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If 0 < r 1, then max lui , 1 AT[(At) + (Ax) 2 ], 0 xi 1, 0 ti < T. ) Another conclusion can be drawn from this theorem. From Section 1-8 we observe that the above boundedness condition implies stability in the sense of John. Thus, for the problem described by Eqns (2-6), the stability condition is At 0 < r (Ax y —< 2* (2-20) t This condition can be relaxed to C2 . 1 ; see, for example, Douglas 116, 18]. 46 PARABOLIC EQUATIONS In the next section we shall see that Eqn (2-20) is also the stability condition of von Neumann (see, for example, Richtmyer [19] and Section 1-8).

F The symbol 1 is used to denote the identity operator, that is lYn = y„. ) p times. 21 FINITE DIFFERENCE OPERATORS From Eqns (1-59) and (1-56) the relations hD = log E = log (1 + A) = — log (1 — V) = 2 sinh -1 8/2 = 2 log [(1 + 18 2) 1/ 2 + 18] (1-60) are obtained by formal operations. Upon expansion of log (1 + A) we obtain the forward difference relation for the first derivative at the relevant point dy 1 _1 [A _ _A2 + 1 h 2 3 a-17x 1 y (1-61) Forward difference approximations for higher order derivatives follow from Eqn (1-60) since — Dkyf i = [log (1 A)]kyi Ak +1 k(3k + 5) Ak+2 , 1 k hk A — 2 24 k(k + 2)(k + 3) Ak +3 + • • yi .