Industrial Mathematics - Alexiades

    Homework on Forward and Backward Euler schemes

To get a sense about how numerical schemes for ODEs work,
apply the Forward Euler and the Backward Euler schemes,
BY HAND, to approximate y(2), using Δt=1/2, for:

1. dy/dt = y − 2t ,   y(0) = 1

2. dy/dt = ty ,   y(0) = 1

Note that Backward Euler can be done by hand for these,
only because f(t,y) is linear in y, so can easily solve the equation for Yn+1.
But this would NOT be the case for, e.g.,   y′ = t y3ey   (try one step!).
In general, an equation must be solved numerically at each time step (using a root finder!).

This is just for gaining insight... NOTHING TO TURN IN.

Forward Euler scheme:
  Y0 = y0,   Yn+1 = Yn + Δt F(n , Yn),   n=0,1,2,...
Backward Euler scheme:
  Y0 = y0,   Yn+1 = Yn + Δt F(n+1 , Yn+1),   n=0,1,2,...