In this problem we discuss the global truncation error associated with the Euler method for the initial
Question:
In this problem we discuss the global truncation error associated with the Euler method for the initial value problem y' = f(t,y), y(t o ) = y o . Assuming that the functions f and f y are continuous in a closed, bounded region R of the t y -plane that includes the point (t o ,y o ), it can be shown that there exists a constant L such that If |f(t,y) - f(t, y - |- where (t, y) and (t, y - ) are any two points in R with the same t coordinate. Further, we assume that f 1 , is continuous, so the solution (t. has a continuous second derivative.
(a) Using Eq.20 show that.
Where ? = 1 = hl and ?= max |? n (t)|/2 on t 0 ? t ? t n .
(b) sume that if E 0 = 0, and if |E n | satisfies Eq. (i) then |E n |? ?h 2 (a n -1)/( ? -1) for a ? 1. Use this result to show that.
Equation (ii) gives a bound for |E n | in term of h, L, n and ?. Notice that for a fixed h, this error bound increases with increasing n; that is, the error bound increases with distance from the starting point t 0 .
(c) Show that (1 +hL) n ? e nhL ; hence
If we select an ending point T greater than t 0 and then choose the step size h so that n steps are required to traverse the interval [t 0 , T], then nh = T ? t 0 , and
Which is Eq.25 Note that K depends on the length T - t 0 of the interval and on the constants L and ? that are determined from the function f.
Equation (20) is below:
And Euler?s formula gives:
If we defin E n =? (t n ) ?y n , then using ? ?= f(t, ? )
Or
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