128a_su09_hw6

# 128a_su09_hw6 - MATH 128A SUMMER 2009 HOMEWORK 6 Problems...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 128A, SUMMER 2009: HOMEWORK 6 Problems 5.3: 1(c), 5.4: 1(c) and 13(c), 5.5: 1(c), modiﬁed versions of 5.6: 1(c) and 4: Use the following methods to solve the problem of 5.2: 1(c) from last week (y = 1 + y/t, t ∈ [1, 2], y (1) = 2,). Put your answers (for each value of t) in a table, rounded to four signiﬁcant digits (x.xxx). (1) Actual solution: y = t(ln t + 2); (2) Taylor method of order 2, h = 0.25; (3) Modiﬁed Euler method, h = 0.25; (4) Adams-Bashforth two-step explicit method, h = 0.25, with exact starting values; (5) Adams-Moulton one-step implicit method (wi+1 = wi + h (f (ti+1 , wi+1 ) + f (ti , wi ))), h = 0.25, with 2 exact starting values; (6) Adams second-order predictor-corrector method (one-step Adams-Bashforth, i.e. Euler, feeding the above Adams-Moulton formula), h = 0.25, with exact starting values; (7) Runge-Kutta method of order 4 (see rk4.m), h = 0.25; (8) Runge-Kutta-Fehlberg method (see rk4.m), tol = 10−4 , hmin = 0.05, hmax = 0.25; RK problem: Assume that the diﬀerential equation y = f (t, y ) takes the form y = g (y ). (Equations of this form are called autonomous.) Find the best choice of coeﬃcients a, b1 , b2 , and c in the following Runge-Kutta method: k1 = hg (wi ) k2 = hg (wi + ak1 ) wi+1 = wi + b1 k1 + b2 k2 What is the order of the resulting method’s truncation error? Multistep Method Problem: Find a0 , a1 , b0 , and b1 so that the following explicit 2-step method has the highest-order truncation error possible: wi+1 = a1 wi + a0 wi−1 + h[b1 f (ti , wi ) + b0 f (ti−1 , wi−1 )] Hint : Let y (t) = c0 + c1 (t − ti + c2 (t − ti )2 + c3 (t − ti )3 + . . .. Match as many terms as possible in the power series y (ti + h) and a1 y (ti ) + a0 y (ti − h) + h[b1 y (ti ) + b0 y (ti − h)]. Note : In spite of the good truncation error, the resulting method is unsuitable for all problems. Date : Due Thursday 7/30. 1 ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online