Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
EGR 599 (Fall 2003) _____________________ LAST NAME, FIRST Problem set #7 1. Given I V V I =L RI and =C GV x t x t In these equations, V(x, t) = voltage, I(x, t) = current, L = inductance, R = resistance, C = capacitance, G = leakage to the ground. Show that V and I each satisfy u 2u 2u = LC + (RC + LC) + RGu (Note: u denotes V or I) 2 2 t x t 2. Find the directional derivative of f(x, y) = 2x2 + y2 at x = 2 and y = 2 in the direction of h = 3i + 2j. Ans: 8.875 3. Solve x u u +y = u; given that u = 3x on x + y = 1 x y Ans: u = 3x 4. If u = 3y2 when x = 0 on x = 0 solve u u (x y 1) = 3x(y x) x y Ans: u = 3(y x)(x 1) + 3(y x)e-x[(y x)e-x + 1] 5. Use the change of variables = x + ct, = x ct to transform the wave equation 2 2u 2u 2 u = c into = 0. Integrate the equation to obtain the solution u(x,t) = F(x + ct) + t 2 x 2 G(x ct). 6. Find the solution to the wave equation that satisfies the following boundary and initial conditions u(0,t) = 0, for t 0 and u(L,t) = 0, for t 0 u(x,0) = sin Ans: u(x,t) = sin 2x u , for 0 x L and (x,0) = 0, for 0 x L t L 2x 2ct cos L L 7. Find a period of the function (b) cos x (a) cos x 2 (c) cos x 3 (d) cos x + cos 2x Ans: (a) 2, (b) , (c) 3, (d) 2 8. Establish the orthogonality of the trigonometric functions over the interval [,]. 9. Plot the Nth partial sums of the Fourier series for N = 1 and N = 5 for the following functions over the interval x . a) f(x) = |x| = 4 2 1 (2k 1) k 0 2 cos( 2k 1) x 1 term 3 2.5 |x| 2 1.5 1 0.5 0 -8 -6 -4 -2 0 x 2 4 6 8 2 4 6 8 5 terms 3.5 3 |x| 2.5 2 1.5 1 0.5 0 -8 -6 -4 -2 0 x b) f(x) = |sin x| = 2 4 1 ( 2k ) k 1 2 1 cos( 2kx) 1 term 1.4 1.2 |sin(x)| 1 0.8 0.6 0.4 0.2 0 -8 -6 -4 -2 0 x 2 4 6 8 2 4 6 8 5 terms 1.4 1.2 |sin(x)| 1 0.8 0.6 0.4 0.2 0 -8 -6 -4 -2 10. Find the Fourier series of f(x) = a 1 2 ( 1) n 1 nx Ans: f(x) = a + 4a cos( ) 2 3 p n 1 ( n ) 0 x 2 x if p x p. p 11. Find the Fourier series of f(x) f(x) = Ans: f(x) = 2c 2c (x p/2) if 0 x p and f(x) = (x + p/2) if p x 0 . p p 8c 2 1 2k 1 cos[(2k + 1) p x] k 0 12. Find the sine series expansion of x(x 1) on the interval 0 < x < 1. Ans: x(x 1) = 8 3 1 2k 1 k 0 3 sin[(2k + 1)x] 13. Find the sine series expansion of sin x on the interval 0 < x < 1. Ans: f(x) = sin x 14. Find the sine series expansion of (sin x)(cos x) on the interval 0 < x < 1. Ans: f(x) = 1 sin 2x 2