12
votes
Accepted
Bounding the discrete $l^p$ norm by the continuous $L^p$ norm for trigonometric polynomials
Yes, this goes back to the work of
Plancherel, M.; Pólya, George, Fonctieres entières et intégrales de Fourier multiples, Comment. Math. Helv. 9, 224-248 (1937). ZBL0016.36004.
(see for instance ...
- 114k
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
8
votes
Generalized Bernoulli numbers
These numbers and corresponding polynomials were introduced by Korobov, see second edition of his book "Number theory methods in numerical analysis". He found some examples, where these &...
- 12.3k
6
votes
Accepted
kth finite difference always positive when kth derivative is?
Yes, this is true. For the function $\Delta(f,h)$ its $(k-1)$-st derivative is strictly positive, since by Lagrange theorem it equals $$f^{(k-1)}(x+h)-f^{(k-1)}(x)=hf^{(k)}(x+\theta h)>0,$$ for ...
- 109k
5
votes
Accepted
Change of variable formulas in discrete calculus?
In terms of the differential operator $\partial_x\equiv d/dx$ one has $f(x+h)=e^{h\partial_x}f(x)$, hence
$$\Delta_h=e^{h\partial_x}-1.$$
Upon Fourier transformation, $\hat{f}(k)=\int_{-\infty}^\infty ...
- 188k
4
votes
What is the inverse of a triangular matrix whose nonzero elements are binomial coefficients? What is the closed-form solution to a recursive relation?
Using the answer of @Toni, the sequence $u_k$ can be related to the Bernoulli numbers $B_k$ for $k\geq 0$,
\begin{align}\tag{1}
u_{k+1}= \frac{B_{k}}{k!}.
\end{align}
After some algebra, the inverse ...
- 3,671
4
votes
Accepted
What is the inverse of a triangular matrix whose nonzero elements are binomial coefficients? What is the closed-form solution to a recursive relation?
This may answer the question, the sequence is the only implicit thing. I consider $Q=\begin{pmatrix}Q_{i,j}\end{pmatrix}_{n\times n}$ for $n\ge 3$, where
$$Q_{i,j}=
\begin{cases}
\dbinom{j}{i-1}, &...
- 785
2
votes
Finite differences of Stirling numbers
Let $c(n,k) = |s(n,k)| = (-1)^{n-k} s(n,k)$. It is well known that for fixed $k$, $S(n+k,n)$ and $s(n+k,n)$ are polynomials in $n$ of degree $2k$ with leading coefficient $(2k-1)!!/(2k)!$. If we ...
- 17k
2
votes
Algorithmically finding mixed-derivative coefficients from finite differences
An algorithm for this purpose has been developed by B. Fornberg in Calculation of Weights in Finite Difference Formulas.
It has been implemented in Mathematica, see the documentation:
- 188k
2
votes
High order difference operator applied to 1/u
You can apply the formula
$$\Delta^p f(x)=\sum_{k=0}^p{p\choose k}(-1)^{p-k}f(x+k)$$
to $f(x)=1/u(x)$. Further simplification will need knowledge how $u(x)$ depends on $x$.
For example, if $u(x)=x$ ...
- 188k
2
votes
Accepted
Finite difference approximation
Given pairwise distinct real numbers $x_0,\dots,x_4$, one can approximate $f'(x_0)$ by a linear combination $a_0f(x_0)+\cdots+a_4f(x_4)$ so that
$$g_j'(x_0)=a_0g_j(x_0)+\cdots+a_4g_j(x_4)$$
for $g_j(x)...
- 128k
2
votes
Change of variable formulas in discrete calculus?
$\newcommand\De\Delta$Elementarily: The relation $f=\De_h^{-1}g$ means that $f(x+h)-f(x)=g(x)$ for all $x$.
For a function $g$ and each real $r$, we know $f_r:=\De_1^{-1}g_r$, where $g_r(x):=g(rx)$, ...
- 128k
1
vote
Accepted
Shrinking the base field of an affine variety
A standard reference is Chapter 6 of "Neron Models" of Bosch, Lütkebohmert and Raynaud, Springer, 1990. The original Grothendieck's exposés in the Cartan Seminar (1960--1961) are available ...
- 1,561
1
vote
Finding numerical solution for nonlinear Poisson-like equation using finite difference method
Since it is a non-linear differential equation you cannot expect to obtain a linear system at the end. Think about using a non-linear solver like a Newton solver instead.
- 11
1
vote
Finite difference for a highly nonlinear equation - The wind within the forest
As noted already in the comments, your boundary conditions seem off. Note that generically for a second-order BVP one expects to impose only two boundary conditions; you have 4.
Once you’re sure you’...
- 1,253
1
vote
Decomposing functions to Taylor-Fourier series
The approach for determining the Fourier series coefficient $a_{j,r}$ for $U_r(x)$ is similar to the approach for determining the Fourier series coefficient $a_{j,0}$ for $U_0(x)$. The Fourier series ...
- 1,126
1
vote
Laplace equation, medium discontinuity and finite difference method
A surface charge will accumulate on the interface where the dielectric constant has a discontinuity. You need to calculate this surface charge and include it into the discretised Poisson equation. ...
- 188k
1
vote
kth finite difference always positive when kth derivative is?
No, this is not true. For instance, for function $(10(1-e^{-1/x^2})-9)+\exp(x)$ and $f(0)=2$ all the consecutive derivatives are positive at $x=0$, while the first difference is not.
For function
$f(...
- 10.1k
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