0
votes
1answer
16 views

How rigorous is multiplying both sides of an eqaution for the differential of a function?

I have to solve this equation: $$ -C_0 f + \frac{1}{2}f^2 +\frac{d^2 f}{d X^2}=A $$ where $C_0$ and $A$ are two real nonzero constant; $f:\mathcal{R}\to \mathcal{R}$ I have seen that the person who ...
0
votes
0answers
32 views

On an interesting boundary condition

So I am tackling an interesting boundary condition, where $B(Du)=0$, for $x\in\Omega$, where $B$ is the signed distance function to $\Omega^*$ (where $\Omega,\Omega^*$ are convex domains in $\Bbb ...
0
votes
4answers
47 views

$f(x,y) = g(\sqrt{x^{2}+y^{2}})$ Prove that f is differentiable at $(0,0)$ iff $g'(0)=0$

$f(x,y) = g(\sqrt{x^{2}+y^{2}})$. Prove that f is differentiable at $(0,0)$ iff $g'(0)=0$ This was a question on my midterm a few days ago. I've been thinking about it for a while and still cannot ...
2
votes
2answers
41 views

An object is travelling in a straight line. Its distance, s meters, from a fixed point at time t seconds is given by the expression

$$s=t^3−t^2−6t$$ a) Find ds/dt when t=3 and interpret this result. b) Find d^2s/dt^2 when t=3 and interpret this result. c) Find the time in seconds when the velocity is 2m/s (d) Using the ...
2
votes
1answer
114 views

Second Derivative of Eigenvalue

Consider the potential $V_t=tx$ and the corresponding Schrödinger operator $H_t=- \frac{\partial ^2}{\partial x^2}+V_t$ on $L^2([0,R])$ with Neumann or Dirichlet boundary conditions. Let ...
1
vote
0answers
52 views

variational derivative

Let $\Omega \subset \mathbb{R}^n,\ n=1,2 \mbox{ or } 3$. Define the following energy $$E=\int_{\Omega} \frac{1}{\varepsilon}\left[f(u)+\frac{\varepsilon^2}{2}|\gamma(n)\nabla u|^2\right]\,dx$$ ...
0
votes
1answer
64 views

Differentiation of the transpose of a vector? [closed]

Suppose $s$ is a scalar, and $x$ is a vector, how would I calculate $$ \left(\frac \delta {\delta x} (x^T s)\right) $$Basically I couldn't find any reliable source letting me know how to ...
1
vote
2answers
35 views

Prove that the average of $D_wD_wf(x_0,y_0)$ over all unit vectors $w$ is equal to $\frac{1}{2} \Delta f(x_0,y_0)$ for any smooth function $f$.

Here is a challenge problem from my math professor: Let $w$ be a unit vector in $\mathbb{R}^2$, and let $D_w$ denote the directional derivative with respect to $w$. Prove that for any smooth ...
1
vote
0answers
33 views

Legendre transform for HJB PDE

I am trying to understand section 2.3 of the following article: http://www.princeton.edu/~sircar/Public/ARTICLES/montreal.pdf . I think I understand that $H_v(t,g(t,z)) =z$, because by definition of ...
2
votes
1answer
94 views

Symmetry in partial derivatives.

I was wondering how the relationship $$x_j \partial_i f(x) = x_i \partial_j f(x)$$ means, that a function has rotational symmetry? I mean with rotational symmetric, that the value of $f$ at a point ...
0
votes
2answers
38 views

Second order quasilinear PDE

Some quick question about PDE's. Only recently started studying PDE's so this might be trivial. The second-order quasilinear elliptic equation is given by: $ -\sum_{i=1}^{n} \frac{\partial}{\partial ...
0
votes
1answer
23 views

derivation of a differential Eq

Look at $F(u) = \frac{\partial u}{\partial t}-\nabla \cdot (a(u)\nabla u)$. My question is, what $F'(u)$ is. I need this for the linearization of a PDE. The idea is to use the newton-approximation. ...
0
votes
2answers
42 views

Help finding the general solution of a (partial?) differential equation.

I've been asked to find the general solution of the differential equation: $$ y^{'} - y^3 = y^3e^x\qquad\text{, satisfying}\quad y(0)=1 $$ To solve it I did the following: $$ y^{'} - y^3 = ...
0
votes
0answers
26 views

Why must a stochastic process be at least second order in terms of differential equations?

A first order differential equation in $q(t)$ has a unique path through each possible value of $q(0)$. This is opposed to a stochastic process (e.g. random walk), where any place might be "hopped ...
0
votes
1answer
25 views

Second Order forward finite difference scheme

Show that $d^2u/dx^2(x_i)=[(-u_{i+3})+(4u_{i+2})-(5u_{i+1})+2u_i]/h^2 +O(h^2)$ provided all terms in the expression are well defined is a second order finite difference scheme for second order ...
2
votes
0answers
207 views

Use energy method to prove uniqueness of a parabolic PDE

A question in my PDE class is Suppose $a(x) = a_{ij}(x)$ takes values in the class of symmetric, positive definite $n \times n$ matrices, $b(x)$ takes values in $\mathbb{R}^{n}$. Consider the PDE ...
0
votes
1answer
252 views

Use energy method to solve modified heat equation

A question in my PDE class is Suppose $a(x) = a_{ij}(x)$ takes values in the class of symmetric, positive definite $n \times n$ matrices. Consider the PDE $$u_t = \sum_{i,j=1}^n ...
1
vote
1answer
249 views

Heat equation with Neumann boundary condition

Background: In our PDE class we explored the heat equation with Dirichlet boundary condition $$u_t - \Delta u = 0 \;\text{ in } \Omega \subset \mathbb{R}^n \;\text{bounded}\\ u = u_0(x) \;\,\text{at} ...
3
votes
1answer
130 views

Integrate $u_t - \Delta u = 0$ to get $\frac12 \frac{d}{dt} \int_{\Omega} u^2 + \int_{\Omega}|\nabla u|^2 = 0$?

In my PDE class, my instructor wrote the following notes: Consider equations $u_t - \Delta u = 0$ in $\Omega$, where $\Omega \subset \mathbb{R}^n$ is bounded. Suppose boundary conditions $u = u_0(x)$ ...
0
votes
1answer
81 views

Partial Differential Equation Conversion

Convert the partial differential equation $u_{x}-3u_{y}=2x$ from $u(x,y)$ to $u(\varepsilon, \eta)$ given $\varepsilon = x$ and $\eta = 3x + y$. Edit: Convert the partial differential equation ...
0
votes
1answer
27 views

Affects of a coordinate transformation

I am attempting to solve a PDE of $f(r,t)$, where $r\in[0,g(t)]$ is a spacial coordinate and $t$ is time. The PDE is coupled to to an ODE for $g(t)$. I wish to simplify the problem by defining a new ...
4
votes
1answer
228 views

Step in derivation of Euler-Lagrange equations of motion

From http://www.mathpages.com/home/kmath523/kmath523.htm Variations in $x,y,z$ and $X$ at constant $t$ are independent of $t$ (since each of these variables is strictly a function of $t$), so we ...
5
votes
2answers
582 views

What is the intuition behind a function being 'weakly differentiable'?

As part of an optimization paper I am reading now, they are talking about a function $g$ being "weakly differentiable". I looked it up on the wiki but I do not have enough context to start cracking ...
2
votes
2answers
353 views

Find derivative of convolution with gaussian

Let $A(\sigma)$, $\sigma > 0$ be an operator that acts on bounded continuous functions $f$ on $\mathbb{R}$ by the rule $$ (A(t)f)(x) = \int\limits_{\mathbb{R}} f(y)\frac{1}{\sqrt{2 \pi ...
0
votes
1answer
548 views

Finding a function given its partial derivatives

I need to find/define a function $G(x_1(t),x_2(t)) : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ such that the following holds: $$ \frac{\partial G(x_1(t),x_2(t))}{\partial x_1(t)} =u_1(t)$$ and $$ ...
0
votes
3answers
244 views

How to prove this partial derivative?

Consider $u:\mathbf{R}\times\omega\rightarrow\mathbf{R}$, where $\omega\subset\mathbf{R}^{n-1}$ is a bounded domain. For each $y\in\omega$ and each $\lambda>0$, consider ...
0
votes
1answer
60 views

why $\left(\nu \nabla{u}\right)\nu=\frac{\partial u}{\partial \nu} \nu$ ?

I think the following question is one simple but I need your help :) So, how can I prove that : $$\left(\nu \nabla{u}\right)\nu=\frac{\partial u}{\partial \nu} \nu$$ ? and second question, why: ...
1
vote
1answer
66 views

Identify the distrionbutional derivative with classical derivative?

I am reading Rudin's Functional Analysis and got quite confused by his proof for theorem 7.25, which he calls Sobolev's Lemma. In proving the theorem, he defines the function $F$, and calculates its ...
0
votes
0answers
53 views

which subarea of math text book study about the theory of smooth function?

In another word, which subarea does the theory of smooth function have? I would like to know the list of book on analysis that i could learn more about smooth function.
1
vote
1answer
82 views

The definition of a directional derivative

We're given that for $e \in \mathbb{R}^2$ the directional derivative of $u$ in the direction of $e$ is, $$\frac{\partial u}{\partial e}(x,t):= \lim_{h \to 0}\frac{u((x,t) + he) - u(x,t)}{h} = ...