0
votes
0answers
31 views

A PDE problem related to the ratio of populations

Let $m>0$ in $\bar{\Omega}$ be a given nonconstant function, where $\Omega\subset \mathbb{R}^n$ is a bounded smooth domain. Then consider the following elliptic modeling problem: $$ \Delta ...
0
votes
0answers
142 views

Find a scalar potential of $v(x, y)= g(y/x)(-1/x, 1/y)$

Let $v(x, y)= g(y/x)(-1/x, 1/y)$ be a vector field on $Ω$ where $Ω := [(x, y) : x > 0, y > 0]$. $g:\mathbb{R}→\mathbb{R}$ is continuous. Find a scalar potential of $v$ in terms of an ...
0
votes
0answers
9 views

Existence of a sequence of regular values converging to a given point

Suppose I have a function $f \in C^{1}(\mathbb{R}^{2}; \mathbb{R})$, $f \geq 0$, with compact support. Am I correct to think that the level sets of $f$ corresponding to regular values (i.e. values ...
0
votes
0answers
15 views

Regularity of the border of a set

How is regularity defined on the border of a set? Suppose I have an open subset $U \subset \mathbb{R}^{n}$ with border $\partial U$, what does it mean for $\partial U$ to be Lipshitz? $C^{1}$? ...
1
vote
1answer
48 views

Express partial derivatives of second order (and the Laplacian) in polar coordinates

$z=f(x,y)$ where $x=rcosθ$ and $y=rsinθ$ Find $ \frac{\partial z}{\partial x}$ and $ \frac{\partial^2 z}{\partial x^2}$ I'm having big troubles with using chain rule, in particularly the second ...
1
vote
1answer
41 views

Writing the squared sine as a Legendre polynomial of cosine

I'm just getting learning about how Legendre polynomials come about when considering product solutions in spherical coordinates with azimuthal symmetry. I'm trying a problem on my own, and I'm a bit ...
1
vote
1answer
35 views

Heat equation: Why are these ratios of functions constant

One can solve the heat equation $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} $$ by a separation of variables such that $u(x, t) = f(x)g(t)$. Substituting this into the ...
1
vote
0answers
19 views

System of PDE's with unknown functions

So by messing around with some stuff in my own research I came across this problem and I have no idea how to proceed. I suspect it may have something to do with solving systems of PDE's but I could be ...
0
votes
1answer
38 views

Integration by parts on all of $\mathbb{R}^n$ with $n>1$

So this came up as I was thinking about the uniqueness of solutions to the wave equation. I have seen proofs for uniqueness on all of $\mathbb{R}$ or on bounded subsets of $\mathbb{R}^n$, but never ...
1
vote
0answers
61 views

a simple calculation

Can anyone see how (1) lead to (2)? \begin{align} ...
1
vote
1answer
30 views

Partial differential equation, mixed derivatives

What can be concluded from following equation: $$\frac{\partial f(x,y)}{\partial x}-\frac{\partial g(x,y)}{\partial y} = 0$$ where $f(x,y)$ and $g(x,y)$ are functions of two independent variables ...
1
vote
2answers
67 views

Evans 's PDE proof

Again, I got stuck. Please help me to understand the following: What is the meaning when you change from integration over the Ball B(x,r) to the surface integration dB(x,s), with another integral ...
1
vote
1answer
15 views

Solving PDE by Canonical form transformation

For reference, the entire equation to be solved for $u(x,y)$ is: $A= -2x^2-8xy-8y^2+42x-14y$ $B= -5x^2-20xy-20y^2+105x-35y$ $C= 3x^2+12xy+12y^2-63x+21y$ $E= 28x+56y$ $K= -14x-28y$ where ...
3
votes
1answer
121 views

Find vector field given curl

I have an equation $\nabla \times \vec{B} = \mu_{0}\vec{J}$, where $\vec{J} = \left\langle f(x,y), g(x,y), 0 \right\rangle$ and need to solve for $\vec{B}$. I've looked elsewhere on here for how to ...
1
vote
2answers
50 views

What is the order of the PDE $\newcommand\pp\partial\frac{\pp^2u}{\pp x^2}+\frac{\pp^3u}{\pp x^2 \pp y}+\frac{\pp^2u}{\pp^2y}=xy\frac{\pp u}{\pp x}$? [closed]

The order of the differential equation $$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^3 u}{\partial x^2 \partial y}+\frac{\partial^2 u}{\partial^2 y}=xy\frac{\partial u}{\partial x}$$ is ...
0
votes
0answers
75 views

Solving the Telegraph Equation using Partial Differential Equations and Sturm-Liouville theory

I've been asked to do the following question, and I've got through the brunt of it (so this is going to be a rather long question...), but I'm just having a bit of trouble applying Sturm-Liouville ...
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
3answers
42 views

Finding the change of variables to transform $u_{tt} - u_{xx} = 0$ into $u_{rs} = 0$

I'm just beginning to introduce myself to partial differential equations and one of the first problems presented in the textbook I have literally no idea how to do. I think the author intended the ...
2
votes
0answers
40 views

Proof of Second Partials Test

How does one rigorously prove the second partials test without firstly assuming that $D(a,b)=AC-B^2$ that states the following: $ A=\frac {\partial^{2}f(a,b)}{\partial x^{2}},B=$$\frac ...
1
vote
1answer
51 views

To show unique solution for the Laplace equation

The problem is in the top while its weak form at the end, source $\hskip 1in$ I know that the solution is unique because the boundary condition is Dirichlet. But I want to show this. How can you ...
1
vote
0answers
29 views

Existence and Uniqueness of Solutions of PDEs

I have been looking into the Cauchy-Kovalevskaya Theorem where one can "establish the local existence of analytic solutions to a system of PDEs". I wanted to see an application (for example, see ...
2
votes
1answer
44 views

Third Order PDE written as a System of (Linear) First Order PDEs

I need to rewrite the PDE $$f_{y}+ff_{x}+f_{xxx}=0,$$ where $f=f(x,y)$ as a system of first order quasi-linear PDEs. I have no idea how to tackle this problem. Any form of help will be appreciated. ...
5
votes
0answers
136 views

Why is $a_{n}(x,y)=a_{n}(y)$?

This particular question is connected (with a slight variation in the definition of $g$) to an earlier question. The link is here. The specifics are: Given that $u(x,y)$ is the solution of a PDE ($x$ ...
1
vote
0answers
74 views

Implicit Function Theorem (Two Variables)

While reading through a paper I came a cross a result (due to the Implicit Function Theorem) that I cannot derive. Let $u(x,y)$ be the solution of a PDE ($x$ and $y$ are independent variables). We ...
1
vote
3answers
54 views

Solving $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 0$ by changing variables

Transform the differential equation $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 0 $ by introducing new variables $x = u+v$ and $y=u-v$. then solve it. I which I could show ...
2
votes
1answer
138 views

Rankine Hugoniot Jump Condition Derivation

I follow the majority of this derivation I just do not understand where the two terms highlighted in green come from.
1
vote
0answers
36 views

Partial Differential Equation with a flux term

I don't understand why $\phi=\frac{1}{2}u^2$ is the flux in this case? Recall how we derived all our equations: Take an interval $[a,b]$ and consider $$\dfrac{\mathrm d}{\mathrm ...
1
vote
0answers
25 views

Non Linear general kinematic wave equation

I am rather confused by this section of my non-linear waves notes. In the parts I have underlined in green $c(u_0(\xi))$ is defined as a constant and then as a variable even though in both instances ...
4
votes
0answers
76 views

Second degree partial differential equation with variable-change

Edit: @Etienne mentioned that I did a typo, writing $u_y' = -xye^{-y}$ instead of $u_y' = -xe^{-y}$. I've corrected that in the calculations and now it's closer to being correct! Though I still miss ...
2
votes
1answer
170 views

Show that $L[v^2] := \Delta(v^2) + \frac{2}{w} \sum_{i=1}^n w_{x_i} (v^2)_{x_i} \ge 0$

Let $u$ be a function such that $$ \Delta u + \lambda u = 0 $$ for some $\lambda \in \mathbb R$, also let $w$ be a function such that $$ \Delta w + \beta w < 0. $$ for some $\beta \in \mathbb R$. ...
3
votes
2answers
45 views

Variable substitution in second order PDE

Consider the the PDE $$A(x, y)\partial_{xx} u + B(x, y)\partial_{xy}u + C(x, y)\partial_{yy}u=h(x, y) $$ Now I want to make a variable substitution $\xi=f(x, y), \eta=g(x,y)$, so I can get $u$ as a ...
0
votes
0answers
18 views

Poisson equation on a square

Studying PDEs from the notes of my professor, and there's a part I don't understand about seeking a solution for the Poisson equation on a square. Let's start from the beginning though. We want to ...
1
vote
0answers
15 views

Question about extending a solution to Monge-Ampere solution

I am interested in solutions to the Monge-Ampere equation for a smooth function $h(x,y)$ of two variables(though I suppose I could try to make do with $C^2$ solutions). The equation is: ...
1
vote
2answers
21 views

Solve this PDE using a change of variable.

Solve the following PDE using a change of variable: $$ \alpha^2 \dfrac{\partial^2 z}{\partial x^2} - \beta^2 \dfrac{\partial^2 z}{\partial y^2} = 0$$ This is my attemp: Let the following change of ...
1
vote
1answer
52 views

When is a given matrix-valued function the Jacobian of something?

Let $F$ be an $n\times m$ matrix of real-valued functions which are defined and smooth on a neighborhood of a point $p\in \mathbb{R}^m$. Under what conditions is it possible to find a smooth function ...
0
votes
1answer
48 views

Differentiation Formula for Moving Regions.

I've run into a few calculations in a series of textbooks/papers that require differentiating an integral with a changing region. In particular, I'd like to know if $f(x,t):\mathbb{R}^d\times ...
0
votes
1answer
29 views

Does every function with $f_x,f_y>0,f_{xx},f_{yy}<0$ with particular condition have to satisfy $f_{xy}/f_{xx} = -x/y$?

Let continuous real functions $f$ of two real variables $x,y$ satisfy the following condition: (Let us define $f_{xx}:=\frac{\partial^2 f}{\partial x^2}$ and $f_x:=\frac{\partial f}{\partial x}$, and ...
0
votes
1answer
47 views

Clarifying definition of outward unit normal

I would like to figure out how to properly define the outward unit normal vector $\nu$ for a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$ ($n \ge 2$). I am ...
1
vote
0answers
76 views

Solving a system of integral-partial differential equations

Hi I am a student in electrical engineering. Currently I am facing a difficult problem solving a coupled integral-differential partial equations arising from mean field game. The problem is similar as ...
1
vote
0answers
37 views

Can you explain this partial derivative approximation?

How do they go from the left side to the approximation on the right? What does $j$ mean in this summation? $$ {\partial f_i^n \over \partial t} \approx \sum_j {\partial f_i^n \over \partial \hat x_j} ...
0
votes
1answer
29 views

existence of the solution of Neumann problem in $\mathbb{R}^3$

Let $D\subset \mathbb{R}^3$. Let $D$ be connected subset of $\mathbb{R}^3$. Show that there is not any solution of the system of equation \begin{equation} \Delta u=f \text{ in } D, ...
0
votes
1answer
21 views

Eliminate unknown function $f$ by obtaining a PDE

Question Let the funcion $z = z(x,y)$ be given by the equation $z = xy + f(x^2 -y)$, where $f$ is an arbitrary $C^1$ function. By forming the first partial derivatives of $z = z(x,y)$: $p = z_x$ and ...
2
votes
1answer
25 views

Value of the origin.

Here let $\Delta u = 0$ in the unit ball and $$u(1 , \varphi,\theta) = \sin^2 \varphi.$$ What is the value of u at the origin? So I know that this problem uses green's first identity and I suppose ...
1
vote
1answer
38 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 ...
2
votes
1answer
31 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
66 views

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
88 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
64 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
52 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, ...
3
votes
0answers
61 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= ...