What is Finite difference: Definition and 143 Discussions
A finite difference is a mathematical expression of the form f (x + b) − f (x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems.
Certain recurrence relations can be written as difference equations by replacing iteration notation with finite differences.
Today, the term "finite difference" is often taken as synonymous with finite difference approximations of derivatives, especially in the context of numerical methods. Finite difference approximations are finite difference quotients in the terminology employed above.
Finite differences were introduced by Brook Taylor in 1715 and have also been studied as abstract self-standing mathematical objects in works by George Boole (1860), L. M. Milne-Thomson (1933), and Károly Jordan (1939). Finite differences trace their origins back to one of Jost Bürgi's algorithms (c. 1592) and work by others including Isaac Newton. The formal calculus of finite differences can be viewed as an alternative to the calculus of infinitesimals.
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 method is formulated by stepping forward in time. The problem is that I obtain instabilities, either at...
I have big problem with finite difference schemes (DS) on Matlab. I need write DS on Matlab, example:
u_x=(u_(i+1,j)-u_(i-1,j))/2, we choose step is 1.
On Matlab:
u_x=(u( :,[2:n,n])-u( :,[1,1:n-1]))/2
And I can write u_y, u_xx, u_yy, u_xy. But now, I need to write for higher order, example...
$$F(z) = \sum_{n=0}^\infty a_n x^n $$
$$\partial_zF(z) = \sum_{n=0}^\infty (n+1)a_{n+1}x^n $$
So, we can begin to piece together some differential equations in terms of generating functions in order to satisfy some discrete recursion relation (which is the desired problem to solve). However I...
Hi guys , i am solving this equation by Finite difference method.
(dt2/dx2 + dt2/dy2 )= -Q(x,y)
i have developed a program on this to calculate the maximum temperature, when i change the mesh size the maximum temperature is also changing,
Should the maximum temperature change with mesh...
hello. I have a MATLAB skeleton provided because i want to model a distribution with a circular geometry. all in all, i want the 3d graph of the code to be some type of cylinder. This is the code:
% flat step condition
for ii=1:nHi,
for jj=1:nHj,
if (X(ii)/R_P)<1 &...
Homework Statement
Plot the transient conduction of a material with k = 210 w/m K, Cp = 350 J/kg K, ρ = 6530 kg/m3
Where the material is a cylinder, with constant cross sectional area and is well insulated. The boundary conditions for the cylinder:
T(0,t) = 330K
T(l,t) = 299K...
The Finite difference scheme:
\begin{equation} y_{n+3}-y_{n+1}= \frac {h}{3}(f_{n}-2f_{n+1}+7f_{n+2})
\end{equation}
Deduce that the scheme is convergent and find its interval of absolute stability(if any)
=> the first characteristic polynomial is then
\begin{equation} ρ(r)= r^3 -r...
Hi,
Are there any open source C or Fortran libraries for solving 3D Poisson'sequation on an irrefular domain? I'm having difficulty finding them.
If not, is there any papers or recipes that would be useful so I could write my own? Speed is not a priority, I just need anything that works...
Hello all,
I hope this is the write sub-forum for this question. I have been looking at the Laplacian of a 2-D vector field. It is explained nicely by this Wikipedia article here. My question is more regarding how these operators work together.
So, in the case of the Laplacian, it tells me...
Hey! :o
I have a implicit finite difference method for the wave equation.
At step 0, we set: $W_j^0=v(x_j), j=0,...,J$
At the step 1, we set: $W_j^1=v(x_j)+Dtu(x_j)+\frac{Dt^2}{2}(\frac{v(x_{j-1})-2v(x_j)+v(x_{j+1})}{h^2}+f(x_j,0)), j=0,...,J$
Can that be that at the step 1 $j$ begins from...
Hi,
I have to use a wide 5 point stencil to solve a problem to fourth order accuracy. In particular, the one I'm using is:
u'' = -f(x + 2h) + 16f(x + h) - 30f(x) + 16f(x - h) - f(x - 2h) / 12h2
or when discretized
u'' = -Uj-2 + 16Uj-1 -30Uj + 16Uj+1 -Uj+2 / 12h2
In addition to...
Hey! :o
I am implementing in a program the finite difference method for the heat equation.
The problem is the following:
$$u_t(x,t)=(g(x,t)u_x(x,t))_x+f(x,t), \forall (x,t) \in [0,1]x[0,1]$$
$$u(0,t)=u(1,t)=0, \forall t \in [0,1]$$
$$u(x,0)=0, \forall x \in [0,1]$$
where $f(x,t)=\pi x...
Hello! :)
I am implementing the finite difference method in a program in C and I got stuck at the rate of convergence.. The formula is \frac{log(\frac{e_{1}}{e_{2}})}{log(\frac{J_{2}}{J_{1}})} , right? where e_{i}=max|y_{j}^{J_{i}}-y(x_{j}^{J_{i}})| , 0<=j<=J_{i} . How can I find the J_{1}...
Homework Statement
Given that we the following elliptic problem on a rectangular region:
\nabla^2 T=0, \ (x,y)\in \Omega
T(0,y)=300, \ T(4,y)=600, \ 0 \leq y \leq 2
\frac{\partial T}{\partial y}(x,0)=0, \frac{\partial T}{\partial y}(x,2) = 0, \ 0\leq x \leq 4
We want to solve this problem...
For possion equation $$u_{xx}+u_{yy}=f$$
I know the general five point scheme is in the form
$$a_{1}U_{i,j-1}+a_{2}U_{i-1,j}+a_{3}U_{i,j}+a_{4}U_{i+1,j}+a_{5}U_{i,j+1}=f_{i,j}$$
But , is there have the form...
I have the differential equation
\frac{dM}{dt}=4\pi \rho(r,t)r(t)^2\frac{dr}{dt}
which is the first term from
M(t)=4\pi\int_0^{r(t)}C(r,t)r(t)^2dr
This describes the change in mass (M) of a sphere from a change in radius (r) given a density (rho) that depends on radius and time (t).
My...
Hi,
I am currently trying to plot and animate a wave function using the Schrodinger equation.
I currently have the following finite difference equation:-
i(\psi(x, t+\Delta t)-\psi(x,t))/(\Delta t)=-(1/2)*(\psi(x+\Delta x, t)+\psi(x-\delta x, t)-2*\psi(x,t))/((\Delta...
Homework Statement
http://puu.sh/1QFsA
Homework Equations
The Attempt at a Solution
I'm actually not sure how to do this question. How do i find Δx^2. I kind of understand the question but I don't know how to prove it. I know that Δy becomes dy when the width becomes...
Hi.
I'm trying to determine the CFL condition for the fourth-order leapfrog scheme. I'm finding 2 as what's published, which does not match what I'm getting.
Does anyone know where I can find a von Neumann (or Fourier) stability analysis of the leapfrog (2,4) scheme (so I can compare my work)...
hi;
I have 3 hyperbolic electrodes ,one as a ring and 2 others as endcap
electrodes which have potential v and 0 respectively.(quadrupole ion trap)
I want to solve potential inside the trap by finite difference method.
I don't know how general equations for unshaped materials will change...
Use finite difference method to solve for eigenvalue E from the following second order ODE:
- y'' + (x2/4) y = E y
I discretize the equation so that it becomes
yi-1 - [2 + h2(x2i/4)] yi + yi+1 = - E h2 yi
where xi = i*h, and h is the distance between any two adjacent mesh points.
This...
Use finite difference method to solve for eigenvalue E from the following second order ODE:
- y'' + (x2/4) y = E y
I discretize the equation so that it becomes
yi-1 - [2 + h2(x2i/4)] yi + yi+1 = - E h2 yi
where xi = i*h, and h is the distance between any two adjacent mesh points.
This is my...
Hi,
I have a question regarding finite difference approximation:
Consider the finite difference approximation
u'(xj-1/2) + λu(xj−1/2) ≈ 1/h*[u(xj ) − u(xj−1)] + λ(θu(xj ) + (1 − θ)u(xj−1))
how can I Find the order of approximation as a function of θ?
I am really new in this field, so...
Hallo, I tried to use 'finite difference' method to solve a Initial Value Problem(IVP). For the two boundaries I used periodical condtion and for the differential operators I used 4th degree center approximations. But as result, I got this thing. Where comes this strange oscillation What do you...
Homework Statement
I'm doing a class on Numerical Solutions of DE and I have my first assignment. The problem is stated:
Consider the following second order boundary value problem:
\epsilon \frac{d^{2}y}{dx^{2}} + \frac{1}{2+x-x^{2}} \frac{dy}{dx}-\frac{2}{1+x}y = 4sin(3x), y(0) = 2, y(2) =...
Hello to everyone,
while solving homework course Nanotechnology and Nanocomponents, I have encountered a problem in FD method that is applied in even potential. In my homework assignment it is explicitly said that it must be done only in x>0 part of the domain, where my problem starts with...
Hello,
I need help writing a MATLAB program to solve a heat transfer problem implicitly. For some reason this is very confusing to me. The problem is stated below. Any help is greatly appriciated. Let me know if you need a little more info.
I need to write a program to solve this...
I was going through my notes on different finite difference methods and came across something I don't quite understand. I have code that will calculate an approximate solution we can call this U_nm that I define on a grid using h and dt for the change in x and time respectively. Now I have...
Hello
I want to resolve a nonlinear partial differential equation of second order with finite difference method in matlab. the equation is in the pdf file attached.
Thanks
Hello all,
I am in the process of solving a finite elements problem involving obtaining deflection of a simple mass-spring-damper 2nd order ODE system with a defined forcing function. While going through my class notes, I came across the idea of the central difference method, which is...
I have a system of non-linear coupled PDEs, taken from a paper from the 1980s which I would like to numerically solve. I would prefer not to use a numerical Package like MatLab or Mathematica, though I will if I need to.
I would like to know if anyone knows how to solve non-linear coupled...
I made a small program to simulate the time development of a 1D wavepacket obeying the Schrodinger equation, mostly in order to learn a new programming language - so in order to not have to invoke big numerical methods packages, I opted for the simplest solution: The standard three-point...
I'm attempting to perform interpolation in 3 dimensions and have a question that hopefully someone can answer.
The derivative approximation is simple in a single direction:
df/dx(i,j,k)= [f(i+1,j,k) - f(i-1,j,k)] / 2
And I know that in the second order:
d2f/dxdy(i,j,k)= [f(i+1,j+1,k)...
Hello all:
For modeling flow (or whatever) in a non-rectangular geometry, can anyone comment on whether the finite element method would be better or worse or the same as the integrated finite difference method?
I'm reading some papers by competing groups (so I can decide which code to...
Problem - Find backward finite difference approximations to first, second and third order derivatives to error of order h^3
Attempt
By Tailor’s series expansion
f(x-h) = f(x) - h f’(x) + h^2/2! f’’(x) - h^3/3! f’’’(x) + …
Therefore, f’(x) with error of order h^3 is given by
f(x-h) = f(x)...
Homework Statement
This is for a course in numerical solutions of PDEs. So far it has all been theory of PDEs which is great since I am the worst programmer in the world. Right now I have to program the following: to solve the simplest hyperbolic equation,
du/dt = du/dx,
using the...
I have a question which states that I have to solve Phi using the finite difference method in matlab.
Phi = Phi_infinity - Phi_t
where Phi_infinity = 1 - e^2
and Phi_t = A(t).B(e)
A has the differential for dA/dt = -c^2. A
and B 1/e.d/de(edB/de)=-c^2.B
where e is a separation...
Circuler grid need to be solved by Finite difference method! pls help me...
hi this is the picture of the problem.. i have studied the rectangular grid but not the circular grid... now pls someone help me to find out the way to solve a heat conduction problem for circle using finite difference...
Homework Statement
y'' + 3y' + 2y = 0, y(0) = 1, y'(0) = 0
Homework Equations
Finite Difference Approximations:
y'' = (y(ii+1) - 2y(ii) + y(ii-1))/h^2
y' = (y(ii+1) - y(ii-1))/(2h)
where h is the finite difference.
The Attempt at a Solution
I wrote the MATLAB code (just to try...
Hello:
I am looking to solve a set of 1D PDEs. I thought the finite difference method would be a good way to go about it. So I decided to pick a simple first order forward difference scheme to obtain preliminary results.
I just have 1 question: According to my scheme, at the last node...
I am writing a multi-region diffusion code. The two regions have different material properties, so the analytical solution shows a discontinuity at the interface between the regions.
As can be seen here:
The numerical code I am running is (Mathematica):
While[converge > .00001...
Hey,
I want to solve a parabolic PDE with boundry conditions by using FINITE DIFFERENCE METHOD in fortran. (diffusion) See the attachment for the problem
The problem is that there is a droplet on a leaf and it is diffusing in the leaf
the boundry conditions are
dc/dn= 0 at the upper...
Hey,
I want to solve a parabolic PDE with boundry conditions by using FINITE DIFFERENCE METHOD in fortran. (diffusion) See the attachment for the problem
The problem is that there is a droplet on a leaf and it is diffusing in the leaf
the boundry conditions are
dc/dn= 0 at the upper...
Hi,
I am trying the solve the Poisson equation in a domain with curved boundaries using the Finite Difference Method (second order accurate). I need to apply the Neumann condition on the curved boundary. I have used bilinear interpolation to do this but this causes the resultant scheme to be...
I'm reading a book (Numerical Techniques in Electromagnetics by Sadiku) & just finished the section on finite difference methods. As what I thought would be an easy exercise, I tried to apply what I'd learned to the telegraphers equations that describe the voltage, V(x, t), and current, I(x, t)...
Hi,
We all know that the finite difference formulae for the derivatives are given by:
\frac{dy}{dx}_{i}=\frac{y_{i}-y_{i-1}}{\delta x}
and
\frac{d^{2}y}{dx^{2}}=\frac{y_{i-1}-2y_{i}+y_{i+1}}{\delta x^{2}}
What would be the formulae for the boundary terms? when i=1? I think I can...