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?
it is Linear Systems Analysis class
1.4-1 Sketch the signals (a) u(t-5)-uſt-7)(b) uſt-5)+u(t-7) (c) lu(t-1)-ut-2)] (d) (t - 4)[u(t - 2) - uſt - 4)]
FInd u(x,t) and lim u(x,t)
Solve the heat problem Ut = Uzx + 5 sin(4x) - sin(2x), 0 < x <7, u(0,1) = 0, u(,t) = 0 u(x,0) = 0
PROBLEM 4. Determine the function u = u(t, x) if Ut = Uzz, t> 0, x € (0, 7), and u(0, x) = cos (x), uz(t, 0) = uz(t, 7) = 0.
7) Let f(t) = u(t - Ta) - ut - 27.) a) Sketch f(t) with proper amplitude vs. time labels b) Now rewrite f(t) in terms of (8) "delta" functions 8) Suppose F(s) *+1) What is the value of f(t) at time t = 0-?
Find the general solution of jutt + 2 ut + 2 u 3 u(0,t)ut)-0for all t s o ater for all x E (0, π), t > 0 Be sure to clearly indicate the following steps in your solution: 1. 2. 3. How to use separation of variables How to solve the resulting elgenfuiction/eigenvalue problem How the superposition principle is used.
A. : Suppose that u(x, t) satisfies Ut = Uzr +1, € (0,2) u(x,0) = 0 u(0,t) = u(2,t) = 0 Solve for u(x,t). What is lim u(x,t)? B. Consider the heat equation in the region 0 < x < 1, but supoose that the system is heated with a source. This is represented by: Ut = Uzz + cos(2), 1 € (0,7) u(x,0) = 1+ cos(2x) U (0,t) = U7(TT,t) = 0 Solve for u(x, t).
4. Compute and plot the results of each of the following convolutions: (a) ut) u(t- 2) (b) a(t-1)、n(t-2) (d) u2) ut) t- 2)] (e) u2) [ul) - u(t - 2)]
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
4. Use the method of eigenfunction expansion to find the solution of the IBVP ut (x, t) u (0,t) u (x, 0) ura' (a, t) + 2t sin (2na:) , 0 < x < 1, 0, u(1,t)=0, t > 0, sin(2π.r)-5 sin (4π.r) , 0 < x < 1. t > 0, = = =
4. Use the method of eigenfunction expansion to find the solution of the IBVP ut (x, t) u (0,t) u (x, 0) ura' (a, t)...