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

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Unformatted text preview: Approximation Consider again the FOOIVP y ( t ) = f ( t,y ( t )) , y ( t ) = y . This covers systems and higher order ODE. If one of our four arrows does not shoot it down, well try numerical approximation of the solution (Note the power series so- lutions are practically approximations). Well talk about three approximation schemes 1) the Euler Method ; 2) the Improved Euler Method , also called the Heun Method ; 3) the Runge-Kutta Method . All three methods consist in a prescrip- tion to move from the starting point ( t ,y ) 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...
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L20 - Approximation Consider again the FOOIVP y ( t ) = f (...

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