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# The computation of y2 and yn are not recognizable in

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Unformatted text preview: unge-Kutta and collocation methods. With u(t) a polynomial of degree s in t for t tn 1, a collocation method for the IVP ; y = f (t y) y(tn 1) = yn 0 ; ; (3.3.12a) 1 consists of solving u(tn 1) = yn ; (3.3.12b) 1 ; u (tn 1 + cih) = f (tn 1 + cih u(tn 1 + cih)) i = 1 2 ::: s 0 ; ; ; (3.3.12c) where ci, i = 1 2 : : : s, are non-negative parameters. Thus, the collocation method consists of satisfying the ODE exactly at s points. The solution u(tn 1 + h) may be used as the initial condition yn for the next time step. Usually, the collocation points tn 1 + ci h are such that ci 2 0 1], i = 1 2 : : : s, but this need not be the case 6, 10, 20]. Generally, the ci, i = 1 2 : : : s, are distinct and we shall assume that this is the case here. (The coe cients need not be distinct when the approximation u(t) interpolates some solution derivatives, e.g., as with Hermite interpolation.) Approximating u (t), t tn 1, by a Lagrange interpolating polynomial of degree s ; 1, we have s X t (3.3.13a) u (t) = kj Lj ( t ;hn 1 ) j =1 ; ; 0 ; ; 0 w...
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