Mathematical Methods for Physics and Engineering - Matematica.NET

27.6 DIFFERENTIAL EQUATIONSWe assume that this can be simulated by a formy i+1 = y i + α 1 hf i + α 2 hf(x i + β 1 h, y i + β 2 hf i ), (27.76)which in effect uses a weighted mean of the value of dy/dx at x i and its value atsome point yet to be determined. The object is to choose values of α 1 , α 2 , β 1 andβ 2 such that (27.76) coincides with (27.75) up to the coefficient of h 2 .Expanding the function f in the last term of (27.76) in a Taylor series of itsown, we obtainf(x i + β 1 h, y i + β 2 hf i )=f(x i ,y i )+β 1 h ∂f i∂x + β 2hf i∂f i∂y +O(h2 ).Putting this result into (27.76) and rearranging in powers of h, we obtain()y i+1 = y i +(α 1 + α 2 )hf i + α 2 h 2 ∂f iβ 1∂x + β ∂f i2f i . (27.77)∂yComparing this with (27.75) shows that there is, in fact, some freedom remainingin the choice of the α’s and β’s. In terms of an arbitrary α 1 (≠1),α 2 =1− α 1 , β 1 = β 2 =12(1 − α 1 ) .One possible choice is α 1 =0.5, giving α 2 =0.5, β 1 = β 2 = 1. In this case theprocedure (equation (27.76)) can be summarised bywherey i+1 = y i + 1 2 (a 1 + a 2 ), (27.78)a 1 = hf(x i ,y i ),a 2 = hf(x i + h, y i + a 1 ).Similar schemes giving higher-order accuracy in h can be devised. Two suchschemes, given without derivation, are as follows.(i) To order h 3 ,wherey i+1 = y i + 1 6 (b 1 +4b 2 + b 3 ), (27.79)b 1 = hf(x i ,y i ),b 2 = hf(x i + 1 2 h, y i + 1 2 b 1),b 3 = hf(x i + h, y i +2b 2 − b 1 ).1027