By J. D. Lambert

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**Additional info for Computational Methods in Ordinary Differential Equations (Introductory mathematics for scientists & engineers) **

**Example text**

I. If, instead, we choose to calculate G from (12iv), then we obtain 'i ill + alIa 2G = oIG(S)lds= o · ' 1- as' + (1 + a)Slds + fl 1_ as' + (1 + a)slds + f' (2 - s}'ds 1 J(l+a)/a for all a. i(1 o +a){II ' [ - as' + (1 + aJs] ds + 5,1 [- as' + (1 + a)s] ds (l +a)/a + and we clearly recover the result G = n(5 2G = f,I+o". [as' Jo _ (I + a)s] ds + + a). f' . a 5,' (2 1 which gives G = -0:(5 + a) - f W + a)'/a'. The differential equation in example 4 of chapter 2 is y' = 4xyt. s)' ds, '. 54 Computational methods in ordinary differential equations In order to bound yIP.

From (19) it is clear that for the method to have order p, p + 1 linear conditions must be satisfied by these parameters. Thus the highest order we can expect from a k-step method is 2k, if the method is implicit, and 2k - I if 'it is -explicit. However, these maximal orders cannot in general be' ,attained without violating the condition of zero-stability, as the following theor~m shows. 2 No 'zero-stable linear multistep method of stepnumber k can have order exceeding k + 1 when k is odd, or exceeding k + 2 when k is even.

1)'/'~O) = I + 0·1 + 0·005 = 1·105. • • 48 Computational methods in ordinary differential equations Since yt3) does not exist at x = 0, we cannot use (5ii) or (5iii). OI)[P'(x l ) - P'(O)] or . I)t + Yl ,- I]. Solving for Yl' we find Yl = 1·106,39. The exact value for -"Xl)' taken from the theoretical solution to the initial value problem is -"Xl) = 1·106,50. We point out that the differential equation in this example is linear; if it were non-linear, the labour in solving (5i) for Yl would be much increased.