0
votes
0answers
8 views

Four dependent variables - How to find the lowest (min) of one of the variables.

I am new here and I have a math problem which I hope I can get some help from the geniuses out there. I have a formula of the form: $E = \int_t P(f)dt$ where $E$ = Energy, $P$ = Power, $f$ = ...
0
votes
1answer
19 views

Differentiating by Partial Differentiation.

Among the methods for finding derivatives, differentiating by partial differentiation looks interesting. Is there any general proof for this method. For instance my text mentions this method. Let ...
2
votes
0answers
30 views

Taking partial derivatives over multiple summations

I have the following equation obtained from one of the models. $\mathcal{H} = \sum\limits_{D} \sum\limits_{W}n(d,w)\sum\limits_{Z} p(z|d,w)[\log{p(d)}+\log{p(z|d)}+\log{p(w|z)]}$ I need to take ...
0
votes
0answers
34 views

Estimate for $f^2$ on a Ball from below

Let $f\in C^\infty(B_R(0))$, where $R>0$. For $0<\sigma<1$ require the following properties on $f$: $$ 0\leq f\leq 1,\ f=1 \text{ on } B_{(1-\sigma)R},\ f=0 \text { on }\partial B_R,\ ...
0
votes
2answers
38 views

Finding the partial derivatives of $V (x, y) = U (x, y)e^{−ax−by}$

I think I did something wrong, so I was hoping someone might be able to show me the solution Two functions $V (x, y)$ and $U (x, y)$ are connected by the equation $$V (x, y) = U (x, y)e^{−ax−by}$$ ...
2
votes
1answer
74 views

ODE $d^2y/dx^2 + y/a^2 = u(x)$

Does the following ODE: $$d^2y/dx^2 + y/a^2 = u(x)$$ have a solution? where $u(x)$ is the step function and a is constant.
3
votes
4answers
85 views

Differential equation which has following solution $y=\frac{1}{1+\exp(ax)}$

Is there any linear differential equation which has following solution $$y=\frac{1}{1+\exp(ax)}$$ $a$ is constant. something like: $$ y'' + by' +cy + \alpha = 0$$ where $b$, $\alpha$ and $c$ are ...
2
votes
1answer
35 views

Solve 2 connected ODEs describing a domain

This problem confused me for a long time. I have 2 ODEs which describe part of our domain. They are connected at middle: $$ \frac{d^2}{dx^2} u = -a, x<x_0 $$ $$ \frac{d^2}{dx^2} u - \frac{u}{b^2}= ...
1
vote
4answers
68 views

Differential equation with the solution of $(1+ax/2)\exp(-ax)$

Is there any linear differential equation which has following solution $$y=(1+ax/2)\exp(-ax)$$ $a$ is constant.
0
votes
1answer
48 views

How do the existence and $L^p$ integrability of weak derivative affect the smoothness of Sobolev functions?

In the definition of Sobolev spaces, let us say the space $W^{1,p}(\Omega)$, where $\Omega$ is an open subset of $\mathbb{R}^n$. What contributes more to the smoothness of a function $f$: 1-The fact ...
0
votes
1answer
23 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
37 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
51 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
43 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
135 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
56 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
74 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
36 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
42 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
107 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
59 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
26 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
50 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
30 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
217 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
271 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
285 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
139 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
87 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
240 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
649 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
378 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
657 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
249 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
67 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
84 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} = ...