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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online