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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41 views

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. ...
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1answer
32 views

How to discretize mixed partial derivatives?

How to discretize $\frac{\partial^3 f}{\partial x\partial y^2}$ at mesh point $(i,j)$? We should use mesh points which are nearest to $(i,j)$.
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1answer
31 views

$\Delta^kn^\alpha$ converges monotonically to zero when $\alpha<k$

I am trying to prove that the $k$-th finite difference of the series $n^\alpha$ converges to zero monotonically as $n\to\infty$ when $\alpha<k$. The differential analogue is ...
3
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4answers
62 views

Elementary Proof of sum of alternating factorials

so I came across the rather interesting identity $$ \sum_{i = 0}^{\infty}\left[\sum_{j=0}^{i}\left[\frac{(-1)^j}{j!} \right] \prod_{j=0}^{i}\left[ (x-j)\right] \right] = x! $$ For positive integers ...
5
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1answer
70 views

Expressing a Recursion in terms of factorials

Given the recursion $$f(n) = nf(n-1) + (n-1)f(n-2) $$ $$f(0) = 1, f(1) = 1$$ How exactly does one express the target function? I know that $$f(n) = nf(n-1)$$ gives rise to $$f(n) = \Gamma(n+1)$$ ...
2
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1answer
46 views

Checking if the Hessian is the derivative of the gradient

Suppose $f: \Bbb R^n \to \Bbb R$. I have a code that computes the gradient of $f$. I have another code that computes the Hessian of $f$ times a vector. Now I want to check if they are correct. ...
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1answer
43 views

Solving ODE with matrices

I have an equation in ODE $M{'}(x)= M(x)*A(x)$. Issue here is $A(x) = C_1+C_2* x $ where $C_1,C_2 $ has dimension $3 \times 3$. And x is a scalar variable Doubt What is M(x)? Can any one give ...
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1answer
32 views

Finite Difference Method Stability with diffusion equation

The diffusion equation is: $ \frac{\partial T}{\partial t} = \alpha \left( \frac{\partial^2 T}{\partial x^2} \right) $ An explicit finite difference approach can be used to solve this, forward in ...
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0answers
25 views

Space-dependent diffusivity and finite-differences

I want to implement a finite difference code of this simple diffusion equation with space-dependent diffusivity: $$\partial_{t}u =D\partial_{x}^{2}u+\partial_{x}D\cdot\partial_{x}u$$ I go for a ...
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1answer
54 views

Second derivative of discrete function

Given function $y[n]$ what is the best way to define the second derivative? Some background to the question: in linear systems we often sample a continuous signal to a discrete one with sample rate ...
2
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0answers
100 views

Finite Difference Spacing of Points for PDE's for Convergence of Explicit Forward-Stepping Scheme

I realize that this question could be pretty broad, but I'm wondering at least what the conditions are for my simulation. I'm developing an Explicit Forward-Stepping Finite Difference scheme to solve ...
2
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1answer
39 views

Finite differences to ODE in polar coordinates

I have an equation of finite differences as follows: $$\frac{X_1(r+\epsilon)-X_1(r)}{ \frac{\epsilon~\beta}{2~r+\epsilon} } + \frac{X_1(r-\epsilon)-X_1(r)}{ \frac{\epsilon~\beta}{2~r-\epsilon} } = ...
0
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2answers
40 views

Avoiding substraction for finite difference with log and exp

I want to approximate the derivative of f(x) Finite difference $f'(x) \approx \frac{f(x+h)-f(x)}{h}$ I was taught that the error from the substraction is blown up for small h. This I can verify ...
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0answers
26 views

heat equation with Interface Crank Nicolson

I am currently working on solving the heat equation with an interface numerically using Crank-Nicolson. There are jump discontinuities at the interface which are dealt with using fictitious values ...
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0answers
32 views

Stability conditions

Below is a problem about stability conditions that I have been struggling with it during an exam: Find the stability conditions for $$A\left ( \frac{\partial^2 u(x,\, y,\,t)}{\partial x^2} + ...
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0answers
14 views

Why more smooth the function the more precise finite difference method?

As the title, Why more smooth the function the better finite difference method? I guess that if the function is smooth we can better approximate with Taylor series, but formally how this helps? ...
2
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0answers
88 views

Boundary integral method to solve Poisson equation

Suggest how to solve Poisson equation \begin{equation} σ ∇^2 V = - I δ(x-x_s) δ(y-y_s) δ(z-z_s) \nonumber \end{equation} by using the boundary integration method to calculate the potential ...
1
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1answer
34 views

Finite differences coefficients

I'm interested in deriving a forward finite difference approximation for the gradient of a function, $f(x)$, at the point $x = x_i$ using $k+1$ points. If the spatial domain is uniformly discretized, ...
3
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1answer
69 views

9 point stencil for Laplacian operator

Given the following 9 point Laplacian \begin{align} -\nabla^2u_{i,j} = \frac{2}{3h^2}\left[5u_{i,j} - u_{i-1,j} - u_{i+1,j} - u_{i,j-1} - u_{i,j+1} - u_{i-1,j-1} - u_{i-1,j+1} - u_{i+1,j-1} - ...
5
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1answer
82 views

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. ...
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11 views

Why is it that if $||\textbf{v}^{n+1}||_h \leq (1+kK)||\textbf{v}^{n}||_h$ then the scheme is stable?

Question I'm having difficulty understanding a lemma in my notes. The lemma is: If there exists a constant K, independent of $h,k,n,\textbf{v}^0$, such that $||\textbf{v}^{n+1}||_h \leq ...
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27 views

Finite Differences and Scale Invariance

The semilinear heat equations $u_t = u_{xx} + u^p$ is invariant under the one parameter family of scalings $ t \to \lambda t, \quad x \to \lambda^{1/2}x, \quad u \to \lambda^{-1/(p-1)}u$. When we ...
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1answer
24 views

Finding the value of Cmax for the Courant–Friedrichs–Lewy condition

According to the Courant–Friedrichs–Lewy condition, in the 1 dimensional case we have: $C=\frac{u\cdot\Delta_t}{\Delta_x} \leq C_{max} $ It is said that $C_{max}$ changes depending on the method ...
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0answers
32 views

Finite Difference Scheme for a PDE on non-rectilinear coordinates

Consider poisson's equation on the domain $0 \leq x \leq 1$ $0 \leq y \leq H(x)$. Change the coordinates to $\xi=x$, $\eta=y/H(x)$. Construct a FDS that gives a positive definite symmetric matrix. ...
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1answer
24 views

Raising the power of forward difference formulas

In a forward difference formula, $D_-$ refers to the backward difference operator. Thus, $(D_- u)(x) = u(x)-u(x-h)$. In the answer key of a problem that I was working on, one of the steps is: ...
0
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1answer
46 views

General solution of a finite-difference equation with real non-commensurable differences.

How to find a general solution of the following functional (recurrence) equation: $$f(x) = c_1 f(x - a_1) + \dots + c_n f(x - a_n), \tag{1}$$ where $c_i, a_i$, $i = 1, ... n$ are arbitrary real ...
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0answers
24 views

Discretization 5 equations and unknowns on a 2d grid

I would like to solve the following set of equations using a finite difference approach with a 2D-grid. $\frac {Dφ}{Dt}$ = (1 − φ)P + Γ with $\frac {Dφ}{Dt}$ = $\frac {\partialφ}{\partial t}$ + V · ...
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46 views

Is there a special name for pi-based finite difference?

Did anybody consider $\pi$-based finete differences, that is the operator $$\Delta_\pi f(x)=f(x+\pi)-f(x)$$ and its corresponding inverse operator? It seems for me that taking the step equal to $\pi$ ...
3
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1answer
97 views

Having trouble understanding the finite element method

I am trying to read some sources on how to implement the finite element method and I am having difficulty putting all the concepts together. I can read and understand the Galerkin approach just fine. ...
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40 views

Finite difference discretization of the Porous Medium Equation - How to prove stability?

The porous medium equation is given by $\delta_t u(x,t) = (u^p \cdot u_x)_x, (x,t) \in \mathbb{R} \times (0, \infty)$ We are working on approximating the solution with finite differences. ...
4
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1answer
26 views

Numerical computation of the $n^{\mathrm {th}}$ derivative of a multivariate function

From a multivariate function $f$, depending on $n\geq 1$ variables, which can be computed numerically, but which does not admit simple analytic expression, I would like to approximate numerically the ...
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82 views

FDTD validation with Poynting Vector

I'm attempting to validate my FDTD optics simulator results. Visually, I can see that my output is nearly identical to that produced by Meep, the only difference being a slight phase shift. Using a ...
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0answers
68 views

Calculate a 5x5 Vandermonde system for a 5 point mesh

This is problem 1.2 in Randall J Leveque's textbook, "Finite Di fference Methods for Ordinary and Partial Di fferential Equations". I'm struggling with how to actually do the computation, I'm not so ...
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0answers
15 views

finding most and least they can afford

The Johnsons have accumulated a nest egg of $30,000$ that they intend to use as a down payment toward the purchase of a new house. Because their present gross income has placed them in a relatively ...
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1answer
61 views

Analytical way of describing centred difference coefficients

I am trying to find an analytical way to describe the finite difference coefficients of various degrees of accuracy of centred difference schemes that approximate the second derivative. For example, a ...
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1answer
91 views

In the numerical solution of the Wave Equation, using finite differences, where do I obtain the spatial values from?

In trying to implement a simplistic numerical solver for wave equations, I have run into a conceptual problem that I haven't been able to solve. Consider a one-dimensional wave equation of a quantity ...
2
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1answer
107 views

Finite element method for the 'Particle-In-a-Box' problem in quantum mechanics

(Apologies in advance for the lengthy question, but it really is needed for a precise description of what I've done!) In suitable units, the 'Particle-in-a-box' problem is described by the following ...
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2answers
129 views

A proof using $\Delta^ny(t)=\sum_{k=0}^{n}{(-1)^k\dbinom{n}{k}y(t+n-k)}$

Please How can I use $\Delta^ny(t)=\sum_{k=0}^{n}{(-1)^k\dbinom{n}{k}y(t+n-k)}$ to prove $\sum_{i=0}^{n}{(-1)^i\dbinom{n}{i}y(i)}=(-1)^n\Delta^ny(0)$ and hence to evaluate ...
0
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1answer
78 views

Summation of falling factorials

I just want to know if I should evaluate $\sum(t+1)^\underline{4}$ the way we evaluate $\sum{t^\underline{4}}$. Thanks.
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1answer
101 views

Negative falling Factorial

Please can someone tell me what is the value of $1^\underline{-2}$? I know that $1^\underline{2}=0$. Thanks.
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0answers
29 views

Questions on Difference operators

Please I really need help on the following short problems on difference operators that I need even some clues on how to go by them: 1) $\sum_{t=1}^{4}{\dfrac{1}{(t+1)(t+2)(t+3)}}= ...
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2answers
189 views

Derive forward Euler method for two-variable function

I need to derive the forward Euler method for solving ODEs and I would like some comments on what I have so far; overdot denote the time derivative: $\dot x \overset{def}{=} dx/dt$. Say we have ...
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198 views

Solve the following boundary value problem using the finite difference method.

Solve $$y''=\frac{1}{2}y'-\frac{1}{2}y+\frac{x^2+3}{2}, ~~~~~y(0)=1, ~~y(4)=24$$ using the second order finite difference approximation order with $h=1$. I know that we use ...
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1answer
38 views

Chain rule in finite calculus for ODEs (RK methods)

SE.Math, I'm reviewing for a test on numerical methods and I am having some difficulty in understanding how to take the second derivative in finite terms. I have: $$ U(t_k) $$ as my underlying ...
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56 views

Find the solution for the given equation in discrete finite differences

Given this equation, determine its solution in finite differences: $y(n+2)+y(n+1)+y(n)=0$ With initial conditions: $y(0)=0$, $y(1)=0$. To determine the general solution I got this: $\lambda^2 + ...
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0answers
63 views

A 2D secant method?

I've recently had occasion (providing an engineering colleague with a little mathematical help) to solve a non linear system $\begin{align*}f(x,y)&=0,\\ g(x,y)&=0.\end{align*}$ If ...
0
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1answer
21 views

von Neumann stability analysis for irregular meshes

All the litterature I have come across about the von Neumann stability analysis is performned on regular grids. Can the analysis be performed analytically on irregular grids, or does it have to be ...
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1answer
70 views

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 ...
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20 views

Proof that Newton expansion over derivatives has the properties of an integral

Let's consider a Newton expansion over consecutive derivatives of a function: $$F(x)=\sum_{m=0}^{\infty} \binom {-1}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$ Can it be proven that such ...
0
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1answer
66 views

Show $f$ concave, $C^2$ implies $f''\leq 0$

Suppose I wanted to show that a concave function $f:(a,b) \to \mathbb{R}$ which is $C^2$ must have negative second derivative at each $x\in (a,b)$. I might try this by finite difference, noting that ...