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→Ordinary differential equations: c_i in [0,1] |
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Suppose that the [[ordinary differential equation]]
:<math> y'(t) = f(t,y(t)), \quad y(t_0)=y_0, </math>
is to be solved over the interval [''t''<sub>0</sub>, ''t''<sub>0</sub>+''h'']. Denote the collocation points by 0 ≤ ''c''<sub>1</sub>
The corresponding (polynomial) collocation method approximates the solution ''y'' by the polynomial ''p'' of degree ''n'' which satisfies the initial condition ''p''(''t''<sub>0</sub>) = ''y''<sub>0</sub>, and the differential equation ''p''<nowiki>'</nowiki>(''t'') = ''f''(''t'',''p''(''t'')) at all points ''t'' = ''t''<sub>0</sub> + ''c''<sub>''k''</sub>''h'' where ''k'' = 1, …, ''n''. This gives ''n'' + 1 conditions, which matches the ''n'' + 1 parameters needed to specify a polynomial of degree ''n''.
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\end{align}
</math>
The collocation method is now given (implicitly) by
:<math> y_1 = p(t_0 + h) = y_0 + \frac12h \Big (f(t_0+h, y_1) + f(t_0,y_0) \Big), \, </math>
where <math> y_1 = p(t_0+h) </math> is the approximate solution at <math> t = t_0+h </math>.
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