19a of the second derivative can be written as 2j xxj

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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 short-hand 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 high-order 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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