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Euler method

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(Changed phi_dot to dPhi/dt)
 
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=== Explicit or Forward Euler ===
=== Explicit or Forward Euler ===
-
:<math> \dot \phi  = f\left( {t,\phi } \right)</math>
+
:<math> \frac{d\phi}{dt} = f\left( {t,\phi } \right)</math>
 +
 
 +
 
:<math>\phi ^{n + 1}  = \phi ^n  + f\left( {t_n ,\phi ^n } \right) </math>
:<math>\phi ^{n + 1}  = \phi ^n  + f\left( {t_n ,\phi ^n } \right) </math>
=== Implicit or Backward Euler ===
=== Implicit or Backward Euler ===
-
:<math> \dot \phi  = f\left( {t,\phi } \right)</math>
+
:<math> \frac{d\phi}{dt} = f\left( {t,\phi } \right)</math>
 +
 
 +
 
:<math>\phi ^{n + 1}  = \phi ^n  + f\left( {t_{n + 1} ,\phi ^{n + 1} } \right) </math>
:<math>\phi ^{n + 1}  = \phi ^n  + f\left( {t_{n + 1} ,\phi ^{n + 1} } \right) </math>

Latest revision as of 17:44, 14 December 2005

Euler Methods

Explicit or Forward Euler

 \frac{d\phi}{dt}  = f\left( {t,\phi } \right)


\phi ^{n + 1}  = \phi ^n  + f\left( {t_n ,\phi ^n } \right)

Implicit or Backward Euler

 \frac{d\phi}{dt}  = f\left( {t,\phi } \right)


\phi ^{n + 1}  = \phi ^n  + f\left( {t_{n + 1} ,\phi ^{n + 1} } \right)



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