PDE Variation of Nonviscous Burger's Eqn.

Ultimately my goal is to find a candidate for the weak solution beyond the time when the classical solution does not exist and determine conditions on the motion of the shock that guarantees it is a ...
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17 views

Solutions to linear elliptic equations with presribed value at zero

Let $Ω ⊂ ℝ^n$ be a open subset containing the origin. Consider the following linear elliptic equation: $-\sum_{jk} a^{jk}(x)\partial_{jk} u + c(x)u = 0$ in $Ω$ and $u(0) ∈ ℝ$, where $c(x) \geq c > ...
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9 views

Question on The Inductive Proof of the Implicit Function Theorem

I'm trying to follow the proof given by Stephen G. Krantz and Harold R. Parks in "The Implicit Function Theorem: History, Theory, and Applications" I'm having trouble following some of the steps. Can ...
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A question on the well-posedness of of p-Laplacian

Can it be shown that the problem \begin{eqnarray} -\Delta_{p} u &=& f(u),\nonumber\\ u|_{\partial\Omega} &=& g, \end{eqnarray} well-posed similar to the case when $p=2$?.
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2answers
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End-of-semester presentation idea on PDE

I am supposed to give a 30-minute class presentation on any PDE subjects as end-of-semester project. Do you have any pet subject you would love to suggest? I have very little applied maths in my ...
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1answer
27 views

For which $p$ the sequence $x^n$ converges in the Sobolev space $W^{1,p}(I)$?

I would like to know for which $p$ the sequence $u(n)=x^n$ converges in the Sobolev space $W^{1,p}(I)$. Is it true that converges only for $p=1$? I find out this looking for which $p$ the Sobolev ...
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28 views

Question about representation of the eigenvalues of second order elliptic operator

let $Lu:=-\text{div}(A\cdot\nabla u)$, where $A$ is symmetric. Eigenvalues of $L$ is $\lambda_1<\lambda_2<\cdots$. By definition. (If exists a nontrivial solution $w$ such that $Lw=\lambda w$, ...
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Domain of $A^{1/2}$ on $L^2(\mathbb{T}^2)$

Let $A$ be a densely defined unbounded self-adjoint operator defined on $L^2(\mathbb{T}^2)$, where $\mathbb{T}^2$ stands for the 2-torus. It is known that $A$ is positive, that is, $\langle Au, ...
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24 views

Existence of Green's Function for Dirichlet Problem

Does anyone know where I can find a proof of the existence of Green's function, $G(x,x_0)$ on any nice enough domain $\Omega \subset \mathbb{R}^n$? Edit: This is for Laplace's problem
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2answers
45 views

Fourier inverse of a function to get dirac

I'm trying to get the dirac function from a fourier inverse tranform: $$\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-iw(x-x_0)}dw$$ It is this last step I am stuck on to get the conclusion. Original ...
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15 views

Uniqueness of solution to linear PDE, 2nd order in time with convection term

I am having troubles proving the uniqueness of the zero solution to the following Cauchy initial value problem for $\rho:\mathbb{R}_{\geq 0} \times \Omega \rightarrow \mathbb{R}$ with $\Omega ...
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1answer
17 views

Integration of Fundamental Solution of Laplace's equation.

I am currently reading Evan's PDE and am getting hung up on many of the more "technical details". This question may be very basic (multivariable calculus). I am given that the fundamental solution of ...
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1answer
44 views

Poincaré type inequality

Consider a sequence of functions $f_n \in H^1(M)$, the first Sobolev space of a complete (possibly noncompact) Riemannian manifold. If we have the normalization $\Vert \nabla f_n\Vert_{L^2} = 1$, ...
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1answer
32 views

Confusion over $\not\equiv$ and $\neq$ when applying boundary conditions

Here $k, A_1, A_2$ are constants Although $A_1$cos$kx+A_2$sin$kx \not\equiv 0$, $A_1$cos$kx+A_2$sin$kx$ can equal zero at certain values of $x$ For example if $A_1=1, A_2=1, k=1$ then ...
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1answer
47 views

A trace inequality with epsilon in Sobolev spaces

We know the standard trace inequality: for a bounded domain with certain boundary regularity, there is a $C>0$ such that $$ \|Tu\|_{L^2(\partial \Omega)}\leq C\|u\|_{H^1(\Omega)}, \quad \quad u\in ...
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nonlinear version of strong maximum principle

page 31 Vazquez has a nonlinear version of strong maximum principle which designated for Quasilinear parabolic PDE. Roughly it says, if $u_0\le v_0$, then either we have $u\equiv v$ or $u<v$ for ...
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Solution to klein-gordon type equation

While studying physics, I ended up having to find solutions for the following partial differential equation: \begin{equation} \left[ \frac{1}{2}\left( \frac{\partial ^{2}}{\partial \alpha ...
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find the fourier transform of $xf(x)$ appended

I've seen the method in which you prove this fourier transform, but what if you don't recognize that $$xf(x) e^{i k x} = \frac{1}{i} \frac{\partial}{\partial k} \Big[ f(x) e^{i k x} \Big] $$ would I ...
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1answer
48 views

Solving Eikonal Equation

The problem is the following: I have the bidimensional eikonal equation with non-constant propagation: $u_x^2+u_y^2=u^2$ The goal is: i) To find the characteristic strips for the parametrization ...
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1answer
22 views

Applying Fourier transform to a gaussian

Let $$G_\beta(w) = e^{\beta w^2}$$ Now I get the process of applying a fourier transform (or inverse) to get a new gaussian: $$G_\beta(x) = G_\beta(0) e^{\frac{-x^2}{4\beta}}$$ but in doing the ...
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1answer
11 views

Show behavior of Fourier Transform

If F(w) is the Fourier transform of f(x), show that F(aw) is the Fourier transform of (1/a)f(x/a). So if I apply a fourier transform to (1/a)f(x/a): $$ \frac{1}{2\pi}\int_{-\infty}^\infty ...
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1answer
37 views

Separation of variables for fourth order PDE

How do I solve: $$u_t = -u_{xxxx} + \pi^2u_{xx},$$ with BCs: $u_x(0,t)=u_x(1,t)=u_{xxx}(0,t)=u_{xxx}(1,t)=0$ and initial condition $u(x,0)=\cos(\pi x)$. We have been told that we can use ...
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1answer
21 views

Laplace operator

The question is that Derive a formula for $\Delta(\frac{f}{g})$ in terms of $f, g, \nabla f, \nabla g, \Delta f, \Delta g$. Naturally, I apply the rules of gradient and divergence, and yield ...