Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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2
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1answer
49 views

Solving Lagrange equation.

Please help me solving the equation. I found one of its solutions: $c_1=(xz)/y$ But another one is given as $c_2=(x^3/y)+x$ in the text book. But I cont find. Thank you.
0
votes
1answer
63 views

basic question in real analysis/Question about Schwartz spaces

Define the Schwartz space by $$ S(R) = \{ f \in C^{\infty} ; \displaystyle\sup_{x \in R} |x^{\alpha} \partial^{\beta f(x)}| < \infty\ \forall (\alpha,\beta) \in N\times N \} $$. Let $f \in ...
2
votes
1answer
94 views

Do I have to use a maximum principle?

Let $\Omega\subset C^0$ a bounded domian in $\mathbb{R}^2$. Let $u\in C^2(\Omega)\cap C(\overline{\Omega})$ be a non negative classical solution of $$ ...
0
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1answer
41 views

Linear PDE of degree 2: general form and an example

As the general form of a linear PDE of degree 2 we wrote $$ (Lu)(x):=\sum_{i,j=1}^{n}a_{ij}(x)\frac{\partial^2 u}{\partial x_i\partial x_j}+\sum_{i=1}^{n}b_i(x)\frac{\partial u}{\partial ...
1
vote
0answers
42 views

Green's formula: $ \int_{a}^{b} [uL(v)-vL(u)]dx=p\left( u\frac{du}{dx}-v\frac{du}{dx}\right)\big|_{a}^{b}$

I'm reading a book on PDE's and they introduce ''Green's formula'' in a rather (to me) abrupt way. It is used to derive that for a steady state heat PDE $u(x)$ is given by: $u(x)=\int_{a}^b ...
2
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0answers
86 views

Find a maximum principle for elliptic PDE of degree 2 in divergence form

In our reading we had the following maximum principle for elliptic PDE of degree 2: Let $u\in C^2(\Omega)\cap C(\overline{\Omega})$ be a solution of the linear Dirichlet task $$ ...
0
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2answers
513 views

Understand 1D FEM solution using quadratics elements

I'm a bit confused about applying the FEM using piecewise linear functions. I think I get understand how to use linear functions. We use the hat function for each element and the solution is ...
3
votes
1answer
166 views

Neumann BV problem on disk (weak vs classical solution)

I am tring to solve $\bigtriangleup u =-1$ such that the normal derivative vanishes at the boundary where the domain is the unit disc. In polar coordinates I got I got $u(r)=-1/4 r^{2} +1/2 \ln(r)$ ...
2
votes
1answer
81 views

find the general integrals of the given P.D.E

I tried to find the general integrals of the given P.D.E in the yellow box. And I found $c_1$. But I cannot find another one, say $c_2$. Please help me finding $c_2$. Thank you.
1
vote
1answer
526 views

Find the integral curves of the equation

Question: Find the integral curves of the equation: $$\frac{dx}{y^2x-2x^4}=\frac{dy}{2y^4-x^3y}=\frac{dz}{2z(x^3-y^3)}$$ I could not find any similar example to understand this type of questions ...
0
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1answer
365 views

PDE - transformation in normal form

Consider the PDE $$ (1+x^2)^2u_{xx}-u_{yy}+2x(1+x^2)u_x=0\text{ in } \mathbb{R}^2. $$ Transform the PDE in normal form (with mixted derivative). And find a general solution of the ...
0
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1answer
45 views

Is this a hyperbolic PDE?

Is the PDE $$ (1+x^2)^2\frac{\partial^2 u}{\partial x^2}-\frac{\partial^2 u}{\partial y^2}+2x(1+x^2)\frac{\partial u}{\partial x}=0\text{ in }\Omega:=\mathbb{R}^2 $$ hyperbolic? ...
2
votes
0answers
162 views

Solutions of Laplace's Equation/Landau & Lifschitz Fluid Mechanics

in Fluid Mechanics by Landau and Lifschitz (a fairly well-known book - I'm just mentioning it for those who might have it) they are discussing "As we know, Laplace's equation has a solution l/r, ...
8
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1answer
312 views

Heat Kernel Property

Let $\phi$ be the Heat Kernel in $\mathbb{R}^n$, i. e. $$\phi (x,t)={(4\pi t)}^{-n/2}\exp\left( - \frac{\mid x \mid ^2}{4t}\right)$$ and let $u$ satisfy Heat equation. Show that: ...
0
votes
1answer
43 views

why negative coefficient robin condition does not unique?

For the 2-dim Laplace equation on unit circle, Robin condition is given by $$u_r(1,\theta)-u(1,\theta)=\beta(\theta).$$ How to give a counter example to show the solution is not unique? By the ...
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2answers
49 views

Is the substitution given the question incorrect?

Show that an equation of the type $$u_{t}+6uu_{x}+u_{xxx}+a'(t)u_{x}=0$$ can be transformed into the KdV equation ($u_{t}+6uu_{x}+u_{xxx}=0$) by the transformation $$\begin{cases} \xi = x+a(t) \\ \tau ...
0
votes
1answer
268 views

Fourier Sine Series extension

If $\phi(x)$ is any function on $(0, l)$, derive the expansion $\displaystyle\phi(x) = \sum c_n \sin\left(\left(n + \frac{1}{2}\right) \frac{\pi x}{l}\right)$ for $0 < x < l$ by the following ...
3
votes
1answer
187 views

Global approximation theorem in Sobolev space

${\bf Global\ Approximation\ Theorem}$(251 page inEvans's PDE book) : If $U$ is ${\bf bounded}$ in ${\bf R}^n$, then for $u\in W^{k,p}(U)$, there exists $u_n \in C^\infty (U)\cap W^{k,p}(U)$ such ...
3
votes
1answer
124 views

Prove the uniqueness of solution of the PDE

Consider the problem $$\begin{cases} u_{tt}=u_{xx}-2\alpha u_x+\alpha^2u,&(x,t)\in\mathbb{R}^2\\ u(x,0)=f(x),\ u_t(x,0)=g(x),&x\in\mathbb{R}, \end{cases}$$ where $\alpha\geq 0$ is ...
2
votes
1answer
59 views

Two different defintions of “energy space”

I am wondering whether the following two constructions actually define the same space. Construction 1 (taken from Lieb-Loss Analysis, 2nd ed., section 8.2). Define $D^1(\mathbb{R}^n)$ to be the ...
3
votes
1answer
313 views

Principle of unique continuation

Let nonnegative function $u$ is a solution of $-\Delta u=\lambda u+u|u|^{2^*-2}$ with $u=\frac{\partial u}{\partial\nu}=0$ on $\partial\Omega$, where $\lambda\leq0$, then u vanishes identically in ...
2
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0answers
30 views

Time continuity of a solution to the generalized porous medium equation

Let $\Phi(x,s)$ be a smooth, non-negative function with $\Phi(\cdot,0)=0$ and $\Phi$ increasing in $s$ and $\partial_s \Phi(x,0)=0$ and $\Omega$ be a bounded smooth domain. Now assume there is a weak ...
4
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1answer
148 views

find a weak solution in an intersection of Sobolev spaces

In using lax-milgram to find a weak solution in an intersection of sobolev spaces the weak solution for $$ -\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0 $$ was ...
1
vote
1answer
481 views

how to prove mean value property for harmonic functions?

For a harmonic function $u(x)$, on domain $\Omega$ where $y \in \Omega \subset \Bbb R^n $, how to show that $$ u(x) = \frac{1}{\omega_n R^{n-1}}\int_{\partial B_R(y)} u(\sigma) d\sigma$$ where ...
0
votes
1answer
67 views

Poisson Equation Limit

How to show the following: If $ \Delta u(x) = |x|^{\alpha} $, with $u(x) \in C^2(R^{n})$, then for any $\beta < \alpha + 2$: $$\limsup_{x \rightarrow \infty} \frac{|u(x)|}{|x|^{\beta}} ...
3
votes
2answers
64 views

Laplace's equation is solved when the functional $E[u] = \int_{\Omega}|\nabla u|^2 $ is minimized

My professor mentioned something like "Laplace's equation is solved when the functional $E[u] = \int_{\Omega}|\nabla u|^2 $ is minimized." I've been trying to understand this statement. If I say that ...
3
votes
1answer
176 views

Differential equation with infinitely many solutions

The problem is to solve for $-1<x<1$ $$y'(x)=\frac{4x^3y(x)}{x^2+y(x)^2}$$ with $y(0)=0$. I need to show that this equation has infinitely many solutions. Note that ...
2
votes
1answer
165 views

On how to get a solution for a nonhomogeneous problem for the heat equation.

Evans PDE book presents the following problem (page 87): Write down an explicit formula for a solution of $$\left\{\begin{matrix} u_t-\Delta u+cu=f &\text{ in }\mathbb{R}^n\times(0,\infty) ...
7
votes
1answer
291 views

References on the Nash-Moser implicit function theorem

To learn, the Nash-Moser implicit function theorem, I tried the document Hamilton (1982) The Inverse Function Theorem of Nash and Moser, but the article is very encyclopedic. I have a ...
3
votes
2answers
71 views

How to solve a system of PDE $u_t+u_x=v, v_t+v_x=-u$

Solve the following initial value problem: $$u_t+u_x=v, \\v_t+v_x=-u, \\u(0,x)=u_0(x), \\ v(0,x)=v_0(x).$$ I did not learn any method to solve a system of PDE so I guess there is a "trick". So ...
1
vote
0answers
94 views

Lamé's equation - elliptic PDE?

I have a specific PDE in 3-dimensional space + time (the right one). u(x,t) is the unknown function (values are in R^3) and F(x,t) the right hand side, mu and lambda are positive constants. Now ...
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0answers
113 views

Change of variables in Fokker-Planck equation

I have the following Fokker-Planck equation: $$\frac{\partial\psi}{\partial t} + \nabla_{r_1} \cdot \left[ u(r_1,t)\psi + \frac{1}{\zeta} \textbf{F} (r_2 - r_1) \psi \right] + \nabla_{r_2} \cdot ...
2
votes
1answer
55 views

Has the property $\int\Phi =1$ (of the fundamental solution $\Phi$ of the heat equation) a physical interpretation?

For each $t>0$ the fundamental solution of the heat equation is given by $$\Phi(x,t)=\frac{1}{(4\pi t)^{n/2}}\exp\left(-\frac{|x|^2}{4t} \right )$$ and satisfies ...
2
votes
1answer
67 views

How to show that $W^{2,\infty}(B_1)=C^{1,1}(\bar B_1)$?

Suppose that $B_1$ is the open unit ball in $\mathbb R^n$, denote $W^{2,\infty}(B_1)$ be the sobolev spaces and $C^{1,1}(\bar B_1)$ is the Holder spaces. It seems the equality ...
0
votes
2answers
69 views

Finding the characteristic ODE from a nonlinear PDE

I am studying for a PDE exam on Tuesday, and I am getting pretty confused about one specific type of problem and I am thinking that perhaps I am misinterpreting the correct procedure to follow. The ...
7
votes
1answer
354 views

Describing co-ordinate systems in 3D for which Laplace's equation is separable

Laplace's Equation in 3 dimensions is given by $$\nabla^2f=\frac{ \partial^2f}{\partial x^2}+\frac{ \partial^2f}{\partial z^2}+\frac{ \partial^2f}{\partial y^2}=0$$ and is a very important PDE in ...
0
votes
1answer
96 views

Very hard question for me

Let $\Omega$ be an open bounded subset of $R^n$, and let $\partial \Omega$ denote its boundary. Let $u$ $\in$ $C^2(\Omega$) $\cap$ $C(\overline \Omega$) be a solution of the Dirichlet problem. ...
2
votes
2answers
553 views

Solve this Dirichlet problem

Show that the Dirichlet problem $$ \left\{ \begin{array}{l} u_{xx}+ u_{yy}=u^3 \ \text{in} \ x^2+y^2 \lt 1 \\ u=0 \ \text{on} \ x^2+y^2 = 1 \end{array} \right.$$ where $u=u(x,y)$, has only the ...
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0answers
77 views

Solution of linear PDE

I want to solve linear equations, which can be found in the article $\rho_t+\dfrac12[\rho(U-v\cos\theta)]_x=0,$ ...
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vote
2answers
84 views

A doubt on Heat equation

I'm studing the Maximum Principle of Heat equation: Let $u\in C(\overline(U_T)\cap C^2_1(U_T))$ in $\mathbb{R}^n$ satisfy $u_t=c\Delta u$ on $(x,t)\in U_T$. Then ...
1
vote
1answer
296 views

Diffusion-advection equation with time-variable coefficients

Is the fundamental solution (Green's function) of the 1D advection-diffusion equation $$\frac{\partial{\phi}}{\partial{t}} = D(t)\frac{\partial{^{2}\phi}}{\partial{x^{2}}} - ...
3
votes
2answers
174 views

Gas Diffusion PDE

Any help on how to get this started would be awesome, I have no idea where to being with this: "Approximate the sensor output for the following model $$u_t+k\cdot u_{xx}=0$$ $$u(x,0)=0$$ ...
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0answers
66 views

Inhomogeneous Wave Equation in 3 dimensions

From section 7.5 in this source, I see that, for $\vec{x} \in \mathbb{R}^3$, if $$ \frac{\partial^2 u\left(\vec{x},t\right)}{\partial t^2} - \nabla^2u\left(\vec{x},t\right) ...
1
vote
1answer
140 views

Reducing wave equation to laplace equation

Question: I want t reduce the following wave equation $$u_n=c^2(u_{xx}+u_{yy}+u_{zz})$$ to Laplace equation $$u_{xx}+u_{yy}+u_{zz}+u_{\tau\tau}=0$$ by letting $\tau=ict$ and $i$ is imaginary. And I ...
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0answers
26 views

Find the limit of the PDE

Consider two following PDEs on functions $\psi_{1,2}(x,\lambda)$, $x \in \mathbb R^n$ depending on parameter $\lambda \in \mathbb C^n$: $$ \Delta_x \psi_1(x,\lambda) + 2i\lambda \cdot ...
6
votes
1answer
131 views

What “standard estimates for the laplacian” do the authors of this paper mean?

I am trying to follow the proof of lemma 2.1 in this paper. The setup. Consider a solution $v$ to the nonlinear equation $$ -\Delta v = ic \partial_1 v + v(1-\vert v\vert^2) ~\mbox{on}~ ...
2
votes
0answers
90 views

Clarification of Hölder norm in terms of oscillation

Let $\Omega\subset\mathbb{R}^2$ be an open bounded set, $B(x_0, \rho)=\{x\in\mathbb{R}^2\ |\ |x-x_0|\leq \rho\}$, $\Omega(x_0, \rho)\equiv B(x_0, \rho)\cap \Omega$, $u\in L^{\infty}\big(\Omega(x, ...
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0answers
47 views

regarding a set of integro-PDE

Let say I have a minimal example that contains most of the mathematics I am looking for. The example is as follows: \begin{align} &\frac{\partial u^0}{\partial t}(x,t)+\frac{\partial^2 ...
4
votes
2answers
235 views

Poincare Inequality implies Equivalent Norms

I am currently working through the subject of Sobolev Spaces using the book 'Partial Differential Equations' by Lawrence Evans. After the result proving the Poincare Inequality it says the following ...
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0answers
313 views

The Hanging Chain Problem, checking if I am right

I am trying to make sure I approached this problem correctly. It's a hanging chain. I'm told by a classmate that the physics doesn't make sense, but it's a really a math problem, so that I guess is ...