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.
Questions tagged [finite-differences]
815 questions
1
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1 answer
Error from central difference seems large
for a function $f(x)=e^{2x}-\cos(2x)$,
at grid points $x \in {-0.3,-0.2,-0.1,0}$
I perform a central difference for the derivative at $x=-0.2$
$$\frac{df}{dx}=\frac{f(-0.1)-f(-0.3)}{2*(-0.1--0.3)}=0.28795$$
The derivative of this function (per…
Frank
- 880
1
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0 answers
Finite difference basic operation to solve boundary value problem
I got a question regarding finite difference method to solve boundary value problem on second order derivative equation.
This is taken from the "Numerical Methods using MATLAB" by Mathews & Fink.
This is the problem from the book:
Solve the…
JIM BOY
- 65
1
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1 answer
Finite Difference method - An example
I am trying to solve
$$\frac{d^2T}{dr^2}+\frac{1}{r}\frac{dT}{dr}+1=0$$
for $0
noname
- 25
- 4
1
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1 answer
Finite difference: problem on edge of Dirichlet and Neumann boundary
I'm trying to solve the non-homogeneious heat equation using a finite difference scheme. The grid on which to solve this looks like this:
N N N N N N N N N
N N
N N
D D
D D
D D
N …
Thijs Steel
- 121
1
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1 answer
Solve finite differences linear equation
Given
$$(\textrm{Pe}-1)u_{i+1} + 2u_i - (\textrm{Pe}-1)u_{i-1} = \frac{h^2}{\mu}$$
where $\textrm{Pe}$ stands for Péclet number, $h$ is the step size of the mesh and $\mu$ is the diffusion (constant)
How to solve such equation?
I'm trying to solve…
BRabbit27
- 767
1
vote
1 answer
On the lax scheme
Does the following lax scheme:
$$\frac{u_{j}^{n+1}-\frac{u_{j+1}^{n}+u_{j-1}^{n}}{2}}{\Delta t}+c\frac{u_{j+1}^{n}-u_{j-1}^{n}}{2\Delta x}=0$$
of
$$\begin{cases}
u_t+cu_x=0 & \text{for} \ (x,t)\in\mathbb{R}\times ]0,T[ \\
u(x,0)=u_0(x)
\end{cases}$$…
Theory Nombre
- 399
1
vote
2 answers
Problem in solving a question related to finite difference.
Show that the $k$-th finite differences of the sequence $1^k ,2^k,3^k ,...$ are each $k!$.
I have tried but I fail when I try it proving using mathematical induction.Please help me.
Thank you in advance.
user251057
1
vote
1 answer
Best numerical scheme for this problem
I have a set of data, $x, y$ and $ z$, each with length n:
$x \rightarrow \{x_{1}...x_{n}\}$
$y \rightarrow \{y_{1}...y_{n}\}$
$z \rightarrow \{z_{1}...z_{n}\}$
$y$ and $z$ are parameterised by $x$:
$y = y(x)$
$z = z(x)$
Problem 1
I wish to find…
1
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1 answer
Why, for simple nth degree polnomials' finite difference tables, does the nth (constant) difference set, equal the nth derivative
For example with the equation $f(x)=x^4+2x^3+4x^2+2x+1$ the fourth derivative is $f''''(x)=24$ and when you construct a
difference table the fourth difference is 24
clear492
- 13
1
vote
1 answer
calculus of finite differences
If D,E,$\delta,\mu$ be the operators with usual meaning and if hD=U, where h is the interval of differencing,How to prove the following relations between operators:-
1]$\frac{U}{\delta}=\frac{2}{\delta} \sinh^{-1}\frac{\delta}{2} =…
Win_odd Dhamnekar
- 1,056
1
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0 answers
First order and second order derivative backwards approximation on non uniform grids
Assume we have a function f(x) and we have samples coming in, except they are not at uniform intervals. (e.g. you have f(x) at x = 0.1, 0.2, 0.7, 1.9, 2.8... etc).
What is the correct way to compute first and second order backwards derivatives? Is…
user2771184
- 139
0
votes
1 answer
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:
$[D_- +…
Twilight Sparkle
- 191
0
votes
0 answers
FDTD validation with Poynting Vector
I'm attempting to validate my FDTD optics simulator results.
Visually, I can see that my output is nearly identical to that produced by Meep, the only difference being a slight phase shift. Using a point source in the middle of a 1k+1 by 1k+1…
3Dave
- 101
0
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0 answers
second-order difference equation (hypergeometric type equation)
We defined the backward and forward difference operators ($\Delta$ and $\nabla$ respectively) by $\Delta f(x)=f(x+1)-f(x)$ and $\nabla f(x)=f(x)-f(x-1)$. We consider the following equation
$$
-B(x)f(x+1)+[B(x)+D(x)]f(x)-D(x)f(x−1)+n(n+\mu +\nu…
Made
- 1,249
0
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0 answers
Derivation of fourth-order accurate formula for the second derivative in FDM for contact node
I was looking for a way to derive a fourth-order error formulation of a second-order derivative at the contact surface of two different materials. Say, solving a thermal diffusion equation with varying thermal conductivity and diffusivity. Thanks in…
