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Apparently some people who can read and write can't pronounce "Euler". |
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In approximating the solution to a first-order [[ordinary differential equation]], suppose one knows the solution points <math>y_0</math> and <math>y_1</math> at times <math>t_0</math> and <math>t_1</math>. By fitting a cubic polynomial to the points and their derivatives (gotten through the differential equation), one can predict a point <math>\tilde{y}_2</math> by [[Extrapolation|extrapolating]] to a future time <math>t_2</math>. Using the new value <math>\tilde{y}_2</math> and its derivative there <math>\tilde{y}^'_2</math> along with the previous points and their derivatives, one can then better [[Interpolation|interpolate]] the derivative between <math>t_1</math> and <math>t_2</math> to get a better approximation <math>y_2</math>. The interpolation and subsequent integration of the differential equation constitute the corrector step.
== Euler
Example of
In this example ''h'' = <math>\Delta{t} </math>, <math> t_{i+1} = t_{i} + \Delta{t} = t_{i} + h </math>
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: <math> y' = f(t,y), \quad y(t_0) = y_0. </math>
First calculate an initial guess value <math>\tilde{y}
: <math>\tilde{y}_{g} = y_i + h f(t_i,y_i)</math>
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