Questions tagged [finite-differences]

A method in numerical analysis which consists of approximating the derivatives of a solution of an ordinary or a partial differential equation. This leads to the solution of a linear system.

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Stability of pde in some $L^p$ norm and stability of a numerical scheme for it equivalence.

I would like to get some light on how to proceed and my confusion. I consider some IBVP of the form $$u_t+L(t,x)u=0, x\in D, t\in [0,T]$$ with some BC and initial data. And I use some numerical method to solve it. First, I start with a question of…
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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.…
Axion004
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Proof of an identity of $n!$

I came up (numerically) with an identity concerning n! and I was wondering about a proof of it. Here it is: \begin{align} \ n! &= \sum_{r=0}^{n} { \binom{n}{r} (-1)^r(k-r)^n } \quad \forall n \in \mathbb{Z}^+ \quad \forall k \in \mathbb{R}…
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Wikipedia wrong? Convergence of finite difference

Update: I have edited the Wikipedia page, so that the mistake no longer appears. On the Wikipedia article for "Finite difference" there is the claim Assuming that $f$ is continuously differentiable, [we have] $$ \frac{f(x+h) - f(x)}{h} - f'(x) =…
Eric Auld
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Who introduced the finite difference notation using $\Delta$?

We all know that Leibniz introduced the differential notation $dx, dy$, and that in developing his calculus for infinitesimal differences he was inspired by his previous work on finite diffences. Today we usually denote such finite differences…
Mikhail Katz
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Nth order central difference for odd $n$

I am interested in the $n^{\mathrm{th}}$-order central difference of an expression $f$. The general form of the $n^{\mathrm{th}}$-order central difference is given by $$\delta_h^n[f](x)=\sum_{i=0}^n(-1)^i {n \choose i}f(x+(\frac{n}{2}-i)h).$$ For…
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Computing Hessian in Python using finite differences

I am computing the Hessian of a scalar field, and tried using numdifftools. This seems to work, but was quite slow so I wrote my own approach using finite differences. Here is my code for the Hessian: def hessianComp ( func, x0, epsilon=1.e-5): …
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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…
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Finite difference method for non-uniform grid

It's been two days that I've been stuck on this problem: Given a regular function $u$, and a non-uniform grid, where every node has a non-constant distance from another, I want to find $u'(x_i)$ and get some information about the error. Here's a…
VoB
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Computing the elements of a Hessian matrix with finite difference

I have a generic function $g(x)$ where $x$ is an 6-dimensional vector. Now I want to compute the Hessian of this function for a point $x_0$. What is the most efficient way to do this? Can I do this with finite differences and which formulas do I…
Derk
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When do the solutions of a system of difference equations converge to solutions of a system of PDEs?

I have a set of rather nasty, nonlinear difference equations roughly of the following form: $$ \frac{a^{(j)}_{i+1}(s)-a^{(j)}_i(s)}{\epsilon}=f^{(j)}(\lbrace a^{(j)}_i(s)\rbrace_{j=1}^m,\lbrace a^{(j)}\prime_i(s)\rbrace_{j=1}^m,\epsilon) $$ I have…
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Why some differential equation can be solved while similar difference equations cannot?

Take an equation $$w'+w-w^2-1=0$$ Its solution is $$w(x)=\frac{\sqrt{3}}{2} \tan \left( \frac{\sqrt{3}}2 C+\frac{\sqrt{3}}2 x\right)+\frac12$$ I wonder why a similar difference equation $$\Delta w+w-w^2-1=0$$ cannot be solved?
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Solving recurrence -varying coefficient

How can one find a closed form for the following recurrence? $$r_n=a\cdot r_{n-1}+b\cdot (n-1)\cdot r_{n-2}\tag 1$$ (where $a,b,A_0,A_1$ are constants and $r_0=A_0,r_1=A_1$) If the $(n-1)$ was not present, this could easily be solved using a…
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Show $f[x_0,x_1,\ldots,x_n]=1$ for divided differences of $x^n$

Let $f(x)=x^n$ for $n \in \mathbb{N}$. Prove that $f[x_0,x_1,\ldots,x_n]=1$, where $\{x_i\}$ are distinct $n+1$ real numbers. I tried doing this by finding an interpolation of the function and by using Newton's form but what I did didn't lead to…
Belgi
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How to solve this finite-difference equation?

How to solve the following finite-differences equation: $$f(x) = f(x-1) + f(x-\sqrt{2}), \quad x\in [\sqrt{2}, +\infty) \,?$$ Let's say $f(x) = f_0(x)$ for $x \in [0, \sqrt{2})$ is a given function. My attempt: Let's write characteristic function…
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