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# sol5 - MATHEMATICS 3161 Fall 2009(2009.9 2009.12 Assignment...

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Unformatted text preview: MATHEMATICS 3161: Fall 2009 (2009.9 - 2009.12) Assignment # 5 (Due Date: Nov. 30) 1. Consider the initial value problem y ′ = ty, y (0) = 1 , hand-calculate the first five iterations of Euler’s method, the first three iterations of Heun’s method and the first two iterations of Runge-Kutta method, with step size h = 0 . 1 (keeping 4 decimal digits). Arrange your results in the tabular form. Solution : Euler’s method: y n +1 = y n + h ( t n y n ) Improved Euler’s method: y n +1 = y n + h 2 ( t n y n + t n +1 ( y n + ht n y n )) Runge-Kutta method: y n +1 = y n + ( K 1 + 2 K 2 + 2 K 3 + K 4 ) / 6 with K 1 = ht n y n , K 2 = h ( t n + h/ 2)( y n + K 1 / 2) , K 3 = h ( t n + h/ 2)( y n + K 2 / 2) , K 4 = h ( t n + h )( y n + K 3 ) , t_n y_n (Euler) y_n(Heun) y_n(R-K) 0.0 1.0000 1.0000 1.0000 0.1 1.0000 1.0050 1.0050 0.2 1.0100 1.0202 1.0202 0.3 1.0302 1.0460 0.4 1.0611 0.5 1.1036 2. For y ′ = (1 + t ) y, y (0) =- 1, a) Find the exact solution of the given initial value problem. b) Use a computer and each of the methods: Euler’s method, Heun’s method and Runge- Kutta method to calculate, respectively, an approximate solutions on the interval [0 , 1] with step size h = 0 . 1....
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sol5 - MATHEMATICS 3161 Fall 2009(2009.9 2009.12 Assignment...

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