This tag is for "partial differential equations". As opposed to "ordinary differential equations".
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18 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 $$V=A(t,x,u)\frac{\partial }{\partial ...
4
votes
1answer
31 views
How to prove that: $\phi^{2}(0) \leq \|\phi\|^2_{L^{2}}+\|\phi'\|^{2}_{L^{2}}$
Let $\phi$ be a function and $\phi \in C^{\infty}(\mathbb{R}_{+},\mathbb{R})$ with compact support and $\mbox{supp }{\phi} \subset [0, \infty)$.
I want to prove that: $$\phi^{2}(0) \leq ...
3
votes
1answer
22 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 ...
1
vote
1answer
22 views
Weak solution to PDE boundary value problems
Let $\Delta$ be the Laplacian operator in dimension $n$ and let $\Delta^2:=\Delta\circ\Delta$. Consider the boundary value problem $\Delta^2 u=f$ on some open subset $U\subset\mathbb{R}^n$ with smooth ...
2
votes
2answers
46 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 ...
0
votes
0answers
14 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): $$ u = f ...
1
vote
0answers
19 views
Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary?
Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary? What smoothness is required of the boundary? I would be grateful for some references to this.
1
vote
1answer
20 views
Finite difference method stability
I have shown that a finite difference method satisfies
$$\underline{u}^{n+1}=((1+6\mu)\mathbb{I}-36\mu A^{-1})\underline{u}^n$$
I don't think that the rest of the question is necessary but it is ...
3
votes
2answers
48 views
How to prove a PDE preserves mass?
My general question is: suppose you are given a PDE (possibly with boundary conditions). What does it mean to say the PDE "conserves mass"?
Specifically, if you are given the PDE
$$- \nabla \cdot ...
1
vote
1answer
36 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 $, ...
2
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0answers
26 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: $$H^1_0(\Omega) \cap H^2(\Omega)\quad \text{and}\quad ...
3
votes
0answers
99 views
+150
Hille Yosida theorem application
Disclaimer: pretty long and specific (contraction semi groups involved).
I have fourth order parabolic equation
$$
u_t + \Delta^2 u = 0
$$
on $U_T = U \times [0,T]$. $U \subset \mathbb{R}^m$ is a ...
1
vote
1answer
32 views
Characterize solutions to Laplace's and Poisson's equation in the unit square with periodic boundary conditions
So I am studying for a qualifying examination and there was this problem from an old exam.
(a) Does the problem $\Delta u = 0$ in the unit square in the plane with u and and all of its partial ...
0
votes
0answers
21 views
Solution to nonlinear heat equation with time variant Neumann type boundary conditions?
The non-linear form of the heat equation can be written as:
$\rho(T) c_p(T) \frac{\partial T}{\partial t}= \frac{\partial}{\partial z} \left ( k(T) \frac{\partial T}{\partial z} \right).$
Assuming ...
3
votes
0answers
26 views
Timestepping PDE with positive eigenvalues
I'm trying to numerically solve a PDE, namely:
$$
\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
31 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, ...
0
votes
0answers
14 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 ...
0
votes
1answer
33 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 ...
1
vote
1answer
29 views
A Semi-infinite right circular cylinder problem
A semi-infinite right circular cylinder whose axis lie along the $z$ axis, has its base on the $x$-$y$ plane.The base is maintained at a constant potential $V_0$ and the side of the cylinder is ...
1
vote
0answers
39 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 $H$ ...
0
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0answers
8 views
Reference request: Finite difference methods on curvilinear (body fitted) grids
I was wondering if someone may be aware of some form of detailed summary (book, tutorial paper) about the use of finite difference methods on curvilinear (body fitted) grids.
I was only able to ...
0
votes
0answers
33 views
turne into dimension one
who can help me to turn this problem into a problem on dimension one ?
how to write (1,1) and (1,2) in dimension one ?
Please ,help me
Thank you
0
votes
0answers
17 views
Von Newman stability analysis for 2D acoustic wave equation explicit
Von Newman stability analysis for acoustic wave equation explicit centered differences: 2nd order time and space (N 2)'th order:
\begin{eqnarray}
U_{jk}^{n+1} = \left( \frac{\Delta t V_{jk} ...
0
votes
0answers
27 views
solving PDE using FEM
I have studied FEM in 1D from here:
http://ocw.mit.edu/courses/mathematics/18-085-computational-science-and-engineering-i-fall-2008/video-lectures/
I was able to understand it. Now I want to study FEM ...
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. ...
2
votes
1answer
47 views
Am I missing something, or does this simple PDE lack an explicit solution due to the nature of its (also simple) boundary conditions?
Short version:
Below, I present a simple PDE with simple boundary conditions (BCs), which has a simple solution. I then modify one of the BCs, and end up with a transcendental equation for the wave ...
2
votes
3answers
57 views
PDEs in biology
I am student who mostly heard lectures on partial differential equations and homogenization. But I really like the idea of working in biology or with biologists - but (with my lack of overview) it ...
1
vote
1answer
25 views
how to calculate $d\Omega(f)$ here
the question was to find $d \Omega(f)$ with :
$$ \Omega : (E,[.]) \to (F,||.||) \\f \to -f'' +f^3$$
$
[f] = |f'(0)| + ||f''|| $
; $ ||f|| = Sup_{[0,1]}|f(x)| $
the answer is given to me like this ...
2
votes
1answer
43 views
The solution to the equation $ - \Delta u = (\lambda - \log u)u$
$\Omega$ is a bounded domain in $\mathbb R^n$ with smooth boundary. Consider the Dirichlet problem:$$-\Delta u = (\lambda - \log u)u ~~~{\rm on}~~~ \Omega~~~ {\rm and}~~~u=0 ~~~{\rm on}~~\partial ...
1
vote
1answer
51 views
Existence of classical solution to quasilinear parabolic PDE
Consider this linear parabolic PDE:
\begin{align*}
u_{t}+a\left(x,t\right)u_x+b\left(x,t\right)u_{xx}+c\left(x,t\right)u+d\left(x,t\right) & =0\\
u\left(x,1\right) & =g\left(x\right)
...
1
vote
1answer
51 views
Use method separation of variables: $\frac{{\partial u}}{{\partial y}} = \frac{{{\partial ^2}u}}{{\partial {x^2}}} - 4u$
I did the dpe:
$$\frac{\partial u}{\partial y} = \frac{\partial ^2 u}{\partial x^2} - 4u$$
$0 < x < \pi $
With boundary conditions: $\begin{array}{l}
u(0,y) = u(\pi ,y) = 0 \\
u(x,0) = ...
1
vote
1answer
39 views
explain $df(tx).x = \sum_{i=1}^n {\partial f\over \partial x_i}(tx)x_i \hspace{1cm} x\in \mathbb R^n$
the question is :
let $U$ be a Neighbourhood of the origine of $R^n$ and :
$x\in U \Rightarrow tx \in U , \forall t\in U $
let f be a numeric function defined in U , and $f(0)= 0$
if we have ...
4
votes
1answer
25 views
Simple Partial Derivative Chain Rule Question
Say $f: \mathbb{R}^n \rightarrow \mathbb{R}$ by some $f(x_1,x_2,\cdots,x_n)$. Further suppose that each $x_i: \mathbb{R}^m \rightarrow \mathbb{R}$ by some $x_i=x_i(\eta_1,\eta_2,\cdots,\eta_m)$. Does ...
0
votes
2answers
53 views
partial differential equation $y^2\frac{\partial ^2 u}{\partial x\partial y} + \frac{\partial ^2 u}{\partial y^2}$
Helle everybody, i have a problem with a partial differential equation, I tried to work but it's not true
$$\begin{array}{l}
y^2\frac{\partial ^2 u}{\partial x \, \partial y} + \frac{\partial ^2 ...
1
vote
0answers
19 views
How to perturb a Palais-Smale functional so that the Palais-Smale condition is preserved?
Suppose that $J_0\colon H \to \mathbb{R}$ is a $C^1$ functional on the Hilbert space $H$ satisfying the Palais-Smale condition, that is:
any sequence $u_n\in H$ such that $J_0(u_n)$ is bounded and ...
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 ...
0
votes
1answer
29 views
Simple PDE classification question
Benjamin-Bona-Mahony equation:
$$\displaystyle u_t+u_x+uu_x-u_{xxt}=0$$
both the paper I was reading and wikipedia claimed that it is nonlinear. It has been some time since I studied classification so ...
3
votes
2answers
91 views
PDE: Why do they have the wrong units?
Take a look, for example, at the telegrapher's equations (let's look at the voltage one). They have the wrong units.
Equation
$u_{x} = Li_{t} + Ri$
*where $u$ is potential in volts $V$, $L$ is ...
1
vote
2answers
70 views
resources to study PDE from
I am an undergrad engineering student. I recently completed my second year, with that said, I have taken several calculus courses. Most recently I completed differential equations and multivariable ...
2
votes
0answers
39 views
Dirichlet eigenvalue problem on the Hilbert cube
I'm trying to solve the Dirichlet problem for the Helmholtz equation
\begin{aligned}-\triangle u & = & \lambda u, & x\in\Omega,\\
u & = & 0, & x\in\partial\Omega,
...
0
votes
0answers
19 views
Reflection of Laplacian eigenfunction
I need to prove a reflection principle for Laplacian eigenfunctions.
Let $OX$ denote the $x$-axis.
Let $U$ be some open, bounded subset of the plane symmetric in $OX$ and $L = OX \cap U$ a line ...
1
vote
0answers
45 views
Regularization by mollifier sequences
A well-known feature used in PDE's is the regularization by convolution with a mollifier sequence $\rho_n$, i.e. $\rho_n(x) := n^d \rho(nx)$ with $x \in \mathbb R^d$, $\rho \in C^\infty_c(\mathbb ...
5
votes
0answers
32 views
The square root of $I-\Delta$
$M$ is a closed Riemannian manifold and $\Delta$ is the Laplace-Beltrami operator of $M$. Can we find a pseudodifferential operator $\Lambda$ of first-order such that $\Lambda^2=I-\Delta$?
My ...
0
votes
0answers
26 views
Discretizing a PDE
I need to discretize the following PDE so that I can run a forward euler method on it:
$$ {\partial u \over \partial t}= {1 \over x}{\partial \over \partial x}{\left(x{\partial u \over \partial ...
2
votes
1answer
40 views
Laplacian of Temperature
I have the following question: In an isotropic medium with constant thermal conductivity, the temperature T(x,y) is independent of time. Show that the laplacian of T is zero. (4 marks)
I don't really ...
0
votes
1answer
29 views
solve a “wave equation” with an extra term
I want to solve the following "wave equation" $$\nabla^2\psi(\vec{r},t) - \frac{1}{c(\vec{r})^2}\frac{\partial^2}{\partial t^2}\psi(\vec{r},t) = R(r)\psi(\vec{r},t)$$ subject to initial conditions ...
2
votes
0answers
28 views
Why can I write the curve shortening flow system as..
Consider the family of curves $$\gamma:S^1\times [0, T)\rightarrow \mathbb R^2, \gamma(u, t)=(x(u, t), y(u, t)),$$ where $u$ parametrizes the trace of the curve and $t$ parametrizes which curve is ...
0
votes
1answer
24 views
partial differential equations and particular solutions
I am trying to solve the Diffusion equation, but with the boundary conditions x(0)=1, x(l)=0. I have been told this is impossible, and I understand that this is because when we come to sum the ...
0
votes
1answer
25 views
The existence of invertible pseudodifferential operator
If $M$ is a closed manifold, then can we find a first order pseudodifferential operator $\Lambda$ acting on ${\cal D}'(M)$ (the distribution on $M$) such that there is a pseudodifferential operator ...
