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m robot Modifying: de:Least Squares Collocation |
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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''].
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, called the '''collocation 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''.
All these collocation methods are in fact implicit [[Runge–Kutta methods]]. However, not all Runge–Kutta methods are collocation methods.
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