ABSTRACT
For partial derivatives ofj(x, t) with respect to x, t = to = constant, I'1t = 0, and Eq. (5.61) becomes
I . Ij(x, to) = 10 + fxlo !1x + -21xxlo M + ... + -/(II)xlo !1x" + . . . (5.63) n.
Equation (5.63) is identical in form to Eq. (5.57), where j~ corresponds to fxlo, etc. The partial derivativefxlo of the functionj(x, t) can be obtained from Eq. (5.63) in exactly the same manner as the total derivative,f~, of the functionj(x) is obtained from Eq. (5.57). Since Eqs. (5.57) and (5.63) are identical in form, the difference formulas forj~ andfxlo are identical if the same discrete grid points are used to develop the difference formulas. Consequently, difference formulas for partial derivatives of a function of several variables can be derived from the Taylor series for a function of a single variable. To emphasize this concept, the following common notation for derivatives will be used in the development of difference formulas for total derivatives and partial derivatives:
d dx (I(x)) = fr (5.64) a ax (I(x, t)) = fx (5.65)
(5.67)
(5.68)
(5.69)
(5.66)
In a similar manner, partial derivatives of j(x, t) with respect to t with x = Xo = constant can be obtained from the expression
j(xo, t) = fo +.1;10 I'1t + -21hllo I'1t2 + .. , + ~ ./(11)11 0 Mil + ... n.
