Approximating Derivatives with Finite Differences

In summary, the conversation discusses approximating the first order derivative y'(xi) at a point xi using a linear system and Taylor expansion. The approach involves using a discrete set of points and solving for the coefficients a in the equation. The dimensions are not specified, but it is assumed to be one dimension.
  • #1
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Homework Statement


Show that the first order derivative y'(xi) in the point xi may be approximated by

y'(xi)= (1/12*h) * (-3yi-1 -10yi + 18yi+1 -6yi+2 + yi+3) - (1/20h) h^4*y^(5) + O(h^5)


The Attempt at a Solution



I think the idea is to setup a linear system and some how use taylor expansion.

y'(xi) = a(-1)*y(xi-1) +
a(0) *y(xi) +
a( 1) *y(xi+1) +
a( 2) *y(xi+2) +
a( 3) *y(xi+3) +

Anyone has any idea on how I can show the above?
 
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  • #2
xi is a point in how many dimensions? 2, 3?
 
  • #3
I would assume one dimension.

xi are discrete points.

If anyone has any ideas on how to solve this please shout ;-)
 
Last edited:
  • #4
If you can give a hint for n-dimensions HallsofIvy then I am sure I can solve it for 1d ;-)
 

1. What are finite differences for PDEs?

Finite differences for PDEs refer to a numerical method used to solve partial differential equations (PDEs). It involves dividing the continuous PDE into a discrete grid, and then approximating the derivatives using finite differences at each point on the grid.

2. How does the finite difference method work?

The finite difference method works by approximating the derivatives in a PDE using a small interval or difference, instead of the actual derivative function. This allows for the PDE to be solved on a discrete grid, making it easier to compute and approximate solutions.

3. What are the advantages of using finite differences for PDEs?

One of the main advantages of using finite differences for PDEs is its simplicity and ease of implementation. It also allows for the solution to be computed at specific points on the grid, making it useful for analyzing the behavior of a PDE at certain points. Additionally, finite differences can handle complex boundary conditions and non-rectangular domains.

4. Are there any limitations to using finite differences for PDEs?

Yes, there are limitations to using finite differences for PDEs. One limitation is that the accuracy of the solution depends on the size of the grid used, and as the grid size increases, so does the computational cost. Additionally, finite differences are only applicable to certain types of PDEs and may not work well for all types of boundary conditions.

5. How do you choose the appropriate grid size for finite differences in PDEs?

The appropriate grid size for finite differences in PDEs depends on the specific problem being solved. Generally, a smaller grid size will result in a more accurate solution, but at the cost of increased computational time. It is important to balance accuracy and efficiency when choosing a grid size for finite differences in PDEs.

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