Euler_Method(3).ppt

6 eulers method φ step size h x y x y true value y 1

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6 Euler’s Method Φ Step size, h x y x 0 ,y 0 True value y 1 , Predicted value 0 0 , , y y y x f dx dy Slope Run Rise 0 1 0 1 x x y y 0 0 , y x f  0 1 0 0 0 1 , x x y x f y y h y x f y 0 0 0 , Figure 1 Graphical interpretation of the first step of Euler’s metho
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7 Euler’s Method Φ Step size h True Value y i+1 , Predicted value y i x y x i x i+1 Figure 2. General graphical interpretation of Euler’s metho h y x f y y i i i i , 1 i i x x h 1
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8 Example 3 A ball at 1200K is allowed to cool down in air at an ambient temperature of 300K. Assuming heat is lost only due to radiation, the differential equation for the temperature of the ball is given by K dt d 1200 0 , 10 81 10 2067 . 2 8 4 12 Find the temperature at 480 t seconds using Euler’s method. Assume a step size of 240 h seconds.
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9 Solution K f h t f h t f i i i i 09 . 106 240 5579 . 4 1200 240 10 81 1200 10 2067 . 2 1200 240 1200 , 0 1200 , , 8 4 12 0 0 0 1 1 Step 1: 1 is the approximate temperature at 240 240 0 0 1 h t t t K 09 . 106 240 1 8 4 12 10 81 10 2067 . 2 dt d 8 4 12 10 81 10 2067 . 2 ,
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  • Fall '19
  • 300K, 32 k, 240 seconds, 106.09 K

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