Using 311 y tn 1 f tn 1 ytn 1 313a 0 di

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Unformatted text preview: 1) y (tn 1) = f (tn 1 y(tn 1)): (3.1.3a) 0 ; ; ; Di erentiating (3.1.1) y (tn 1 ) = ft + fy y ](t ;1 y(t ;1 )) = ft + fy f ](t ;1 y(t ;1 )) : 00 0 ; n n 1 n n (3.1.3b) Continuing in the same manner y (tn 1 ) = ftt + 2fty f + ft fy + fyy f 2 + fy2f ](t ;1 y(t ;1 )) 000 ; n n (3.1.3c) etc. Speci c methods are obtained by truncating the Taylor's series at di erent values of k. For example, if k = 2 we get the method yn = yn 1 + hf (tn 1 yn 1) + h ft + fy f ](t ;1 y ;1) : 2 2 ; ; ; n n (3.1.4a) From the Taylor's series expansion (3.1.2), the local error of this method is dn = h y ( n): 6 3 000 (3.1.4b) Thus, we succeeded in raising the order of the method. Unfortunately, methods of this type are of little practical value because the partial derivatives are di cult to evaluate for realistic problems. Any software would also have to be problem dependent. By way of suggesting an alternative, consider the special case of (3.1.1) when f is only a function of t, i.e., y = f (t) y(0) = y0: 0 This problem, which...
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