A rod of length L coincides with the interval (0, L] on the r-axis. Set up the boundary-valueproblem problem for the temperature u(x, t) in the following scenarios:(a) The left end is held at temperature uo, and the right end is held at temperature u1. Theinitial temperature is zero throughout.(b) There is heat transfer from the left end into a surrounding medium at temperature 20°,and the right end is insulated. The initial temperature is f(r) throughout.(c) The ends are insulated, and there is heat transfer from the lateral surface of the rod intothe surrounding medium held at 50°. The initial temperature is 100° throughout.

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Answered: A rod of length L coincides with the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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The dynamic equilibrium of a one-story building is described by the following equation: x(t) = x.(t) + X(t) where x(t) = displacement in meters, t= time in seconds, x.(t) = e-0.4[A cost + B sin t], is the complementary function, and X(t) is the particular integral (a) Determine the homogeneous second-order DE, assuming X(t) = 0. (b) Determine the non-homogeneous second-order DE, assuming X(t) = 10. (c) Determine the particular solution of the DE in (b) assuming x(0) = 0, and x'(0) = 0.
1. Given z = f (x, y) = V + y² + 4y. (a) Find its domain and range. (b) Sketch the level curves for z = 0 and z = 1.
2. Consider the DE: + e dx – e dy = 0. (a) For what values of a and B is this DE a homogeneous DE? (b) Use the values a, B you suggested in (a) and find the solution that satisfies the initial condition y(1) = 0.
3. A greenhouse has a glass dome in the shape of the paraboloid z=8-2x²-2y2 and a flat wooden floor at z = 0. Let S be the closed surface formed by the dome and the floor, oriented with outward unit normal. Suppose that the temperature in the greenhouse is given by T(x, y, z) = x²+ y²+3(z − 2)². The temperature gives rise to a heat flux density field F(x, y, z) -kVT = where k is a positive constant that depends on the insulating properties of the medium. Assume that k = 1 on the glass dome and k = 3 on the wooden floor of the greenhouse. (a) Sketch S, clearly labelling any intercepts and the direction of the normal vector. (b) Write down an expression in terms of x, y and z for the vector field F on the greenhouse. (c) By direct calculation (do not use any integral theorems), find the total heat flux F.ndS across the greenhouse in the direction of the outward unit normal.
Determine the steady state temperature distribution in a rectangu- lar plate of size 20cm x 20cm by solving the Laplace equation: 0, urith boundary conditions: T(0, y) = T(20, y) = 20°, T(x, 20) = 40°, and T(x,0) = 0°. Take Ax = 5cm = Ay. %3D
1. Here the the volume of between 0 and z (in meters) is πT V(z) = (12z - (z − 1)³ – 1) . The barrel is being filled with liquid in such a way that at a certain moment in time (in seconds), the rate of change of the volume occupied by the liquid is 127 m³/s and the rate of increase of the level z is 3 m/s. (a) Find the level z at which this occurs. (b) Find the volume of liquid at that time.
Consider a two-dimensional circular metal plate of radius a, as represented by the shaded region below, and suppose that the boundary is held at 0°C around the circumference. Let u(r, 0, t) be the temperature distribution of the metal plate. At time t = 0, the metal plate is suddenly intensely heated up in such a way that its initial temperature distribution can be idealised as u(r, 0,0) = f(r) : 108 (r-2). = a State clearly any symmetry, and its reasoning, that could simplify the problem. [Hint: Observe the dependence on 0.] Then, using the method of separation of variable, determine u(r, 0, t) for t > 0. You may assume that the relevant separation constant is negative and you may leave your final answer in terms of the Bessel functions Jo and J₁.
Find the steady-state temperature u(r,θ) in a semicirular plate of radius r=2 if: u(2,θ) = {1, 0<θ<π/2 and 0, π/2<θ<π} and the edges θ = 0 and 0 = π are insuated.

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