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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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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21 views

Numerical methods and Matlab

I am solving parabolic partial differential equation using Matlab and Finite difference method. I am new to Matlab so I do not know how to write ICs/BCs in Matlab numerically. If some one help me out ...
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1answer
12 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: ...
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11 views

Interpolation and finite differences question. [closed]

$u_x$ is a function of $x$ for which fifth differences are constant and $u_1$ $+$$ u_7$ $=$ $–786$ $ ,$ $u_2$ $+$ $u_6$ $=$ $686$, $u_3$ $+$ $u_5$ $=$ $1088$. Find $u_4$. Answer is $570.9$
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8 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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41 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 $\pi$ ...
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1answer
69 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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24 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. ...
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1answer
20 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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42 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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55 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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12 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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20 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
53 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 ...
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1answer
74 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
106 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 ...
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1answer
49 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
63 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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27 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
99 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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147 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
33 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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34 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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45 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 ...
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17 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
56 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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18 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 ...
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1answer
64 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 ...
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2answers
111 views

Understand 1D FEM solution using quadratics elements

I'm a bit confused about applying the FEM using piecewise linear functions. I think I get understand how to use linear functions. We use the hat function for each element and the solution is ...
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1answer
37 views

Computation of $n$-th order difference of falling factorial

I was reading a difference equation textbook and came across a problem. The question asks to compute ${\Delta}^nt^{\underline3}$ for $n=1,2,3,...$, where $t^{\underline3}$ is the falling factorial ...
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1answer
136 views

Applying two-point forward to two-point forward formula

What do you get when you apply the two-point forward finite difference formula for the first derivative of $f(x)$ to the two-point forward finite difference formula for the first derivative of ...
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1answer
19 views

What's the derivative $\frac{dE[F]}{dF_i}$ of this function $E[F] = |\frac{dF}{dx}|^2$?

The function is $E[F] = |\frac{dF}{dx}|^2$ where we take $\frac{dF}{dx}$ to be the discrete derivative defined by $F_{i+1} - F_i$. Could someone walk through why $\frac{dE[F]}{dF_i} = ...
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1answer
198 views

Discretize second order derivative using the finite difference scheme

I have some problems in Discretize the second order derivative of this equation after I find u(x) by integrating it, I have problem in discretize it ,, I'm not mathematical person and I try to read ...
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65 views

Catalan numbers via partial difference equations?

It is known that Catalan numbers can be characterized in the following way: let $f(n,k)$ be a function of two integer variables, such that the following recurrence holds: $$f(n+1,k+1) = f(n,k) + ...
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447 views

Finite difference approximation of heat equation with source term

I am using the implicit finite difference method to discretize the 1-D transient heat diffusion equation for solid spherical and cylindrical shapes. The general equation is: $$ ...
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2answers
125 views

Is there a mean value theorem for higher order differences?

The standard mean value theorem tells us $\frac{f(x+h)-f(x)}{h} = f'(c)$ for some $c$ between $x$ and $x+h$. Rewriting this, we may see it as $\frac 1h\Delta_h f(x) = f'(c)$. This makes me wonder if ...
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1answer
51 views

Numerical solution of a difference equation

I would like to solve in $f$ the equation $f(x) - f(x-d) = g(x),$ where $g$ is a given function and $d$ a given constant delay. We can assume $f(x) = 0$ for negative $x$. When $g$ is sampled (with a ...
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1answer
133 views

Finite difference for variable conductivity

I'm trying to discretize a portion of the heat equation for a sphere and for a cylinder where: $r$ = radius, $T$ = temperature, and $k$ = thermal conductivity. for the cylinder shape: ...
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1answer
187 views

Finite difference applied to a product

I believe that the finite difference $$\frac{f(x_0 + \frac12 \Delta x) - f(x_0 - \frac12 \Delta x)}{\Delta x}$$ approximates $f'$, and has limit $f'(x_0)$ as $\Delta x \to 0$. Am I correct in ...
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1answer
24 views

Stability of difference equation considering only positive values

I'm analyzing the stability of such system difference equation with the constraint that $y_n \geq 0$ $\forall n \geq 0$ : $y_n = B y_{n-1} + D y_{n-2} \enspace (1)$ Using variable transform, the ...
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55 views

Grid Function Norm

I am studying about the global truncation error in finite difference methods and I have a question about calculating the error in a Boundary Value Problem (BVP). If we take a simple 1-D problem, the ...
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1answer
110 views

Related to Applying Runge-Kutta Method

I have an initial value problem (henceforth IVP) as follows: $$\frac{d \Phi(t)}{dt}= A(t)\Phi(t)$$ subject to the initial condition $\Phi(t_0)=I$, where $\Phi(t), A(t), I$ are square matrices of same ...
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1answer
124 views

Discrete Bessel Functions

I am reading the paper "Discrete Bessel Functions" by R.H. Boyer (Journal of Mathematical Analysis and Applications. Vol 2, Issue 3, June 1961, pg. 509-524) and he begins the paper by discussing ...
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163 views

Stability of the BTCS scheme for the heat equation in a disk

Consider the $1$-D heat equation: $$ u_t = a \Delta u = au_{xx} \\ u(0,t) = u(1,t) = 0 \\ u(x,0) = u_0(x) $$ where $a > 0$ is constant and $u_0$ is given. It is a classic result that the implicit ...
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32 views

How to show $\Delta_{\xi_i}^{\alpha_i}\Delta_{\xi_i}^{\alpha_j}=\Delta_{\xi_i}^{\alpha_i+\alpha_j}$?

Hi I need some help with the following problem: I know the following equality holds: ...
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1answer
142 views

A finite difference method for robust convergence despite large time steps in first order ODE

Suppose we're looking at a first order ODE of the form $$ \frac{dx}{dt}=-\lambda x+ b u $$ where $\lambda$ and b are functions of $x$ and $u$ is an 'energy generating' term which is a function of $x$ ...
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81 views

solution of difference equation

I am trying to solve the following difference equation: ...
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2answers
35 views

finding value of formula.

I am little bit confusing how to calculate $δβ/δρ$ value if I have set of values like this. I have the values of $β$ and $ρ$ like this. $$ \begin{array}{l|l} β & ρ\\ \hline 0,324 ...
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1answer
104 views

Finite Difference without boundary conditions

I'm working through the paper where the Finite Difference method is employed to solve the PDE $\displaystyle \frac{\partial u(x,t)}{\partial t} = a \cdot \frac{\partial^2 u(x,t)}{\partial x^2} + b ...