3. An infinite bar has initial temperature distribution: T(x,0)-T0 [s(x-l) +δ(x + 1)] Find T(x,t) for...
Use the 1D diffusion equation to find T(x,t):
2. A bar of length L has an initial temperature distribution T(x,0) = a + br ( x = L are the ends of the bar). The ends are insulated. Find T(x,t) for t > 0. 0 and си oFlu сх
4. Consider the process X+ = Vaw (t/a), where a is a positive constant. Calculate Var[X/(t+u) - X+(t)], where u > 0. Is X, a Brownian motion?
5. Consider the following IBVP (initial boundary value problem utt - Curr = 0, 0<x<1, t>0, with boundary conditions u(0,t) = u(1, t) = 0, > 0 and initial conditions (7,0) = x(1 – 2), 14(2,0) = 0, 0<x< 1. Use separation of variables method to find an infinite series solution of this problem. Do a complete calculation for this problem.
Find the Laplace Transform
(d) f(t) = te, 0<t<1, et, t > 1. l
Consider a diffusion process {Xt,t > 0} with constant drift μ(z)-a and constant volatility σ(z) > 0 (a) Please derive the conditional distribution of Xt| Xo.
Find vo(t) for t > 0 Q#3 and
Q#4
3. Find vo(t) for t0 4 H 2Ω 4Ω 4 A Answer: vo(t)4e2t [At 0 4. Find i(t) for t> 0 5 mA 1 mH Answer: it) -5e7.5*10°1 [mA] t20
Problem 07.055 - RL circuit with dependent source Find y(t) for t= 0 and t> 0 in the given circuit. Assume L = 1.5 H. 32 § 89 4ie A 20 200 24 V 20 V + V. The voltage for t = 0 is The voltage for t> 0 is v(t) = ett u(t) v.
(1 point) Consider the following initial value problem: 4t, 0<t<8 \0, y" 9y y(0)= 0, y/(0) 0 t> 8 Using Y for the Laplace transform of y(t), i.e., Y = L{y(t)} find the equation you get by taking the Laplace transform of the differential equation and solve for Y(s)
Solve the y"+ 4y = initial value problem s 1 if 0<xsa To if x>,T ylo)= 1, g(0)=0
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