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Unformatted text preview: of the exponential function and the shift operator of Table 2.1.1 to write this in the shorthand form
E uj = j+1 =
u xD e uj : We can, thus, infer the identity between the shift, exponential, and derivative operators
E = e xD (2.1.12) : Additional relationships can be obtained by noting that j = ( ; 1) j , which implies
that = ; 1 or = 1 + . Treating the operators in (2.1.12) as algebraic quantities,
we nd
1
(2.1.13a)
= ln = ln(1 + ) = ; 2 2 + 1 3 ;
3
where the series expansion of ln(1+ ), j j 1, has been used. A similar relation in terms
of the backward di erence operator can be constructed by noting that r = 1 ; ;1 thus,
1
= ln = ;ln(1 ; r) = r + 2 r2 + 1 r3 +
(2.1.13b)
3
These identities can be used to derive highorder nite di erence approximations of
the rst derivative. For example, retaining the rst two terms in (2.1.13a)
1
; 2 2] j
j
u E E u E xD ::: E x x < E xD E ::: xDu u 2.1. Di erence Operators 7 or ( j+1 ; j ) ; 1 ( j+2 ; 2 j+1 + j )]
2 xDuj u or u ; uj Duj u u u +2 + 4uj +1 ; 3uj : 2 x 2
x...
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 Spring '14
 JosephE.Flaherty

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