Derive an O(At²) accurate forward difference derivitive for the function f(t) at time t usingthe following three measurements or data points:f(t), f(t+At), and f(t+24t)Please show that this scheme is really O(At²) accurate using the Taylor series expansions.

Answered: Derive an O(At²) accurate forward…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Derive an O(At²) accurate forward difference derivitive for the function f(t) at time t using
the following three measurements or data points:
f(t), f(t+At), and f(t+24t)
Please show that this scheme is really O(At²) accurate using the Taylor series expansions.

Expert Solution

Step 1

Let $f (t)$ be any function at time $t$ and let $∆ t > 0$ representing the positive number close to zero

Using the Taylor expansion of the function $f$ about the base point $t$ , we get

$f (t + ∆ t) = f (t) + ∆ t f^{'} (t) + \frac{{(∆ t)}^{2}}{2!} f^{''} (t) + \frac{{(∆ t)}^{3}}{3!} f^{'''} (t) + . . . (1)$

Replacing $∆ t b y 2 ∆ t$

$f (t + 2 ∆ t) = f (t) + 2 ∆ t f^{'} (t) + \frac{4 {(∆ t)}^{2}}{2!} f^{''} (t) + \frac{8 {(∆ t)}^{3}}{3!} f^{'''} (t) + . . . (2)$

Multiplying $(1)$ by 4, we get

$4 f (t + ∆ t) = 4 f (t) + 4 ∆ t f^{'} (t) + \frac{4 {(∆ t)}^{2}}{2!} f^{''} (t) + \frac{4 {(∆ t)}^{3}}{3!} f^{'''} (t) + . . . (3)$

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Derive an O(At²) accurate forward difference derivitive for the function f(t) at time t using the following three measurements or data points: f(t), f(t+At), and f(t+2^t) Please show that this scheme is really O(At²) accurate using the Taylor series expansions.