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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6
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1answer
105 views

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 ...
1
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0answers
35 views

Elementary proof about nth differences of nth powers of integer

In a post on Math.SE., a proof sketch was proposed for the proposition below: The sequence of $n$th differences of the sequence of $n$th powers of positive integers, is the constat sequence $n!$. ...
0
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0answers
5 views

Finite differences on a hexagonal/triangular lattice with Cartesian coordinates

So, I've been thinking recently about how to approximate the Laplacian operator using finite differences on a non-square lattice. For example, on a typical square lattice, in a Cartesian coordinate ...
0
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0answers
12 views

Turning a difference equation in two unknowns into an eigenvalue problem

We were given a differential equation to solve numerically, {y''-3y'+2k^2*y=0, y(0)=y(1)=0}. We used the finite difference approximations to convert it into a diference equation. The following ...
0
votes
1answer
22 views

Unsure of 2d finite difference method with second order term?

I have the following equation for the price of Black Scholes Euro option - (1) $$-\frac{\delta C}{\delta t} = -\alpha\frac{\delta^2 C}{\delta x^2} + [r - \delta + \frac{\sigma^2}{2}]\frac{\delta ...
0
votes
1answer
16 views

Backwards finite difference for mixed partials at higher order

I'm trying to understand what is the general method for calculating a backwards difference for a mixed partial of $n$ variables. Let's start with one variable: The forward and backward finite ...
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0answers
19 views

Ideas on how to improve stability in solving PDE

I am solving a non-linear second order system of PDEs in two variables. The equations are too complicated to write out here, but an essential feature is that there is a propagating wave which then ...
0
votes
0answers
10 views

Stability of Difference Equations

Suppose $\dot{x}(t) = h(x(t))$ is a well posed ODE and $x^*$ denotes an asymptotically stable equilibrium point. Let $B$ be an open set containing $x^*.$ Also suppose that $B$ lies in the region of ...
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0answers
15 views

Proper writing of IBVP PDE & Finite Difference Implementation

I've seen a few examples (see slide 4 ) of the 2D heat equation described as $$ \begin{cases} u_t(t,x,y) = \nabla^2 u(t,x,y), \quad t > 0, \quad (x,y) \in (0,L) \times (0,H), \\ u(t,0,y) = ...
0
votes
0answers
29 views

eigenvalues inequality finite differences

I have $x,y\in[0,1]^2$, $a\in[0,A]$ $t\in[0,T]$ and the mesh points $x_j = j \, \Delta x, j=0,\ldots,J$; $y_l = l \, \Delta y, l=0,\ldots,L$; $a_k = k\Delta a, k=0,...,K$ and $t_n = nh, n=0,\ldots,N$ ...
2
votes
1answer
73 views

What is this type of function called? How can I translate it to a different origin?

A factory produces 1 widget per week. A builder builds 1 factory each week. A construction firm trains 1 new builder each week. Partially-produced things do not produce anything. Starting with 1 firm, ...
0
votes
1answer
24 views

Having trouble with discretization and boundry value problems

I have the following homework question: Consider the boundary value problem $y''(x) + 5y'(x) − (2 + x)y(x) = e^x$ on $x ∈ (0, 2)$ with boundary conditions $3y(0) + y'(0) = 5$ and $y'(2) = 7$. ...
2
votes
1answer
34 views

How do I write the generic finite difference approx of f'(x) using Lagrange interpolating polynomial approximation?

I have the following homework problem: (10 points) Differentiation Formulas by Lagrange Interpolating Polynomials. (a) Write the generic finite difference approximation to f'(x) using the Lagrange ...
0
votes
0answers
6 views

What is the parameter of the Streamline-Diffusion Method?

Recently, I want to apply streamline-diffusion(SD) method to solve Burgers Equation. \begin{align} u_t + uu_x - \varepsilon u_{xx} &= 0,\\ u(0,t) = u(X,t) &= 0,\\ u(x,0) &= u_0(x),\ \ \ ...
0
votes
1answer
42 views

Expressing a function's value using finite differences

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ and let $x = (x_0, x_1, x_2, \dots)$ be a sequence of pairwise distinct real numbers. For every $n \in \{1, 2, \dots\}$ and every ordered $(n+1)$-tuple ...
0
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0answers
21 views

How to make a 9-point two-dimensional stencil for a elliptic operator?

I want use a finite difference schem to discretizate the elliptic operator: $$ \nabla \cdot \left( k(x, y) \nabla p\right), $$ where $k(x, y)$ a positive scalar function and $p$ is the unknow. We can ...
3
votes
0answers
23 views

Stability of a scheme for one way wave equation.

Multiplying the scheme \begin{equation*} \dfrac{v_m^{n+1} - v_m^n}{k} + \dfrac{a}{2}\left(\dfrac{v_{m+1}^{n+1} - v_m^{n+1}}{h} + \dfrac{v_m^n - v_{m-1}^n}{h}\right) = 0 \end{equation*} by $v_m^{n+1} + ...
0
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0answers
24 views

Show that $\Delta^{-1}\sin\alpha x = - \cos(\alpha x - \frac{α}{2})/(2\sin \frac{\alpha}{2}) + c(x)$ for fixed $\alpha\in \Bbb{R}$.

Show that $\Delta^{-1}\sin\alpha x = - \cos(\alpha x - \frac{α}{2})/\left(2\sin \frac{\alpha}{2}\right) + c(x)$ for fixed $\alpha \in \Bbb{R}$. I can find a few tables for anti-difference tables ...
0
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0answers
32 views

Finite Difference Method for Heat Equation with Neumann Boundary

I have read the book of Morton and Mayers. In chapter6, it said that the explicit finite difference scheme of a heat equation, $\frac{U^{n+1}_j-U^{n}_{j}}{\Delta ...
-1
votes
0answers
26 views

Laplace equation in polar coordinates [closed]

Hy, I'm new in here but i have a question please help me. I want to solve a Laplace PDE in a polar coordinate system with finite difference method, but I have a problem with boundary conditions at r = ...
0
votes
2answers
38 views

Given this operator what is inverse operator?

Given operator $$\Delta_{sym}[f(x)]=\frac{f(x+\varepsilon)-f(x-\varepsilon)}{2\varepsilon}$$ what is inverse operator in terms of summations? For instance, given operator ...
21
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2answers
514 views

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 ...
0
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0answers
25 views

How to transform $\frac{\partial^2 u}{\partial t^2}=\frac{\partial^3 u}{\partial x^3}$ into a system of first order PDE's and finite difference matrix

So I have this equation: $\frac{\partial^2 u}{\partial t^2}=\frac{\partial^3 u}{\partial x^3}$ and I need to transform it into a system of first order PDE's. I was thinking like this: ...
0
votes
1answer
23 views

In the finite difference formulas, how can we pick h to give a certain tolerance?

There's lots of questions on here about finite differences. In particular, picking the 'best' h value. But what if I want to find the biggest 'h' which bounds to a given tolerance? On first glance, I ...
0
votes
0answers
14 views

Finite Differences Scheme for Poisson Nernst Planck Equations

I want to solve numerically a simple version of Poisson Nernst Planck equations in 1-D on the domain $[0,\ell]$: $C_t=-j^c_x=(C_x +\rho\phi_x)_x$ $\rho_t=-j^\rho_x=(\rho_x +C\phi_x)_x$ ...
0
votes
0answers
25 views

Riccati Finite Difference Equation:

Question: Consider the finite difference equation in Zeilberger notation: $$D_{1,x} [y] = a_0(x) + a_1 (x) y + a_2 (x)y^2 $$ Which in functional form is: $$ y(x+1) - y(x) = a_0(x) + a_1 (x) y + ...
1
vote
1answer
20 views

Proof that strictly tri-diagonally dominant matrix has an inverse

We are given the following theorem of which we need only know the result. Theorem Suppose an $n\times n$ matrix $A= (a_{ij})$ is tri-diagonal with $a_{i,i-1}a_{i,i+1} \neq 0$, for each ...
0
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0answers
25 views

Finite Difference method for nested derivatives

I'm moderately experienced with finite difference methods, but I'm hoping that somebody has better intuition than I do. Suppose I have the following expression which I want to evaluate via finite ...
2
votes
1answer
49 views

Construct a symmetric Poisson matrix from $\nabla \cdot (\rho\nabla p)$

Let $\rho$ and $p$ be 2D scalar fields. How do I construct a symmetric Matrix $A$ so that is satisfies \begin{equation} Ap = \nabla \cdot (\rho\nabla p) \end{equation} in finite differences? I am ...
0
votes
0answers
19 views

Computing the CFL number for non-linear advection

Consider $u_t -(1-2t)u_x=\phi(t,x) $ with $u(x,0)=u_0(x)$ for integrable $\phi$. The solution for this PDE is given by $ u(x,t)= u_0(x+t+t^2) + \int_0^t \phi(x(z),z) dx$. For a consistent scheme ...
0
votes
1answer
19 views

Writing the implicit scheme using matrices

\A= \begin{bmatrix} 2 & -1 & 0 & . & . & . & 0 \\ -1 & 2 & -1 & . & . & . & . \\ 0 & . & . & . & . & . &. \\ ...
0
votes
1answer
62 views

Closed form solution to $\frac{1}{a-1}= \log a$

I want to find a function that satisfies $$\Delta [f(x)]=f'[x]$$ Obviously the solution is the exponential function $f(x)=a^x$ with $a$ in between $2$ and $e$ because $\Delta[2^x]=2^x$ and ...
2
votes
1answer
79 views

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} ...
0
votes
0answers
17 views

Finite differences

I don't know finite differences but I have reached a step in a proof where I need them. In particular, I need to know the $3k$th differences of $4\cdot 1^{3k}, 4\cdot 2^{3k},...,4\cdot (3k)^{3k}$. I ...
3
votes
0answers
64 views

Finite difference solution of steady-state diffusion equation with variable material properties

I'm trying to use a finite difference method to solve the steady-state neutron diffusion equation in a nuclear reactor: $$ D(x) \nabla^2 \phi(x) + \left( \frac{\nu(x)}{k} \Sigma_f(x) - \Sigma_a(x) ...
0
votes
0answers
48 views

Solution to the heat equation using a finite difference scheme

I have used a difference scheme and Fourier Analysis to find an expression for the solution to the heat Equation. My problem is plotting my solution. $w_{k,j,m}=(1-\Delta t\mu_k)^msin(k\pi x_j)$ ...
0
votes
0answers
48 views

Laplace problem with elliptic BC - FINITE DIFFERENCES

I am experiencing some trouble in solving the problem $u-\Delta u = f$ with periodic boundary conditions. First question: Is the problem always solvable? If I am not mistaken $-\Delta u = f$ ...
0
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0answers
24 views

Order of accuracy of an scheme

How does one find order of accuracy for the following scheme (see picture below). The picture shows two equations instead of the typical one equation which confuses me a lot. You can read about the ...
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0answers
30 views

Irregular grid for finite differences PDE solution

I got a project in a class I'm taking. In this project, I need to solve fluid dynamics and heat equations. Up to here, the problem is not so complicated, however the chamber shape is the problem: as ...
1
vote
1answer
42 views

Calculus over integers for derivative/integral of factorial?

One usually introduces the Gamma function to define a derivative of the factorial. However couldn't one define a derivative over integers like $$ f'(n) = \frac{f(n+1) - f(n)}{1} ? $$ Such a discrete ...
0
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0answers
11 views

Discretization of Variable Coefficients

I'm trying to discretize something of the form $$ a(x)\frac{df}{dx}. $$ Now, with $x_{i}\equiv i \Delta x$ and $a_{i}\equiv a(x_{i})$ (similarly for f), this is what I think it should be: $$ ...
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0answers
67 views

Approximate Laplace Operator with Central Difference in Polar Coordinates

I'm trying to approximate the Laplace operator in polar coordinates with the central difference quotient and I know how to do this in cartesian coordinates, but with polar coordinates I just feel ...
2
votes
3answers
53 views

Finite differencing of the diffusion (heat) equation

I am attempting to code a problem for a meteorology class. Our initial equation was as follows: $$\tfrac{\partial u}{\partial t} = \nu \tfrac{\partial^2 u}{\partial x^2} (*)$$ We were then assigned ...
0
votes
1answer
39 views

In 1D FDTD, Do we expect power oscillations at PEC reclecting boundary?

Basically I am trying to check if my 1D FDTD code works fine and how to write quantity that is conserved all the way. In 1D FDTD should we expect that the power is conserved when the pulse is being ...
0
votes
0answers
15 views

Deriving solution of Poisson equation by considering special integral & finite differences

I know how to solve the Poisson Equation on the unit disc $\{ (x,y) \in \mathbb R^2 : x^2 + y^2 < 1 \}$ by separation of variables, and then rearranging to get the Poisson Kernel. But I am asked to ...
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0answers
44 views

Diffusion of a chemical species inside a Y-shaped tube

I'm trying to model diffusion of a chemical species X inside a Y-shaped tube, whose diameter (thickness) is constant everywhere. The diffusion constant of X is $D$ ($\mu$m$^2$/s), so the concentration ...
0
votes
1answer
18 views

Question about Convergence Definition for Finite Difference Scheme

I have a question about the Convergence Definition for Finite Difference Scheme. The definition is given by Convergence: for one-step schemes approximating a IBVP to be convergent we compare ...
1
vote
0answers
41 views

How to improve stability of numerical solutions to partial differential equations

This is a quite general question, but I am working with a system of partial differential equations in two variables. There is one time direction $t$ and one spatial direction $z$ and the numerical ...
0
votes
0answers
83 views

Stability of Lax-Wendroff Approach for Advection Equation

The Problem: I am attempting to solve the following problem in 1D over a periodic region: "In one dimension, the mass density $\rho$ is advected with velocity $v$, so that it follows the equation: ...
0
votes
0answers
23 views

Discretisation of a product of two functions

Suppose I have two functions, $f(x,t)$ and $g(x,t)$, and for an upwind scheme I want to use the quantity $\partial_x (fg)$ to solve the advection equation $$ \frac{\partial f}{\partial t} + ...