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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.
- Spring '09