15
votes
Accepted
How to compute $\sin(\frac{d}{dx})f(x)$?
As noted in the OP, $\sin (d/dx) = (\exp (id/dx) - \exp (-id/dx))/(2i)$, which casts the operator as a combination of two shift operators,
$$
\sin (d/dx) f(x) = \frac{1}{2i} (f(x+i) - f(x-i))
$$
The ...
- 6,696
12
votes
Is there a connection between representation theory and PDEs?
This goes back to the beginning of the subject of unitary representations of locally compact noncompact groups. Wigner was looking for all possible generalizations of the Dirac equation to higher spin,...
- 3,768
12
votes
Is there a connection between representation theory and PDEs?
There is e.g. a book
Differential Galois Theory by M. van der Put and M. F. Singer, where in Appendix D one can find things on the PDE case (the book is mostly about ODEs).
In mathematical physics ...
- 14k
12
votes
Accepted
Review paper/book on Finite Difference Methods for PDEs
There are many well-written books/notes on this topic including:
J. C. Strikwerda, Finite difference schemes and partial differential equations, SIAM, 2004.
R. J. LeVeque, Finite difference methods ...
- 6,045
9
votes
Accepted
Which matrices can be realized as the Dirichlet-to-Neumann map for a given domain?
Let $A$ be this matrix. Because of the formula
$$\int_D\sigma\nabla u\cdot\nabla v\, dx=\sum_{i,j}a_{ij}U_iI_j,$$
($U$ for voltages of $u$, $I$ for currents of $v$), we see three necessary conditions:
...
- 52.3k
9
votes
Is there a connection between representation theory and PDEs?
Peter Olver has an interesting book on Symmetry and PDEs. Another area to consider (that is particularly important for geometric PDEs) are exterior differential systems. Here are some notes on the ...
- 2,478
7
votes
Representing a nonlinear elliptic PDE as an energy minimization problem
As already said by Kosh, you are trying to solve an inverse problem in the calculus of variation: in its classical formulation, given a system of PDE, the problem consists in finding a functional ...
- 6,357
7
votes
Which matrices can be realized as the Dirichlet-to-Neumann map for a given domain?
In dimensions 3 and higher, and without any constraints on $\sigma$, one can apparently obtain any symmetric matrix $A = (a_{ij})$ such that $a_{ij} < 0$ when $i \ne j$ and $a_{ii} = -\sum_{j \ne i}...
- 17.2k
7
votes
Accepted
How to generate a random function with conditions?
You can simulate functions $u$ vanishing at $\infty$ using the formula
$$u(x)=u_N(x):=\sum_{n=0}^N \xi_n l_n(x),$$
where $N$ is a natural number, the $\xi_n$'s are independent standard normal (or ...
- 128k
6
votes
Which matrices can be realized as the Dirichlet-to-Neumann map for a given domain?
This question (for two dimensional domains) was answered by Curtis, Ingerman and Morrow, "Circular Graphs and planar Resistor Networks" (1998). Let $a$ be the $n \times n$ response matrix. As already ...
- 156k
6
votes
Accepted
Conditions on the velocity ensuring that a flow moves points along the boundary of a manifold
In the situation you described, if $M$ is properly embedded (i.e., topologically embedded and closed), the flow of a vector field takes $\partial M$ to itself if and only if the vector field is ...
- 1,500
6
votes
Is there a connection between representation theory and PDEs?
The book "D-Modules, Perverse Sheaves, and Representation Theory " by Ryoshi Hotta, Kiyoshi Takeuchi and Toshiyuki Tanisaki is the perfect source for this topic. The introduction gives a very nice (...
- 26.5k
5
votes
Representing a nonlinear elliptic PDE as an energy minimization problem
In general, the problem can be complicated because some nonlinearities arise as Lagrange multipliers. Therefore, an energy functional (whose Euler-Lagrange equations coincide with the nonlinear PDE) ...
- 364
4
votes
Solving a differential system
For simplicity, let's assume initially that the support of $\mu$ is the whole real line, so that $F$ is a homeomorphism $ \mathbb{R} \to (0,1) $. Let's denote $b:=F(0)=\mu(-\infty,0]=1-\mu[0,+\infty)\...
- 60.5k
4
votes
Accepted
Navier-Stokes equations and machine learning
One of the earliest papers is Application of machine learning algorithms to flow modeling and optimization (1999).
A model reduction can be accomplished by projecting the Navier-Stokes
equations ...
- 188k
4
votes
What are dissipative PDEs?
Q: Is there a rigorous mathematical definition or an easy characterization of a dissipative PDE?
Dissipative Operators and Hyperbolic Systems of Partial Differential Equations by R.S. Phillips gives a ...
- 188k
3
votes
Accepted
$H^s$ norm of non-integer power of functions
In Christ-Weinstein, JFA 100 (1991) 87-109 you can find the fractional chain rule (Proposition 3.1)
$$
\|F(u)\|_{\dot H^s_r}\le C
\|F'(u)\|_{L^p}\|u\|_{\dot H^s_q}
$$
where $s\in(0,1)$, $p,q,r\in(1,\...
- 9,007
3
votes
Construct examples satisfying some inequalities
This question could be asked on MSE.
Using the rewording of the problem by Gerry Myerson, the analysis below shows it is not possible for some $n$. For $n=2$ it is possible, but for $n\geq 3$ it is ...
- 3,209
3
votes
Accepted
Numerical iterative methods for Poisson equation
In the first case, when you are given a finite set of points, your problem is not well defined. There are in general arbitrarily many solutions if you are just given a finite set of function values.
...
- 224
3
votes
Accepted
Lumped mass matrices and bubble functions for tetrahedral elements
The lumped elements recently underwent some improvements, which are explained in detail in the following thesis
Sjoerd Geevers, Finite element methods for seismic modelling (English), University of ...
- 206
2
votes
Accepted
A mathematical motivation for Lax-Friedrich type of Numerical Fluxes
The most natural way to derive Lax-Friedrichs's scheme is to consider the discretisation where the approximate state $U$ is constant ($\equiv U_i^n$) in cells $((i-1)\Delta x,(i+1)\Delta x)\times((n-1)...
- 52.3k
2
votes
Solving numerically an equation involving exponentials
Pretty much any textbook method should work on a monotonic and convex function. Bisection, for instance, if you want to keep it simple (once you manage to find upper and lower bounds for the solution, ...
- 20.2k
2
votes
Accepted
Iterative method for $p$-Laplacian
I know this method under the name lagged diffusivity. I learned it from the paper
Vogel, Curtis R., and Mary E. Oman. "Iterative methods for total variation denoising." SIAM Journal on Scientific ...
- 12.7k
2
votes
Accepted
Numerical methods for IDE
I must premise that I am not a specialist in numerical analysis, therefore I may be not right when talking about more popular methods in this field pertaining the solution of IDEs. Said that, however, ...
- 6,357
2
votes
Accepted
Weird claims and conclusions in "Introduction to Shape Optimization"
Not an answer, but too long for a comment. The general idea with this stuff seems to be to pair your family $\mathscr{D}$ of admissible domains with a(ny) suitable normed-space $\mathscr{V}$ of vector ...
- 778
2
votes
$H^s$ norm of non-integer power of functions
In general this depends on the value of $\alpha$, $d$ and $s$. For instance, when $d=1$, we have
$$ \||u|^\alpha u\|_{\dot{H}^s}\lesssim\|u\|_{\infty}^\alpha
\|u\|_{\dot{H}^s}.$$
For a proof, see for ...
- 333
2
votes
How to compute $\sin(\frac{d}{dx})f(x)$?
Differentiation is a linear operator so you are basically asking when a real function (like sine) can be applied to an operator
to obtain a new one. This is precisely the concept dealt with under ...
- 406
2
votes
Where can I find the paper by Tappert and Hardin on split-step Fourier transform method?
The original-original was two talks in 1973 at a Boston SIAM conference, just a descriptive paragraph for each. About a year later there was a more formal talk for Ocean acoustics specialists given ...
- 21
2
votes
What are dissipative PDEs?
In some sense, the so-called "entropy" does not have a unified definition in the mathematics community and sometimes the notion of "entropy" is identified (loosely speaking) as ...
- 730
2
votes
Inconsistency in determinability of the solution of a linear first order PDE
It may help to review quickly why the backwards Euler method works for finite dimensional ODEs, before moving to the infinite dimensional case.
Finite Dimensional Backwards Euler
Consider the ...
- 39k
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