Define the k-th difference of a sequence $\{a_n\}$ inductively as follows: The $1$-th difference is the sequence $\{b_n\}$ given by $b_n=a_{n+1}-a_n$ The “$k+1$”-th difference is the sequence $\{b_n\}$ given by $b_n=c_{n+1}-c_n$, where $\{c_n\}$ is the $k$-th difference of the sequence $\{a_n\}$. Prove that, given the sequence $\{a_n\}$ such that $a_n=n^k$, for a fixed $k \in […]

I was looking in a numerical analysis book and found the statement: if $y=x^n$ and the difference is $h=1$ then $\Delta^ny = n!$ and $\Delta^{n+1}y = 0$. Here $\Delta y = y(x+1)-y(x)=(x+1)^n -x^n$, ($\Delta$ is the difference operator). $\Delta^{2} y = \Delta\Delta y = \Delta(y(x+1)-y(x)) = \Delta y(x+1)-\Delta y(x) = $ $((x+2)^n -(x+1)^n) – ((x+1)^n-x^n)$ […]

The domain we want to obtain the solution on is $x \in [0,1]$. Let’s write the second difference equation corresponding to this differential equation problem $\frac{y_{i-1}-2y_i+y_{i+1}}{h^2} = 1$ (is that step valid by the way?). The first condition is $ y_0 = 0 $. For the second condition we use the first forward difference: $\frac{y_1-y_0}{h} […]

I scanned the internet and could not find further representations of the central difference approximations past the fourth derivative. Are there published results past the fourth derivative? Ideally I am looking to find up to the $12$th derivative. It doesn’t sound like an enjoyable exercise by pencil and paper. Is this information known? Central difference […]

Define the forward difference operator $$\Delta f(x) = f(x+1) – f(x)$$ I believe that if $f(x)$ is a polynomial with integer coefficients, $\Delta^k f(x)$ is divisible by k!. By linearity it suffices to consider a single monomial $f(x) = x^n$. I’ve checked this for small values of $n$ and $k$, and believe that a simple […]

In the context of discrete calculus, or calculus of finite differences, is there a theorem like the chain rule that can express the finite forward difference of a composition $∆(f\circ g)$ in simplified or otherwise helpful terms? It’s probably not possible for a general function, but it might be possible with some restrictions. I’m also […]

Can you help me explain the basic difference between FDM, FEM and FVM? What is the best method and why? Advantage and disadvantage of them?

I was working on a problem involving finite differences and came across the following sum as a general solution to a problem I was working with $$ 1 + x + \frac{1}{1 + a}x^2 + \frac{1}{1 + a} \frac{1}{1 + a + a^2}x^3 + \frac{1}{1 + a} \frac{1}{1 + a + a^2} \frac{1}{1 + a […]

The subject presented here is some content of the Wikipedia page about Platonic solids combined with my own experience on Finite Elements.To start with the latter, there is a standard piece of Finite Element theory concerning triangles on MSE. The concept of isoparametrics is introduced herein. A reference to the same theory is found in: […]

When differentiating we usually take a limit and drop the infinitesimal terms. But what if not to drop anything? First, we extend the real numbers with an infinitesimal element $\varepsilon$ which has its own inverse $1/\varepsilon=\omega$. And define the full derivative of a function formally as follows: $$D_{full}[f(x)]=\frac{f(x+\varepsilon)-f(x)}{\varepsilon}$$ Now we can compute full derivatives of […]

Intereting Posts

Proving two equations involving the greatest common divisor
Computing an awful integral
Do endomorphisms of infinite-dimensional vector spaces over algebraically closed fields always have eigenvalues?
Probability $P(A < B)$
how to prove $\sum_{k=0}^{m}\binom{n+k}{n}=\binom{n+m+1}{n+1}$ without induction?
Finding vector $x$ so that $Ax=b$ using Householder reflections.
Multiplication of Rational Matrices
Can a polynomial size CFG over large alphabet describe a language, where each terminal appears even number of times?
A simple mathematical expression for the periodic sequence $(1, -2, -1, 2, 1, -2, -1, 2,\dots)$
Construction of an infinite number type and other ideas
Factoring a hard polynomial
Uniqueness for antipodal points of maximum distance on closed convex surface
First order differential equation : $\frac{dy}{dt}+kty(t) = \frac{\sin(\pi t/10)}{\pi}$
Angle bracket and sharp bracket for discontinuous processes
Is R with $j_d$ topology totally disconnected?