Question

5. Following an approach similar to what was performed in lecture for the first forward finite-divided-difference equation with local truncation error Oſhº), please derive the following expression for the first backward finite-divided-difference equation with location truncation error O(h?). f'(x) – 3 f(xi) – 4 f(xi-1) + f(xi-2) 2 h

Answer 1

We will use Taylor's series expansion of a function f at Xi with step size h.

Notation :- X(i-1)= Xi-h

X(i-2)=Xi-2h

Backward difference estimate of the first derivative f (x) with errot Oche) is: $1%) = f(x-) - 4 +(0; -) + 34();) ah Hayler series expansion in : flove + h) = f(x) +ht' (3+"(2)+5? {"(x3) +-- 3. flos -) = fx) - ht'g) + (y * {"4]+[% " flei 4h) = f(%)+(-26) (-am f"(x) + -- 3 Consider flog-2h) - 4 f(26-2) + 3 floco) Scanned with CamScanner

[112) – an "(42) + at ful x6.) - Sub " (40)*-* -A[flxs) -h f'(25) + f (45) - f f "(%)+ --1 + 34(:) = f(x) (1 +4+3) + h f'(>:) (-2+4) + ?" (as) the (4 4) + a 4" (2:) (-8 + 4) +--- - Ot dh f'(xi) +0 -4 42 F"(x0) +- - = Bh [f"(**) - 2 mol f1 (2)+ ] = 2h ['1%) + Olhe) if(04-2h) - 4 f («; -h) +38(06) = 2h [f'(>23)0(kes) - 1143-2h) - 4 fl4; )+31(25) = f'(44) + 0(he) ñ f(2;-2h) -4 f (*;-h)+ 33120) 2h Scanned with CamScanner