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MAE 192B MIDTERM EXAM ENGINEERING MATHEMATICS FALL 2002 November 5, 2002, 10:00-11:50 1.m., Room 5272 Boelter Hall (TOTAL : 100 points) Additional Bonus Problem Worth 15 Points- max. Score could be 115/100 ________________________________________________________________________ Note: determine all the constants you generate in your solutions 1. (Total Point 35) Solve Laplace’s equation inside a circular annulus ( a < r < b ) subject to the boundary conditions: u (a,θ ) = f (θ ) , and u (b,θ ) = g (θ ) (1) Hint: you may consider the solution as a superposition of two solutions u1 and u 2 , each satisfies Laplace’s equation and one homogeneous boundary condition 2. (Total Points 35) Consider a slightly damped vibrating string of length L that satisfies the PDE: ∂ 2u ∂ 2u ∂u ρ 0 2 = T0 2 − β , (2) ∂t ∂t ∂x where β is a fictional coefficient which is relatively small ( β 2 < 4π 2 ρ 0T0 / L2 ). Determine the solution (using the separation of variable method) which satisfies the boundary conditions: u (0, t ) = 0, and the initial conditions u ( x,0) = f ( x), and u ( L, t ) = 0, ∂u ( x,0) = g ( x), ∂t (3) (4) 3. (Total Points 30) Consider the heat equation in two dimensional rectangular region 0 < x < L, and 0 < y < H, ∂ 2u ∂ 2u ∂u = k 2 + 2 ∂t ∂y ∂x (5) subject to the initial condition: u ( x, y,0) = f ( x, y ). (6) -1- Solve the initial value problem and analyze the temperature as t → ∞ if the boundary conditions are: ∂u ∂u (0, y, t ) = 0, ( L, y , t ) = 0, u ( x,0, t ) = 0, u ( x, H , t ) = 0, (7) ∂x ∂x Bonus Problem : Points from this problem will be added only after seriously attempting the above problems 4. (Total Points 15) If a vibrating string of length L , satisfying the following PDE, initial condition (IC), and boundary conditions(BC’s): 2 ∂ 2u 2 ∂ u , = c ∂t 2 ∂x 2 BC’s: u (0, t ) = 0, PDE: u ( L, t ) = 0, ∂u ( x,0) IC: u ( x,0) = f ( x), = g ( x), (8) ∂t is initially unperturbed ( f ( x) = 0) , with the initial velocity g ( x) given , show that: u ( x, t ) = 1 x + ct G (x )dx . 2c ∫x −ct (9) where G ( x) is the odd periodic extension of g ( x) . 1 Hint: sin a sin b = [cos(a − b) − cos(a + b)] 2 -2-