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: Remember that y(tn + l ) = y(t" + h), and deduce that

(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: Remember that y(tn + l ) = y(t" + h), and deduce that

for some t; in the interval (tn, tn+ 1 ) .

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

Where we assume that the nth approximation is exact y(tn) = yn. Hence, if |y'' (t)| ‰¤ M on [tn, tn+1] then en+1 ‰¤ M h2/2

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

Differential Equations and Linear Algebra

2nd edition

Authors: Jerry Farlow, James E. Hall, Jean Marie McDill, Beverly H. West

ISBN: 978-0131860612

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