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5 0 ) provides an excellent balance of simplicity, high-order accuracy and economy, we will concen- trate on it. However, the development t o follow can be extended t o any of the various Runge-Kutta formulas available. Consider again ( 2 . 51) In ( 2 . 5 1 ) , set y' = v so that ( 2 . 52) i y' = v v' = f(x,y,v) . But ( 2 . 53). 60) m3 = h f ( x i t l I yi t vih t tz,h yi mO 2‘”it2’ rn = hG(x. t-, h y t-kg = hF(xi t;,h yi t-kg v +-)#mO 2’ i 2 4, vi t m y) l t-vih t moh 2 mlh 7, vi t m2) . 56). 56).
14), t h e v a l u e s of y(a) and y ' ( a ) a r e t a k e n from ( 2 . 2 ) while t h e v a l u e s of y " ( a ) , y " ' ( a ) , ... 1). structing y 1 After con- and y ' one then g e n e r a t e s i n a n analogous fashion 1' a n d y; y2 and y ' from y and y ' 2 1 1' y 3 from y so forth, until one terminates t h e procedure. 15) - (2016) n h2 h3 yktl= y t h y ' t - y " t - y " ' + . . k k 2 k 3 ! + k 2 k 3! k k = 0,1,,.. + Lnyn! ( nkt l ) : k = O,l,... 1) assumed to be exact. 18) Y(0) = 0 , Y'(0) = 1 .
3! tG (nt2)(u), Let G[a,h] be a set of grid points. < x t h. 10), ORDINARY DIFFERENTIAL EQUATIONS 30 5 n t hntl y @ ) ( a )t - (ntl)! ~ ( ~ ~ " ( ae <) ,4 < a t h . 12) w e call a n nth order approximation t o y ( a t h ) , a n d , from (2. 14) 3 hn ( n t l ) h2 h iv (a) t o *t--Y y ' = y ' ( a ) t h y " ( a ) t - y " ' ( a ) t -y (a) 1 2 3! n! 14), t h e v a l u e s of y(a) and y ' ( a ) a r e t a k e n from ( 2 . 2 ) while t h e v a l u e s of y " ( a ) , y " ' ( a ) , ... 1). structing y 1 After con- and y ' one then g e n e r a t e s i n a n analogous fashion 1' a n d y; y2 and y ' from y and y ' 2 1 1' y 3 from y so forth, until one terminates t h e procedure.
Discrete Numerical Methods in Physics and Engineering by Greenspan