0
votes
0answers
29 views

Partial Derivative With Respect to $t$

What is $\frac{\partial v}{\partial t}$ if $v$ can be defined as $v(x,t,\zeta)=w(x(3t)^{-1/3},\zeta (3t)^{1/3})$?
0
votes
0answers
6 views

Modified Symplectic Euler

Simple harmonic motion: $y'= -z $, $z'= f(y)$ and the modified Symplectic Euler equation are $$y'=-z+\frac {1}{2} hf(y)$$ $$y'=f(y)+\frac {1}{2} hf_y z$$ deduce that the coresponding approximate ...
0
votes
1answer
28 views

Implicit Euler Scheme and stability

Find the fixed points of the implicit Euler scheme \begin{equation} y_{n+1}-y_{n}= hf(t_{n+1},y_{n+1}) \end{equation} when applied to the differential equation $y'=y(1-y)$ and investigate their ...
0
votes
0answers
20 views

A predictor-corrector method

A predictor-corrector method for the approximate solution of $y'=f(t,y)$ uses \begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P \end{equation} as predictor and \begin{equation} ...
2
votes
1answer
26 views

Checking a solution of a PDE

I have the following PDE: \begin{equation} -yu_x + xu_y = 0 \quad\text{where } u(0, y) = f(y) \end{equation} I derived a solution as follows: \begin{align} -yu_x + xu_y =& 0 \\ \iff& ...
0
votes
1answer
33 views
+50

Solving the LaPlace PDE by complete induction

I have a problem with an induction step that appears to be bothersome for me to comprehend it. Problem: Show that $f(x_1, \dots ,x_n)=(x_1^2+ \dots + x_n^2)^{1- n/2}$ for $(x_1^2 + \dots + ...
1
vote
1answer
59 views

Solution to Anisotropic Heat Equation

I am trying to find the solution to a 1-D anisotropic heat equation. The domain is a line segment of length L (i.e., it's a line segment extending from $x = 0$ to $x = L$). The form of the equation ...
0
votes
1answer
35 views

Cauchy-Reimann equations and harmonic functions

I have the following question which I am supposed to use the Cauchy-Riemann equations to prove: Let u and v have continuous second derivatives satisfying: $\frac{\delta u}{\delta x} = \frac{\delta ...
1
vote
1answer
24 views

Wave Equation Descending from 2D to 1D

I am stuck deriving the familiar d'Alembert formula using the Method of Descent, going from 2D to 1D. After using Kirchhoff's Formula, writing the solution independently of $y$ and some manipulations, ...
2
votes
0answers
24 views

Conservation of momentum for nonlinear Schrodinger equation

I am having trouble proving the following momentum conservation law. Given smooth compactly supported solution $u(x,t)\in \mathbb{C}$ where $(x,t)\in \mathbb{R}^n\times \mathbb{R}$ of $iu_t+\Delta u= ...
1
vote
1answer
16 views

characteristic curves tangent and gradient

In the PDE $aU_x+bU_y = 0$.This is equivalent to $\frac a{\sqrt {a^2+b^2}}U_x+ \frac b{\sqrt {a^2+b^2}}U_y = 0$. $\langle\frac a{\sqrt {a^2+b^2}}, \frac b{\sqrt {a^2+b^2}}\rangle$.$\nabla U=0$. Then ...
0
votes
1answer
23 views

how to introduce time into calculus of variations for image processing?

I'm studying some topics about calculus of variation applied to image processing. I'd like to understand how to introduce time parameter to evolve an image in an iterative way. For example, let's ...
1
vote
1answer
25 views

Coordinate method for PDE

Solving the PDE $au_x+bu_y+cu=0$ The PDE is transformed by the coordinate method via, $\begin{cases}x'=ax+by\\y'=bx-ay\\\end{cases}$. What I don't understand is how should I know I have to pick ...
0
votes
0answers
16 views

Hadamard variational formula Evans chapter 6 problem 15

This is Evans' chapter 6 problem 15. Consider a family of smooth, bounded domains $U(\tau) \subset \mathbb{R}^{n}$ that depend smoothly upon the parameter $\tau \in \mathbb{R}$. As $\tau$ changes, ...
1
vote
0answers
12 views

Prove that the Laplacian of the integral of a certain function is $0$

Let $f(x)$ be a continuous function. Define $$g(x,y)=\int_a^b\frac{yf(t)}{(x-t)^2+y^2}dt$$ Show that $\nabla^2g=0$
0
votes
0answers
19 views

Definition of outward normal velocity

Does anyone know the definition of outward normal velocity? Since I read some articles related to porous medium equation, but I don't know what outward normal velocity means. Also I googled it but ...
2
votes
2answers
43 views

What is wrong with the calculation about a wave equation here?

Consider the following wave equation in "negative" time: $$u_{tt}=\Delta u, \quad x\in {\Bbb R}^3, \ \color{red}{t<0}$$ with initial conditions $$ u(x,0)=g(x),\quad u_t(x,0)=0,\quad ...
0
votes
0answers
30 views

laplace-beltrami and angular momentum operator

Im trying to understand this formula: In $\mathbb{R}^n$ we say that the laplacian can be seen as (radial part and angular part). : $$\triangle=(\frac{\partial^2}{\partial ...
0
votes
1answer
33 views

If $u(0,t)=0$ then does $u_t(0,t)=0$

This came up while I was working with PDEs. What I'm asking isn't part of the question itself but it would definitely simplify my work if it was true. So if I had some function $u(x,t), x\in [0,l], ...
1
vote
2answers
27 views

Showing $ u =0 $ for a particular laplace equation.

Given open $\Omega\subset \mathbb{R}^n $ with smooth enough boundary, if for $x_0 \in \Omega$ there exists $ u \in C^2(\Omega\setminus\{x_0\}) $( we may as well take $u$ as smooth as we want) that ...
2
votes
1answer
54 views

Weak Laplacian of $\|x\|^\alpha$

Let $\alpha> 0$ and consider the function $\|\mathbf x\|^\alpha = (x^2 + y^2)^{\frac{\alpha}{2}}$ defined on $\mathbb R^2$. I want to compute the Laplacian $\Delta (\|\mathbf x\|^\alpha)$ in the ...
2
votes
1answer
32 views

Harmonic map into sphere

Let $B$ be the unit ball and $S$ the unit sphere in $\mathbb{R}^3$. Consider the map $u: B\rightarrow S$ defined as: \begin{equation} u^j(x)=\frac{x_j}{|x|}\quad\forall \ j =1, 2, 3. \end{equation}I ...
0
votes
1answer
58 views

How to show $\int_Q \varphi^2 (\Delta v)v_t = \int_Q |\nabla v|^2 \varphi \varphi_t - 2\int_Q (\nabla v \cdot \nabla \varphi) (\varphi v_t)$

Let $Q=\bigcup_{t \in (0,T)}\Omega \times \{t\}$. I have seen this identity for all $\varphi \in C_c^\infty(Q)$ such that $0 \leq \varphi \leq 1$: $$\int_Q \varphi^2 (\Delta v)v_t = \int_Q |\nabla ...
0
votes
1answer
39 views

is this intengral bounded?

Im just stuck beacause I dont know if this integral in bounded, I was trying to make a change of variable but I cant get to anything: (edited what need is that f is bounded for a fixed x) ...
0
votes
1answer
34 views

Sign of Laplacian at critical points of $\mathbb R^n$

Suppose we are in $\mathbb R^n$. What can we say about the sign of $\Delta u(\vec x)$ if u($\vec x$) has a local max/min at $\vec x$? I've tried looking at the reverse of the second partial derivative ...
1
vote
1answer
42 views

Proving Uniqueness of Solution to PDE on Boundary of Volume

The problem at hand is to show that, $\ \nabla^2u = m^2u $ has a unique solution where $\ u$ is specified on the boundary of some $\ V \subseteq \mathbb{R}^3 $. Using the Divergence Theorem I ...
0
votes
1answer
75 views

upper bound of some weird function with an exponencial

well im trying to find the proof of the following statement, but I cant go foward anymore: Let $\Omega= \mathbb{R}^{n} \times (0,\infty) $ $$w(x,t)= \sum^{\infty}_{k=0} ...
1
vote
1answer
54 views

Multivariable differential equation

Given $u=f(2x-y)+g(x-2y)$, show that $$2 \frac{\mathrm{d}^2 u}{\mathrm{d}x^2} + 5 \frac{\mathrm{d}^2 u}{\mathrm{d}x\,\mathrm{d}y} + 2\frac{\mathrm{d}^2 u}{\mathrm{d}y^2} = 0.$$ I'm not even sure where ...
1
vote
2answers
37 views

How can I calculate $ \int_0^t\int_{-\infty}^{+\infty}\frac{1}{\sqrt{4\pi (t-s)}}\exp\left({-\frac{(x-y)^2}{4(t-s)}}\right) y\ dy\ ds $?

I got the following integral $$ \int_0^t\int_{-\infty}^{+\infty}\frac{1}{\sqrt{4\pi (t-s)}}\exp\left({-\frac{(x-y)^2}{4(t-s)}}\right) y\ dy\ ds\tag{*} $$ when I solve the heat equation using the heat ...
0
votes
0answers
117 views

Change of Variables for PDE and equality of mixed partials.

I've come across some difficulty in working through a general coordinate transform of a PDE. The following boils down to transforming the differential operator $\partial x_1 \partial x_2$ into an ...
1
vote
1answer
44 views

Integration by parts with complex numbers

Suppose $u$ is a complex-valued wave function $u(x,y,z,t)$. Also, suppose you have the integral $\int(u\overline{u}_{t}+u_{t}\overline{u})dx$. I need to get ...
0
votes
2answers
39 views

Finding a minimum value of constrained minima.

What is the minimum value of $x^3y$ on the circle $x^2+y^2=4$? Thanks.
2
votes
2answers
84 views

What is meant by an open boundary when specifying boundary conditions of PDEs?

When speaking about boundary conditions of PDEs, one speaks about Dirichlet, Neumann or Cauchy boundary conditions specified over the boundary which can be closed or open. For example, we say that ...
7
votes
2answers
110 views

Very simple partial differential equation

I am solving $$ \frac {\partial f}{\partial x} = \frac y{x^2 + y^2} \\ \frac {\partial f}{\partial y} = \frac {-x}{x^2 +y^2} $$ As $y$ was held constant when the partial derivative with respect to ...
0
votes
0answers
34 views

Derivative in non orthogonal coordinates

I am trying to transform irregular shape in common Cartesian coordinates ($x-y$) into a regular shape in a generalized coordinates(e.g.,$u-v$), in which the transform can be defined as $u=u(x,y)$ and ...
10
votes
1answer
94 views

Non-divergence form of a 2nd order PDE

This might be a trivial question but I'm very rusty with regards to calculus and am new to PDEs. How would you write the following second order quasilinear equation in it's non-divergence form: The ...
0
votes
1answer
51 views

Finding the Boundary Conditions for a Laplace's Equation in Polar Coordinates

I have solved Laplace's equation in Polar Coordinates for the scalar electric potential in a circle of radius R and have the solution $$ \phi(r,\varphi) = \phi_{0} + ...
3
votes
2answers
94 views

Solution of a partial differential equation.

Find $u$ if $$\dfrac{\partial^2 u}{\partial x^2} = 6xy, \,\,u(0,y) = y, \,\,\dfrac{\partial u}{\partial x}(1,y)=0.$$ I have tried by laplace transformation $$\displaystyle ...
3
votes
2answers
40 views

Solve Helmholtz equation

$$U_{xx}+U_{yy}+k^2U=0$$ Solve by separation of variables by assuming $u(x,y)=X(x)Y(y)$ with the following conditions: $$ U(0,y)=0,\,\, U(2,y)=0,\,\, U(x,0)=0,\,\, U(x,1)=0, $$ This is ...
1
vote
1answer
41 views

To Show $ x.\nu > 0 $ on convex domain.

$\Omega\subset \mathbb{R}^n $ be a convex domain, $\nu$ be the unit outward normal of the smooth boundary, then I have to show that $ x.\nu > 0 $ for all $ x \in \partial \Omega$. Here is my try. ...
4
votes
1answer
47 views

Conservation Law for Heat Equation on Infinite Domain

Let $$ u_{t}(x,t) = \Delta u(x,t) \space \space \space \text{ for } \space t \ge 0 \space \space\space\space , \space x \in \mathbb{R}^{n} $$ and $$ u \rightarrow 0 \space \text{ as } \space ||x|| ...
2
votes
0answers
57 views

Change of variables for linear differential operators

I am trying to find an expression for a change of variables (invertible and $C^{\infty}$) of a linear differential operator in $\mathbb R ^d \newcommand{\t}{\tilde} \newcommand{\p}{\partial}$. i) ...
1
vote
1answer
35 views

Composition of linear differential operators is a linear differential operator

I will use multi-index notation: $$ \newcommand{\p}{\partial} P = (p_1, \dots, p_d), |P| = p_1 + \dots + p_d, \p^P u = \dfrac{\p ^{|P|} u}{\p x_1 ^{p_1} \dots \p x_d ^{p_d}} .$$ Let $A = \sum_{|P| ...
1
vote
1answer
34 views

Change of Variables from Sphere to Plane

Say we are in the space $\mathbb{R}^{n}$. Consider the $n-1$ dimensional surface measure $dS$ on the boundary of the upper half sphere. We can define the coordinates on this sphere by ...
0
votes
1answer
63 views

$C^2$ function on $\mathbb{R}^2$

Let $T:\mathbb{R}^2 \mapsto \mathbb{R}$ be a function that its second partial derivative is continuous ($T$ is $C^2$). If $T(0,y)=0$ and $T_x(0,y)=0$ for all $y$, then show that $$T(x,y)=x^2 S(x,y)$$ ...
5
votes
1answer
117 views

PDE : Mixture of Wave and Heat equations

Today I was given the following equation : $$\frac{1}{c^2}u_{tt} + \frac{1}{D}u_t = u_{xx}$$ with initial conditions : $u(x,0) = 1$ if $|x|<L$ and $0$ otherwise, $u_t(x,0) = 0$. So fairly simple ...
3
votes
1answer
105 views

Laplace's Equation with Neumann BC

Hi fellow math enthusiasts, I am currently working on some research to do with the electric field induced within the brain via magnetic stimulation. I am trying to solve the partial differential ...
0
votes
2answers
28 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 ...
1
vote
1answer
56 views

Difference quotient (pde)

Let $u: U\subset\mathbb{R}^n\rightarrow\mathbb{R}$. The Difference quotient of $u$ is defined by $D_k^hu(x)=\dfrac{u(x+he_k)-u(x)}{h}$ with $h\in\mathbb{R}$, $0<|h|<\textrm{dist}(V,\partial ...
1
vote
1answer
53 views

On a system of PDE

I would like to know what is the set of solutions to the following PDE. I think it consists of just constants, but I need help to prove. Let $f_1(p_1,p_2)$ and $f_2(p_1,p_2)$ be two functions. The ...