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
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Finite difference method solves the wave equation for one set of bounday conditions, but does not when I change them
I wrote a finite difference algorithm in Matlab to solve the wave equation which is derived here.
When I ran my code, the plotted graphs of the numerical and analytical solution deviated, which is the problem I am trying to solve. The finite…
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0 answers
Convert difference equation to differential equation
Suppose I have the following third-order difference equation:
Y[t]+aY[t-1]+bY[t-2]+cY[t-3]=G d^t
I need to convert it into a differential equation and I followed this method:
y' = y(t+1)-y(t);
y'' = y'(t+1)-y'(t) =…
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0 answers
Can the diffusive flux be represented for grids with non-constant thickness?
The figure below shows a water column that is separated into $3$ parts: Grid1, Grid2 and Grid3. Their thickness are $\operatorname{H1}$, $\operatorname{H2}$, and $\operatorname{H3}$. The concentration $(\operatorname{C1}, \operatorname{C2},…
T X
- 147
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0 answers
Approximate $f(x,y,z)-f(0,0,0)$ given $f(x,y,0)$, $f(x,0,z)$ and $f(0,y,z)$
I am trying to approximate $f(x,y,z)-f(0,0,0)$ given $f(x,y,0)$, $f(x,0,z)$ and $f(0,y,z)$. I am ready to make any simple assumption for missing information as long as it fits the given information. I am stuck trying to achieve this. My current line…
Discretizer
- 100
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1 answer
finite difference formula and its stencil
I would like to prove the following finite-difference formula for functions $u$ and the flux $f$:
$$\frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t}+\frac{-f\left(u_{i+2}^{n+1}\right)+8 f\left(u_{i+1}^{n+1}\right)-8…
georg
- 186
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1 answer
Second Order Homogeneous Linear Difference Equation
So i have this equation i want to solve
$$yx_{j-1} + (1+y^2)x_j + yx_{j+1} = 0 $$
with conditions
$$(1+y^2)x_n + yx_{n-1} = 0 $$
and
$$(1+y^2)x_1 + yx_2 = y $$
I tried solving this by guessing that $x_j = Aw^j$, but have not found solution. Can…
Keweik
- 101
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votes
1 answer
General reciprocity formula for finite differences
Define $$\Delta^ra_i= \sum_{j=0}^{r}\binom{r}{j}(-1)^{r-j}a_{i+j}.$$ For consistency also let $$\Delta^0a_i=a_i.$$
My problem is the following: for two sequences $a_i$ and $b_i$, $i=0,\ldots,\nu$, show that…
Chamberlain Mbah
- 167
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1 answer
Intermediate finite difference method
why sometimes do we use intermediate finite difference method
\begin{equation}
\frac{\partial y}{\partial x}=\frac{y_{i+1/2}-y_{i-1/2}}{\Delta x}
\end{equation}
instead of
\begin{equation}
\frac{\partial y}{\partial x}=\frac{y_{i+1}-y_{i}}{\Delta…
armin
- 1
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0 answers
Finite difference method for irregular domain
when we are working with an irregular domain in solution of PDEs using finite difference method, can we divide the irregular domain into several rectangular sub-domains and then solve the problem?
the boundary condition is known and when the firs…
armin
- 1
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0 answers
Finite Differences - determine degree of the equation when the abscissas aren't consecutive
The question is:
Given the set of points,
(2, -8)
(4, -240)
(6, -1272)
(7,-2373)
(8, -4046)
(10, -9960)
(12, -20688)
determine the degree of the equation.
However I am given values at 2, 4, 6 etc so the method of finite differences cant' be used so…
Voltar
- 1
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1 answer
Finite Differences Method
I am confused when following the instructions where i have gone wrong and why its not coming out right when considering a first order ODE
$ y′(x) = y(x), \quad \forall x ∈ [0, 1] $
$ y(0) = 1.$
I am trying to prove $ y_i = (1+h)^i $ for $ i=0, ... ,…
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votes
1 answer
The relation $(1+\lambda \Delta )^{-1}y_n=(a-1)a^{-n}\sum_{k=1}^{n-1}a^ky_k+Aa^{-n}$.
I want to prove the above relation for $\lambda=a/(a-1)$ where $a\ne 1$, and $A$ is an arbitrary constant, without induction on $n$.
And $\Delta z_n = z_{n+1}-z_n$ for some sequence $z_n$.
So I tried to geometrically expand this term and I…
MathematicalPhysicist
- 4,191
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0 answers
Given derivative order and accuracy order, is there a general form of the central finite difference?
I'm trying to find a general formula for the central finite difference approximation given an nth order derivative with pth order accuracy. I've looked up on Wikipedia, and their formula doesn't seem to work unless I'm reading it wrong (it also…
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1 answer
A finite difference
I read that the $n$-th finite difference of the sequence
$1^n, 2^n, 3^n,\dots$
is $n!$, but I'm not able to prove this. Could someone give an idea of why this is true?
Tian An
- 817