2
votes
2answers
45 views

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

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
27 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
32 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
35 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
39 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
22 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
40 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
133 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
70 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
52 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
85 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
30 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
24 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
73 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
111 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
44 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
16 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
18 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
50 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
42 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
36 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
71 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
36 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
25 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
20 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
37 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
29 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
80 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
58 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
43 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
46 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
30 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
29 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
29 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
38 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
26 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
53 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
48 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
38 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
35 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
59 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
39 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
65 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
42 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) ...