This site is being phased out.

Finite differences

From Mathematics Is A Science
(Redirected from Difference equation)
Jump to navigationJump to search

Given a function $f$ of one variable. We are to "discretize" it and its derivatives.

First order

A forward difference is an expression of the form

$$\Delta_h[f](x) = f(x + h) - f(x). $$

The increment $h$ of the variable may be variable or constant.

A backward difference uses the function values at $x$ and $x-h$, instead of the values at $x+h$ and $x$:

$$ \nabla_h[f](x) = f(x) - f(x-h).$$

A central difference is given by

$$ \delta_h[f](x) = f(x+\tfrac12h)-f(x-\tfrac12h). $$


The limits of these as $h \rightarrow 0$ give you the derivative of $f$, if it exists.

Compare to 1st order discrete differential forms.

Second order

2nd order central difference: $$ \delta_h^2[f](x) = f(x+h) - 2 f(x) + f(x-h) . $$

Similarly we can apply other differencing formulas in a recursive manner.

2nd order forward difference: $$\Delta_h^2[f](x) = f(x+2h) - 2 f(x+h) + f(x) .$$

Compare to 2nd order discrete differential forms.


Several variables

These are analogous to partial derivatives.

For $f_{x}(x,y)$: $$f(x+h ,y) - f(x-h,y). $$

For $f_{y}(x,y) $: $$f(x,y+k ) - f(x,y-k). $$

For $f_{xx}(x,y)$: $$f(x+h ,y) - 2 f(x,y) + f(x-h,y). $$

For $f_{yy}(x,y)$: $$f(x,y+k) - 2 f(x,y) + f(x,y-k). $$

For $ f_{xy}(x,y)$: $$f(x+h,y+k) - f(x+h,y-k) - f(x-h,y+k) + f(x-h,y-k). $$

These are used as replacements of the partial derivatives in order to discretize PDEs.