Articles of finite differences

The k-th difference of the sequence $n^{k}$ is constant and equal to $k!$

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 […]

$\Delta^ny = n!$ , difference operator question.

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) = $ $((x+2)^n -(x+1)^n) – ((x+1)^n-x^n)$ […]

Given the Cauchy's problem: $y'' = 1, y(0) = 0, y'(0) = 0$. Why finite difference method doesn't agree with recurrence equation?

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} […]

Looking for finite difference approximations past the fourth derivative

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 […]

Show that the $k$th forward difference of $x^n$ is divisible by $k!$

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 […]

Chain rule for discrete/finite calculus

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 […]

What is the difference between Finite Difference Methods, Finite Element Methods and Finite Volume Methods for solving PDEs?

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?

Infinite sum involving ascending powers

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 […]

Linear isoparametrics with dual finite elements

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: […]

Has anybody ever considered “full derivative”?

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 […]