4. (10 points) Find the solution to the wave problem Ut = c+421 +COSI, <0, t>0,...
4. [10] Find the solution to given initial-boundary value problem: 4uxx = ut 0<x<TI, t> 0 u(0,t) = 5, uit, t) = 10, t> 0 u(x,0) = sin 3x - sin 5x, 0<x<T
7. Find the solution of the heat conduction problem 100uzz = ut, 0 < x < 1, t > 0; u(0,t) 0, u1,t 0, t>0; In Problem 10, consider the conduction of heat in a rod 40 cm in length whose ends are maintained at 0°C for all t0. Find an expression for the temperature u(,t) if the initial temperature distribution in the rod is the given function. Suppose that a
PROBLEM 1 IS SUPPOSED TO BE A WAVE EQUATION NOT HEAT EQUATION 1. Find the solution to the following boundary value initial value problem for the Heat Equation au 22u 22 = 22+ 2 0<x<1, c=1 <3 <1, C u(0,t) = 0 u(1,t) = 0 (L = 1) u(x,0) = f(x) = 3 sin(7x) + 2 sin (3x) (initial conditions) (2,0) = g(x) = sin(2x) 2. Find the solution to the following boundary value problem on the rectangle 0 <...
PDE question Consider the one dimensional wave equation on the half line: Ut(x,0) = g(x) Utt - Uzx= 0 0 < < u(0,t) = 0 u(x,0) = f(x) (a) What is the solution? (b) For the particular initial conditions 12 - 2 25254 f(x) = { 6- 4<r<6 otherwise g(x) = 0 sketch the solution u(x, t) for t = 0, 2, 4, 6.
5. (20 pts). Solve the following initial-value problem: Ut + 2uuz - 0<x<, 0 <t<oo 0 1 <1 > 1 u(t,0) = Then draw the solution for different values of time.
7. (a) Find the solution of the heat conduction problem: Suxx = ut, 0<x< 5, u(0, 1) = 20, tu(5, 1) = 80, 1>0 u(x,0) = f(x) = 12x + 20 + 13sin(tor) - 5sin(3 tex). (b) Find lim u(2, t). (c) If the initial condition is, instead, u(x,0) = 10x – 20 + 13sin( Tox) - 5sin(3 7ox), will the limit in (b) be different? What would the difference be?
(1 point) Solve the nonhomogeneous heat problem Ut = uzz + 4 sin(5x), 0< I<T, u(0, t) = 0, u(T, t) = 0 u(x,0) = sin(3.c) u(x, t) = Steady State Solution lim, , u(x, t)
4. (10 points) A damped wave train is expressed by f(t) = Ae-iwot when t2t> 0; and f(t) = 0 ortherwise. a) Please find the power spectrum of this light; b) Please plot and discuss the spectrum when to = 5
(1 point) Suppose we have ut = aʻuzz, 0 < x <1,t > 0, boundary conditions are u(0,t) = u(1,t) = 0, and the initial condition is u(x,0) = sin(T2). What will be the behavior of u(x, t) as time increases. There may be more than one correct answer. You do not need to solve the equation to answer this question. A. The solution behaves unpredictably. B. The solution increases to infinity. OC. For a fixed t, the solution will...
(1 point) Solve the nonhomogeneous heat problem Ut Uzz + 3 sin(3.c), 0<x<1, u(0,t) = 0, u(T,t) = 0 u(2,0) sin(52) u(x, t) = Steady State Solution lim oo u(a,t) =