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finalsolution

finalsolution - Numerical Analysis in Engineering ME 140A...

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Numerical Analysis in Engineering ME 140A, Fall 2007 Final Solution December 15, 2007 Problem 1 Use the trapezoidal rule to integrate I = 2 0 3 0 xy dx dy (1) Use 3 intervals of equal size in the x-direction, and 2 intervals of equal size in the y-direction. Compare your result with the analytical result. Discuss the comparison. Solution: I = 2 0 y 3 0 x dx dy (2) First, use composite trapezoidal rule in x-direction: I = 2 0 y 0 + 1 × 1 + 1 × 2 + 1 2 × 3 dy = 9 2 2 0 y dy (3) Now, use composite trapezoidal rule in y-direction: I = 9 2 0 + 1 × 1 + 1 2 × 2 = 9 (4) Next, we can find the integral analytically as below: I = 2 0 y 3 0 x dx dy (5) = 2 0 y x 2 2 | 3 0 dy (6) = 9 2 y 2 2 | 2 0 (7) = 9 (8) Comparing the results, we get exact agreement between the numerical and ana- lytical result since the given function is linear both in terms of x and y. There- fore, the trapezoidal rule can reproduce the functions correctly. 1
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Problem 2 The following first order ODE dy dx = 2 x y + x 2 y , y (0) = - 2 (9) can be solved analytically using the separation of variables technique. Toward
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