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# 21 the necessary and su cient conditions for a runge

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Unformatted text preview: gin, we expand f (tn 1 + cih Yi) using (3.1.6) to obtain f (tn 1 + cih Yi) = f + ft cih + fy (Yi ; y(tn 1)) + 1 ftt (cih)2 + 2fty (cih)(Yi ; y(tn 1))+ 2 13 ; ; ; ; fyy (Yi ; y(tn 1))2] + O(h3): ; All arguments of f and its derivatives are at (tn 1 y(tn 1)). They have been suppressed for simplicity. Substituting the exact ODE solution and the above expression into (3.2.3a) yields ; y(tn) = y(tn 1) + h ; s X i=1 ; bi f + ft cih + fy (Yi ; y(tn 1)) + O(h2)]: ; The expansion of Yi ; y(tn 1 ) will, fortunately, only require the leading term thus, using (3.2.3b) s X Yi ; y(tn 1) = h aij f + O(h2): ; ; j =1 Hence, we have y(tn) = y(tn 1) + h ; s X i=1 bi f + ft cih + hffy s X j =1 aij + O(h2)]: Equating terms of this series with the Taylor's series (3.1.8) of the exact solution yields (3.2.5c) with l = 1, (3.2.5c) with l = 2, and (3.2.7) with k = 2. We have demonstrated the equivalence of these conditions when (3.2.5b) is satis ed. Remark 1. The results of Theorem 3.2.1 and conditions (3.2.5) and (3.2.7) apply to both explicit and implicit methods. Let us conclude thi...
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