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Results tagged with ap.analysis-of-pdes
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user 11260
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
25
votes
Why is the Vandermonde determinant harmonic?
The harmonicity of $V$ can be understood and placed in a more general context by identifying $V$ as the Doob h-transform$^\ast$ of $n$ independent and identically distributed diffusion processes, see …
- 188k
17
votes
Accepted
Who is Petrov of the Petrov-Galerkin method?
The Petrov you are looking for is: Georgii Ivanovich Petrov (1912-1987), biographies are here and here and here.
I quote from the third biography:
G.I. Petrov was a prominent Russian scientist in …
- 188k
16
votes
Accepted
What is an "integrable hierarchy"? (to a mathematician)
An integrable hierarchy is another name for a system of commuting Hamiltonian flows. The word "hierarchy" is used because a countably infinite number of commuting flows is obtained recursively.
[For t …
- 188k
15
votes
Accepted
The origin of Discrete `Liouville's theorem'
Mathematica (Cluj), 6, 146-151 (1932)
In a recent article from this journal, Mr. Bouligand has indicated the
possibility of a modified proof of a theorem of Mr. Picard: "a
harmonic function …
- 188k
15
votes
Where do some "energy identities" in PDE theory come from?
In addition to the variational approach based on Noether's theorem, there are other ways to find conservation laws for nonlinear PDE's:
The symmetry/adjoint symmetry pair method extracts a conservati …
- 188k
15
votes
Early successes of Schwartz distribution theory
Following the citation for the 1950 Fields medal I would argue that putting the Dirac delta function on a firm ground was the early success of the theory of distributions.$^\ast$
An extensive list of …
- 188k
14
votes
Accepted
Spectrum of matrix involving quantum harmonic oscillator
The Hamiltonian
$$H=\begin{pmatrix}
\alpha^2+a^\ast a&\alpha a+\beta a^\ast\\
\alpha a^\ast+\beta a&\beta^2+a^\ast a
\end{pmatrix}
$$
is known in the physics literature as the anisotropic Rabi Hamilto …
- 188k
13
votes
Can you hear the shape of a drum by choosing where to drum it?
As mentioned in the comments, knowing both eigenvalues and eigenfunctions gives you enough information to find the shape of the domain, so to make this problem more challenging one might ask what mini …
- 188k
13
votes
"Wild" solutions of the heat equation: how to graph them?
Eq. 1.1 of this 1994 paper gives an explicit example in the form of a series expansion that seems tractable for numerical approximation. At least, I had no difficulty plotting a few terms of the serie …
- 188k
13
votes
Accepted
Caccioppoli-Leray Inequality for De Giorgi's theorem proof
I made a trip to the library and scanned the relevant pages from Miranda's 1955 book:
page 152-153 and page 154-155
the references are:
[3] J. Leray, J.Math. pures et appl. 17, 89-104 (1938)
[8] R. Ca …
- 188k
12
votes
Asymptotic behavior of a certain oscillatory integral
$$I(x):=\int_{0}^{\infty}\frac{e^{i r}}{r^{\frac{1}{2}}}\int_{0}^{\infty}\frac{e^{-s}}{s^{\frac{1}{2}}}\frac{r}{sx+\sqrt{sxr}+r}ds dr.$$
To aid the asymptotic analysis, I regularize $I(x)$ by multiply …
- 188k
12
votes
Reference request: Software for producing sounds of drums of specified shapes
The full physics problem is complex, the vibrating membrane displaces the air, which causes a backreaction and signifantly modifies the response. Moreover, the response also depends sensitively on whe …
- 188k
11
votes
Accepted
Does current follow the path(s) of least (total) resistance?
Perhaps to resolve this issue it helps to work out a simple example.
Take a region $D$ consisting of the strip $|x|<1$, $0<y<1$, and a $y$-independent conductivity profile
$$\sigma(x)=\begin{cases}
1 …
- 188k
11
votes
Accepted
Kernel of the Laplacian + a function
Q: Can we conclude that $Lu=\Delta u+ fu=0$ has only zero (or constant solutions) if we assume $f$ non-constant?
A: No, a counter example in one dimension is the Mathieu equation, which has non-consta …
- 188k
10
votes
Accepted
Linear PDE, analytic continuation, Green's function and boundary conditions
Q: Do I have to consider both problems (real $\xi$ or imaginary $\xi$) totally independently and work hard twice?.
A: A single calculation suffices, you could just do the inverse Fourier transform of …
- 188k