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: […]

After seeing the neat little identity $(n+3)^2-(n+2)^2-(n+1)^2+n^2=4$ somewhere on MSE, I tried generalising this to higher consecutive powers in the form $\sum_{k=0}^a\epsilon_k(n+k)^p=C$, where $C$ is a constant and $\epsilon_k=\pm1$. I discovered a relatively simple algorithm to generate these patterns: simply take $(n+2^p-1)^p$ and subtract $(n+2^p-2)^p$ (using $n\to n-1$) to get a polynomial of degree $p-1$. […]

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 […]

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