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Unformatted text preview: MATH 573 ASSIGNMENT 9 w f (x, w, z ) . Heun’s method for the system of diﬀerential and F (x, Y ) = 1 z f2 (x, w, z ) equations Y = F (x, Y ), with initial condition Y (x0 ) = Y0 , is given by: h h Yn+1 = Yn + F (xn , Yn ) + F (xn + h, Yn + hF (xn , Yn )). 2 2 Find approximations to w(h) and z (h) for the system 1. Let Y = w = z, where a, b, c are given constants. 2. Consider the approximation of the initial value problem y = f (x, y ), by a class of explicit linear multistep methods of the form: yn+1 = a0 yn + a1 yn−1 + h[b0 fn + b1 fn−1 ]. If the constant a0 is considered ﬁxed, determine values of the constants a1 , b0 , and b1 (expressed in terms of a0 ), such that the local truncation error of the method will be O(h3 ). 3. Determine which of the following methods are convergent. State a reason for your conclusion. i) yn+1 = yn + hfn+1 , ii) yn+1 = 3 yn − 1 yn−1 + hfn , 2 2 iii) yn+1 = 3yn − 2yn−1 + h[fn+1 − 2fn ]. y (x0 ) = y0 , z = −cw, w(0) = a, z (0) = b, 1 ...
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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