Lecture 3

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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) + : : : 2 (3.1.6) 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 form 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): 2 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.

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