17 similarly substituting 313 into 312 the taylors

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Unformatted text preview: exact solution is 2 y(tn) = y(tn 1) + hf + h (ft + ffy ) + O(h3): (3.1.8) 2 All that remains is a comparison of terms of the two expansions (3.1.7) and (3.1.8). The constant terms agree. The O(h) terms will agree provided that ; b1 + b2 = 1: (3.1.9a) The O(h2) terms of the two expansions will match if cb2 = ab2 = 1=2: (3.1.9b) A simple analysis would reveal that higher order terms in (3.1.7) and (3.1.8) cannot be matched. Thus, we have three equations (3.1.9) to determine the four parameters a, b1 , b2 , and c. Hence, there is a one parameter family of methods and we'll examine two speci c choices. 4 1. Select b2 = 1, then a = c = 1=2 and b1 = 0. Using (3.1.5), this Runge-Kutta formula is yn = yn 1 + hk2 (3.1.10a) ; with k1 = f (tn 1 yn 1) ; k2 = f (tn 1 + h=2 yn 1 + hk1=2): ; ; ; (3.1.10b) Eliminating k1 and k2, we can write (3.1.10) as yn = yn 1 + hf (tn 1 + h=2 yn 1 + hf (tn 1 yn 1)=2) ; ; ; ; ; (3.1.11a) or yn ^ 1=2 ; = yn 1 + h f (tn 2 ; 1 ; yn 1) yn = yn 1 + hf (tn 1 + h=2 yn ^ ; (3.1.11b) ; ; 1=2 ; ): (3.1...
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