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March 29

taylor series expansions...

hi, i'm doing a course in which my lecturer keeps doing these taylor expansions which i cant work out for love nor money (sadly, i have neither). could someone (anyone!) talk me through what shes done? we start wth the Local truncation Error of the Trapzium Rule:

$T_{j+1}=y(t_{j+1})-y(t_{j})-{\frac {1}{2}}hf(t_{j},y(t_{j}))-{\frac {1}{2}}hf(t_{j+1},y(t_{j+1}))$

She then says cheerfully "expanding in taylor series gives:"

$T_{j+1}+y(t_{j})+hy'(t_{j})+h^{2}y''(t_{j})+{\frac {1}{6}}h^{3}y'''(t_{j})-y(t_{j})-{\frac {1}{2}}hy'(t_{j})-{\frac {1}{2}}h\left\{y'(t_{j})+hy''(t_{j})+{\frac {1}{2}}h^{2}y'''(t_{j})\right\}+O(h^{4})$

she then simplifies this expression. what i dont understand is (basically) the second line (from -(1/2)hy'(t_j)). so if someone could explain where those numbers come from i would be so grateful! thank you so much! (the things that look like i's are in fact j's. not sure what wikipedia has done..) 130.88.123.216 16:16, 29 March 2007 (UTC)[reply]

Taylor series might help. --Ķĩřβȳ ♥♥♥Ťįɱé Ø 16:59, 29 March 2007 (UTC)[reply]

I used LaTeX to reformat your equation for readability and because some of the primes were becoming italics. Unfortunately, the second thing you gave is not an equation and the first one is, so I'm not sure what relationship between them we're supposed to be seeing. Also, what is f here? Are we solving the ODE

y'(t)=f(t,y(t))

? (Or are we applying the trapezoid rule for quadrature?) --Tardis 17:27, 29 March 2007 (UTC)[reply]

For easier reading, here are those expressions again:

{\begin{aligned}T_{j+1}&{}=y(t_{j+1})-y(t_{j})\\&\quad {}-{\tfrac {1}{2}}hf{\big (}t_{j},y(t_{j}){\big )}-{\tfrac {1}{2}}hf{\big (}t_{j+1},y(t_{j+1}){\big )}\\&{}=\left\{y(t_{j})+hy'(t_{j})+h^{2}y''(t_{j})+{\tfrac {1}{6}}h^{3}y'''(t_{j})\right\}-y(t_{j})\\&\quad {}-{\tfrac {1}{2}}hy'(t_{j})-{\tfrac {1}{2}}h\left\{y'(t_{j})+hy''(t_{j})+{\tfrac {1}{2}}h^{2}y'''(t_{j})\right\}+O(h^{4})\end{aligned}}

I have made some minor alterations and corrections. --KSmrq^T 17:39, 29 March 2007 (UTC)[reply]

It seems

y

is a solution of

y'(t)=f(t,y(t))

. Thus

f{\big (}t_{j},y(t_{j}){\big )}=y'(t_{j})

and

f{\big (}t_{j+1},y(t_{j+1}){\big )}=y'(t_{j+1})

. Substitute those into the expression for

T_{j+1}

to get

T_{j+1}=y(t_{j+1})-y(t_{j})-{\tfrac {1}{2}}hy'(t_{j})-{\tfrac {1}{2}}hy'(t_{j+1})

To simplify this formula, we use Taylor expansion of $y(t_{j+1})$ and $y'(t_{j+1})$ around $t_{j}$ . Those expansions are $y(t_{j+1})=y(t_{j})+hy'(t_{j})+{\tfrac {1}{2}}h^{2}y''(t_{j})+{\tfrac {1}{6}}h^{3}y'''(t_{j})+O(h^{4})$ (there fraction 1/2 is missing in your writing) and $y'(t_{j+1})=y'(t_{j})+hy''(t_{j})+{\tfrac {1}{2}}h^{2}y'''(t_{j})+O(h^{3})$ Now you can substitute in these Taylor expansions and all the terms upto $h^{3}$ will cancel. Stefán 20:51, 29 March 2007 (UTC)[reply]

stefan, you're a god. 87.194.21.177 17:43, 30 March 2007 (UTC)[reply]

Mapping technique with arrows

This is general idiocy on my part, but I cannot for the life of me remember the name of a certain type of map/chart, and searching for it has drawn a blank. Basically, it is a map using arrows to show proportional amounts of something going somewhere. For instance, say there was a map showing imports of corn to country A, and it imported 10 tonnes of corn from country B. The arrow may be a centimetre thick, but, by comparison, country C exports 20 tonnes of corn to country A, and so has a two centimetre thick arrow. All I want is the name of this, and preferably a Wikilink, to describe the map I have just drawn. Thanks. J Milburn 19:53, 29 March 2007 (UTC)[reply]

Ok, doesn't matter now. I improvised. Thanks anyway. J Milburn 22:51, 29 March 2007 (UTC)[reply]

I've found the term "flow map" for the type of chart that combines the thickness-of-arrows visualization with conventional topographic map positioning.[1] Flow maps were popularized by Charles Joseph Minard, famous for his flow map of Napoleon's ill-fated Russian campaign. The term "flow map" is also in use for a function describing the evolution of a dynamical system; see e.g. Invariant measure, Poincaré recurrence theorem, and Random dynamical system. --Lambiam ^Talk 07:06, 30 March 2007 (UTC)[reply]