assign7-09

# Assign7-09 - 2 h z h n h 2 f x nh z h n Hint Use the fact that z h n = y h 2 2 n 3 For the Initial Value Problem y = f x y = xy y(1 = 1 ﬁnd an

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MATH 573 ASSIGNMENT 7 1. a) Let y be the solution of the initial value problem y 0 = f ( x, y ) , y ( x 0 ) = y 0 . Let M 2 = max | y 00 ( x ) | and e n = y ( x n ) - y n , where { y n } is the approximation produced by Euler’s method. If e 0 = 0, d f y ( x, y ) 0, and h is suﬃciently small so that 1 + hd 0, show that | e n | ≤ h 2 ( x n - x 0 ) M 2 . HINT: Use the Mean Value Theorem to write f ( x n , y ( x n )) - f ( x n , y n ) = f y ( x n , ξ n )( y ( x n ) - y n ) . b) For the problem y 0 = - 2 y, y (0) = 1 , compare the above error bound with the one obtained in class when x n = 10. 2. Let y h n and y h/ 2 2 n denote the approximations to y ( x ) at x = x 0 + nh produced by Euler’s method with constant step sizes h and h/ 2, respectively. a) Using the fact that the error satisﬁes y ( x ) - y h n = hu ( x ) + O ( h 2 ) for an appropriately chosen function u , show that y ( x ) - [2 y h/ 2 2 n - y h n ] = O ( h 2 ) , i.e., applying Richardson extrapolation to Euler’s method gives improved accuracy. b) Show that y h/ 2 2 n +2 = y h/ 2 2 n + h 2 h f ( x 0 + nh, y h/ 2 2 n ) + f ( x 0 + [ n + 1 / 2] h, y h/ 2 2 n +1 ) i . Hint: Combine two steps of Euler’s method with step size h/ 2. c) Let z h n = 2 y h/ 2 2 n - y h n . Show that if y h/ 2 2 n = y h n , then we get the modiﬁed Euler formula: z h n +1 = z h n + hf ( x 0 + [ n + 1

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Unformatted text preview: / 2] h, z h n + ( h/ 2) f ( x + nh, z h n )) . Hint: Use the fact that z h n = y h/ 2 2 n . 3. For the Initial Value Problem y = f ( x, y ) = xy, y (1) = 1 , ﬁnd an explicit formula in terms of x and y for T 3 ( x, y ) in the Taylor algorithm of order 3. 1 2 MATH 573 ASSIGNMENT 7 4. The general 1–stage implicit Runge-Kutta method is deﬁned by ( * ) y n +1 = y n + chk 1 , where k 1 = f ( x n + hb, y n + hbk 1 ) , and c and b are constants to be determined from the condition that the Taylor series expan-sions of both sides of equation ( * ) are to agree to as many terms as possible. Show that to get agreement through terms of O ( h 2 ), we must have c = 1, b = 1 / 2. HINT: Determine the expansion for k 1 by assuming that k 1 has the form A + hB + O ( h 2 )....
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## This note was uploaded on 10/16/2010 for the course MATH FIN 621 taught by Professor Paulfeehan during the Spring '09 term at Rutgers.

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Assign7-09 - 2 h z h n h 2 f x nh z h n Hint Use the fact that z h n = y h 2 2 n 3 For the Initial Value Problem y = f x y = xy y(1 = 1 ﬁnd an

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