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# L20 - Approximation Consider again the FOOIVP y(t = f(t y(t...

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Approximation Consider again the FOOIVP y 0 ( t ) = f ( t, y ( t )) , y ( t 0 ) = y 0 . This covers systems and higher order ODE. If one of our four arrows does not shoot it down, we’ll try numerical approximation of the solution (Note the power series so- lutions are practically approximations). We’ll talk about three approximation schemes 1) the Euler Method ; 2) the Improved Euler Method , also called the Heun Method ; 3) the Runge-Kutta Method .

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All three methods consist in a prescrip- tion to move from the starting point ( t 0 , y 0 ) to the next point ( t 1 , y 1 ) , from there to the next point ( t 2 , y 2 ) , then to ( t 3 , y 3 ) , . . . . We connect these points by straight lines, ob- taining a polygon, which we hope is an approximation to the real solution. The Euler Method The rule to get from one point ( t, y ) to the next point ( t, y ) is this: Choose a Step Size h . Then set t def = t + h and y def = y + f ( t, y ) × h t y = t + h y + f ( t, y ) h 2
( t, y ) t y ( t n , y n ) ( t 1 , y 1 ) t n ( t n +1 , y n +1 ) The slope is f ( t, y ) The solution has y ( t ) = f ( t, y ( t )) t 1 = t 0 + h t 0 t n +1 = t n + h t 0 t 1 = t 0 + h

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L20 - Approximation Consider again the FOOIVP y(t = f(t y(t...

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