Euler’s Errors We will investigate the local discretization error in applying the Euler ap...

Euler’s Errors We will investigate the local discretization error in applying the Euler approximation given by equations (5) and (6).

(a) If y(t) is the exact solution of y' = f(t, y), use the chain rule to calculate y″(t) and explain why it is continuous.

(b) Recall the following from calculus:

Taylor’s Theorem

Any continuously infinitely differentiable function can be expanded in polynomial form about a value as follows:

where f(n)(t0) means the n th derivative of f(t) with respect to t, evaluated at t0.

If the summation is stopped at k instead of ∞, the remainder is

where t* is some value of t between t and t0.

NOTE: Some care must be taken, because the power series expansion of an infinitely differentiable function may converge only on a finite interval around t0.

Remember that y(tn+1) = y(tn + h), and deduce that

for some  in the interval (tn, tn+1).

(c) Subtract equation (6) from equation (8) to conclude that the local discretization error en+1 is given by

where we assume that the n th approximation is exact: y(tn) = yn Hence, if |y″(t)| ≤ M on [tn, tn+1], then en+1 ≤ Mh2/2.

(d) How small must h be to guarantee that this local discretization error is no greater than some prescribed ε?