ps4[1] - Applied Mathematics E2101y Introduction to Applied...

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Applied Mathematics E2101y Introduction to Applied Mathematics Problem Set #4, due Thursday, 5 March 2009 All references are to Zill & Cullen (3rd edition). Chapter 2, § 2.6: The Euler Method #2 In addition to the two step sizes for h listed, also try to get to the endpoint in just one step: h = 0 . 2. Comment on the quality of the three numerical approximations for y (0 . 2): which do you expect to be best? Chapter 6, § 6.1: The Improved Euler Method #11 Integrate only to x = 0 . 2 in each case. [Hint: For the analytic solution, let u ( x ) = x + y - 1 and derive a separable equation for u ( x ), recalling that du 1+ u 2 = arctan u .] Chapter 6, § 6.2: The Runge-Kutta Method #1 Integrate only to x = 0 . 2. Chapter 6: Supplementary Problems (a) Consider Euler’s method for y = Ay , y ( x 0 ) = y 0 , where A is a constant, and show that y n = y 0 (1 + Ah ) n . Show that, in the limit as h 0, the solution of the Euler difference equation approaches the solution of the continuous problem from which it is derived. (b) Suppose that we have a convergent method with local truncation error that scales like Ch p as h 0. John offers an improved method with a constant
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