This tag is for "partial differential equations". As opposed to "ordinary differential equations".
0
votes
0answers
4 views
Forward Time Centered Space Scheme on unit square, Stability Analysis
I'm stumped on the following problem:
Show that
$u(x,y,t)=\exp(1.68t)\sin(1.2(x-y))\cosh(x+2y)$ solves
$\frac{\partial u}{\partial t}-2\frac{\partial ^2 u}{\partial x^2}-\frac{\partial ^2 ...
1
vote
2answers
30 views
Improvment of $W^{1,p}$ regularity of a elliptic equation solution.
I'm looking for some reference for results like
$$ \mbox{div}(A(x) \nabla u) = 0, \ \ u \in H^1=W^{1,2} \Rightarrow u \in W^{1,p}, p>2 $$
where $A(x)$ is elliptic, this is, $Id\lambda \le A(x) \le ...
0
votes
0answers
23 views
Hyperbolic PDE classification
Considering the following equation
$$u_t + A u_x = 0,\quad t> 0$$
where
$$ A = \begin{pmatrix} 1 & \sin t \\ \sin t & -1 \end{pmatrix} $$
Naturally, the system is hyperbolic if $A$ consist ...
1
vote
1answer
50 views
Dimensions analysis in Differential equation
Differential equation of solitary wave oscillons is defined by,
$$ \Delta S -S +S^3=0 $$
How can we write this equation as,
\begin{equation}
\langle(\vec{\nabla}S)^2\rangle+\langle S^2\rangle-\langle ...
1
vote
2answers
23 views
spherically symmetric configurations
$$\Delta S -S +S^3=0$$ How this Differential equation can be written in this form:
\begin{equation}
\frac{d^2S}{d\rho^2}+\frac{D-1}{\rho}\,\frac{dS}{d\rho}
-S+S^3=0
\end{equation}
Which is ...
1
vote
0answers
23 views
Uniformly quasi-isometric patches over classes of Riemannian manifolds
Suppose $(M^d,g)$ is a closed, connected Riemannian manifold.
Is there a constant $R > 0$ such that for all $z \in (M,g)$, for all $x \in B_R(z)$, \begin{equation} \frac{1}{2} \lVert \xi ...
5
votes
2answers
247 views
Learning differential/Riemannian geometry for PDEs
I know there have been threads on which books to learn DG/RG from but hopefully this is sufficiently different to avoid closure.
Can anyone recommend a book to learn DG/RG (whichever is appropriate) ...
7
votes
0answers
202 views
+150
Hille Yosida theorem application
Disclaimer: pretty long and specific (contraction semi groups involved).
I have fourth order parabolic equation
$$
u_t + \Delta^2 u = 0
$$
on $U_T = U \times [0,T]$. $U \subset \mathbb{R}^m$ is a ...
1
vote
1answer
30 views
PDE initial value problem
Show that the solution of the initial value problem for
$u_t+u_x=\cos ^2 u$
is given by
$u(x,t)=\tan^{-1} \{ \tan [u_o(x-t)]+t\}$,
where $u_0(x)$ is the initial condition.
My attempts at a ...
1
vote
0answers
34 views
Sobolev spaces - about weak derivative
Let $U$ a bounded and open subset of $R^n$. Let $u \in H^{1}(U)$ a bounded function , $v \in H^{1}_{0}(U)$ a non negative function. Consider $\varphi : R \rightarrow R$ a convex and smooth ...
1
vote
1answer
18 views
Inverse fourier transform of a function which is a fundamental solution
Let $f:\mathbb{R}^3 \to \mathbb{R}$ be $f(x)=(1+|x|^2)^{-1}$.
I need to calculate $\mathcal{F}^{-1}(f)$.
I've proven that $f\in L^2(\mathbb{R}^3)$ and I know that the fourier transform is an ...
0
votes
0answers
25 views
sobolev spaces - product of two functions
I am working in a exercise, to my solution works I need the following affirmation is true:
Let $\varphi : R \rightarrow R$ a convex and smooth function. Let $u \in H^{1}(U)$ a bounded function and $v ...
1
vote
0answers
27 views
optimal control -Taylor expansion - PDE problem
I am trying to follow perturbation analysis in this paper (Optimal control of fluid limits of queuing networks and stochasticity corrections) and I am stuck at one point.
For the given control ...
0
votes
1answer
42 views
Green's function. Basic
Can anyone give some advice about books where I could find introductory information about Green's function. What are the methods of constructing Green's function.
Actually, Green's function for 3D ...
1
vote
1answer
144 views
Mean Value Property of Harmonic Functions Proof Step
I'll only include the step that throws me off unless more info is requested, but this is from LC Evans PDEs book:
$$ \displaystyle \lim_{t \to \, 0^+ } \left[ \frac{1}{n\,\alpha(n) \, t^{n-1}} \int_{ ...
2
votes
0answers
35 views
What method can be used for finding Green function for Fokker-Planck equation?
Let's have an equation
$$
u_{t} - (xu)_{x} - \frac{1}{2}u_{xx} = 0, \quad u(x, 0) = g(x), \quad -\infty < x < \infty , \quad 0 < t < \infty .
$$
I need to find a Green function for it. ...
0
votes
0answers
20 views
Behaviour of the equations $u_t = \Delta^3 u$ and $u_t = -\Delta^2 u$
In a paper I'm reading there's a reference to a 'S. D. Eidelman: Parabolic Systems (1969)' which seems to be out of print and impossible to get hold of.
In it, apparently, it is shown that for $u_t ...
3
votes
1answer
291 views
Fourier Sine Transform
There is a question from my book which I find hard. Here it goes:
Consider
$$\frac{\partial u}{\partial t}=k\frac{\partial^2 u}{\partial x^2}-v_0\frac{\partial u}{\partial x} ...
1
vote
1answer
26 views
Finding a strong enough solution to a specific PDE problem.
Let $U\subset \mathbb{R}^n$ with smooth boundary $\partial U$. And consider the expression
$$\Delta u = f.$$
$$\text{+"convenient boundary conditions"}$$
In my specific case $f\in H^2_0$. Under ...
3
votes
0answers
37 views
Maximum principle for heat equation
$u_1$ and $u_2$ solve
$u_t-u_{xx}=f(u)$ for some $C^{\infty}$ function
on $(a,b)\times[0,\infty)$
On the vertical boundaries: $u_1(a,t)=u_2(a,t)=u_1(b,t)=u_2(b,t)=0$
and there exists ...
3
votes
1answer
72 views
Property of the trace of matrices
Let $A(x,t),B(x,t)$ be matrix-valued functions that are independent of $\xi=x-t$ and satisfy $$A_t-B_x+AB-BA=0$$ where $X_q\equiv \frac{\partial X}{\partial q}$.
Why does it then follow that ...
1
vote
2answers
43 views
The harmonic conjugate of $\Im e^{z^2}$?
It is obvious that $e^{z^2}$ is analytic, right? So the harmonic conjugate of $\Im e^{z^2}$ is $\Re e^{z^2}$, isnt' it?
However, the solutions manual I'm consulting gives the answer as $\Im ...
0
votes
1answer
29 views
Writing a 2nd order PDE as a system of equations
I want to turn this 2nd order equation into a system of first order equations but I am unsure about whether I can get rid of the $u$ or not
$$u_{xy}-u_x+u_y+10u u_{xx}$$
To write this as a system of ...
3
votes
0answers
70 views
Weak solution to PDE boundary value problems
Take $\lambda>0$ and let $\Delta$ be the Laplacian operator in dimension $n$ and let $\Delta^2:=\Delta\circ\Delta$. Consider the boundary value problem $\Delta^2 u+\lambda u=f$ on some open subset ...
1
vote
1answer
39 views
Exponential decay of Heat equation solution
I'm refereeing a paper and the authors go to great lengths to prove the following fact.
Let $W(t,x)$ be the solution to the linear heat equation on the half-line: $\partial_t W = D \partial_{xx} W $, ...
0
votes
1answer
239 views
Duhamel's Principle question
I need to prove the following form of Duhamel's principle, but don't know how to...
Can you help me please?
Let u(x,t) be the solution of the following initial-boundary value problem for the ...
0
votes
0answers
21 views
What does this mean: Symmetry of the KDV generated by a vector field
What is
a symmetry of the KDV
$$\frac{\partial u}{\partial t}=6u\frac{\partial u}{\partial x}-\frac{\partial^3 u}{\partial x^3}$$ generated by $$V=A(t,x,u)\frac{\partial }{\partial ...
2
votes
1answer
124 views
How can I find the limiting value of a time-dependent PDE?
I've managed to reduce a question in probability to the following simple looking PDE:
$$
u_t = -t u_x + \frac{1}{2} u_{xx}, {\rm ~for~} x>0, \, t \in \mathbb{R} \;,
$$
with a limiting initial ...
2
votes
2answers
54 views
Harmonic Function bounded by a linear function
Let $u$ be a harmonic function on $\mathbb C$. Suppose that for each $\epsilon > 0$, there is a constant $C_\epsilon$ such that
$$u(z) \leq C_\epsilon + \epsilon |z| .$$
I am trying to show that ...
2
votes
1answer
70 views
Pde: $u_{t}+6uu_{x}+u_{xxx}=0$
Help me please to solve this pde.
$u_{t}+6uu_{x}+u_{xxx}=0$
with boundary conditions $\lim_{x \to \pm \infty }u(x,t)=\lim_{x \to \pm \infty }u_{x}(x,t)=\lim_{x \to \pm \infty }u_{xx}(x,t)=0$
a. Show ...
3
votes
1answer
23 views
Compatible PDEs
If we have an overdetermined system of pdes what does one have to check to be sure that they are compatible? Suppose each pde is derived from a different Hamiltonian and we have that the Poisson ...
4
votes
1answer
34 views
How to prove that: $\phi^{2}(0) \leq \|\phi\|^2_{L^{2}}+\|\phi'\|^{2}_{L^{2}}$
Let $\phi$ be a function and $\phi \in C^{\infty}(\mathbb{R}_{+},\mathbb{R})$ with compact support and $\mbox{supp }{\phi} \subset [0, \infty)$.
I want to prove that: $$\phi^{2}(0) \leq ...
0
votes
0answers
18 views
Laplace equation with time-like boundary conditions
For simplicity suppose that $\Omega = (a,b)\times(c,d)$. Than solve laplace equation i.e. $$\Delta u = 0$$ in $\Omega$ with boundary conditions(they are give as if $y$ is time coordinate): $$ u = f ...
3
votes
2answers
49 views
How to prove a PDE preserves mass?
My general question is: suppose you are given a PDE (possibly with boundary conditions). What does it mean to say the PDE "conserves mass"?
Specifically, if you are given the PDE
$$- \nabla \cdot ...
3
votes
2answers
158 views
Good reference texts for introduction to partial differential equation?
As the title, are there any good reference texts for introduction to partial differential equation?
2
votes
0answers
21 views
Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary?
Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary? What smoothness is required of the boundary? I would be grateful for some references to this.
1
vote
1answer
25 views
Finite difference method stability
I have shown that a finite difference method satisfies
$$\underline{u}^{n+1}=((1+6\mu)\mathbb{I}-36\mu A^{-1})\underline{u}^n$$
I don't think that the rest of the question is necessary but it is ...
1
vote
1answer
37 views
Characterize solutions to Laplace's and Poisson's equation in the unit square with periodic boundary conditions
So I am studying for a qualifying examination and there was this problem from an old exam.
(a) Does the problem $\Delta u = 0$ in the unit square in the plane with u and and all of its partial ...
0
votes
1answer
34 views
Definition of regular point of a boundary with planar brownian motion
This is an exercise in G.Lawler's book Conformally invariant processes in the plane.
First he defined regular point of a boundary using brownian motion:
Suppose $D$ is a domain in $\mathbb{C}$ with ...
1
vote
1answer
146 views
Heat equation separation of variables with different boundary conditions
I need to solve the following heat equation using separation of variables
$
\left\{\begin{matrix}
u_{t}=c^{2}u_{xx} \; \; ,0<x<L, t>0 \\
u(0,t)= 0 \\
u_{x}(L,t)=0 \\
u(x,0)=f(x)
...
2
votes
0answers
27 views
Another kind elliptic energy estimate
I was reading Evans' PDE,in the corresponding chapters Evans use elliptic energy estimate and Lax-Milgram theorem to prove the existence of uniformly elliptic equation and parabolic and hyperbolic ...
0
votes
2answers
42 views
Relationship between sobolev spaces
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. I would like to understand something about the relationships between these two spaces: $$H^1_0(\Omega) \cap H^2(\Omega)\quad \text{and}\quad ...
0
votes
0answers
25 views
Solution to nonlinear heat equation with time variant Neumann type boundary conditions?
The non-linear form of the heat equation can be written as:
$\rho(T) c_p(T) \frac{\partial T}{\partial t}= \frac{\partial}{\partial z} \left ( k(T) \frac{\partial T}{\partial z} \right).$
Assuming ...
3
votes
0answers
26 views
Timestepping PDE with positive eigenvalues
I'm trying to numerically solve a PDE, namely:
$$
\partial_t \binom{u(x, t)}{v(x, t)}
= x \left(\begin{array}{cc}
0 & -1 \\
1 & 0
\end{array}\right) \cdot
\partial_x \binom{u(x, t)}{v(x, t)}
...
2
votes
0answers
32 views
No flux boundary condition on PDE on surface (Laplace-Beltrami)
What would a Neumann BC on a PDE posed on a surface look like? In the flat case, we have
$\nabla u \cdot N = 0$
where $u$ is the solution of the PDE and $N$ is unit normal vector.
In a surface case, ...
1
vote
0answers
40 views
Relation of the kernels of one bounded operator and its extension
Sorry for this long and formal post. The application in PDEs is mentioned just at the end.
Let
$$V \hookrightarrow H \text{ and } Q_H' \hookrightarrow Q',$$
where $V$ and $Q$ are Banach and $H$ ...
1
vote
1answer
53 views
How to determine if the sums and products of harmonic functions is also harmonic?
Suppose I know that $u + iv$ is non-constant and analytic on a domain $D$, then I know that $u$ and $v$ are harmonic on $D$ and not both constant; but how do I then determine whether $3u^2v - v^3 + ...
0
votes
1answer
35 views
Corollary to mean value property for harmonic functions?
For $\Omega \subset\mathbb{R}^n$ open, and $u_i:\Omega \to \mathbb{R}$ a sequence of harmonic functions which are uniformly bounded. Prove that for any multi-index $\alpha$ and for any $K \subset ...
0
votes
0answers
16 views
Finite-Element Method: Question on stability
I am trying to determine the stability of the PDE
http://mathurl.com/cazterh
given the finite-element scheme
http://mathurl.com/cetadmr
and constant s
http://mathurl.com/bcfq5us
My problem is ...
2
votes
2answers
51 views
Differentiating a boundary condition at infinity
A typical boundary condition for an initial boundary value problem is
$$ \lim_{x\rightarrow\infty} T(x,t) = T_\infty.$$
For example, this might be the temperature at the end of a very long rod. ...
