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Unformatted text preview: . To this end, recall
the formula for the Taylor's series of a function of two variables
F (t + y + ) = F (t y) + Ft + Fy ](t y) + 1 2 Ftt + 2 Fty + 2Fyy ](t y) + : : :
3 The expansion of (3.1.5) requires substitution of the exact solution y(t) into the formula
and the use of (3.1.6) to construct an expansion about (tn 1 y(tn 1)). The only term
that requires any e ort is k2, which, upon insertion of the exact ODE solution, has the
k2 = f (tn 1 + ch y(tn 1) + haf (tn 1 y(tn 1))):
; ; ; ; ; ; To construct an expansion, we use (3.1.6) with F (t y) = f (t y), t = tn 1, y = y(tn 1),
= ch, and = haf (tn 1 y(tn 1)). This yields
; ; ; ; k2 = f + chft + haffy + 1 (ch)2 ftt + 2ach2 ffty + (ha)2f 2fyy ] + O(h3):
All arguments of f and its derivatives are (tn 1 y(tn 1)). We have suppressed these to
simplify writing the expression.
Substituting the above expansion into (3.1.5a) while using (3.1.5b) with the exact
ODE solution replacing yn 1 yields
; ; ; y(tn) = y(tn 1) + h b1 f + b2 (f + chft + haffy + O(h2))]:
; (3.1.7) Similarly, substituting (3.1.3) into (3.1.2), the Taylor's series expansion of the...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.
- Spring '14