the below is the previous question solution:
the below is the previous question solution: 1. Recall the following boundary-value problem on the interval [0, 1] from...
4. Consider the following initial value problem of the 1D wave equation with mixed boundary condition IC: u(z, t = 0) = g(x), ut(z, t = 0) = h(z), BC: u(0, t)0, u(l,t) 0, t>0 0 < x < 1, (a)Use the energy method to show that there is at most one solution for the initial-boundary value problem. (b)Suppose u(x,t)-X()T(t) is a seperable solution. Show that X and T satisfy for some λ E R. Find all the eigenvalues An...
Let u be the solution to the initial boundary value problem for the Heat Equation, 0,uột, 2) = 40ều(t, z), t + (0, 0, z + (0,5); with initial condition u(0, x) = f(x), where f(0) = 0 and f'(5) = 0, and with boundary conditions u(t,0) = 0, 0,ult, 5) = 0. Using separation of variables, the solution of this problem is u(t, 2) = Čem () w.(2), n= 1 with the normalization conditions 0,() = 1, W. (2–...
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z) 28?u(t,z), te (0,00), z (0,3); with initial condition u(0, z)fx), where f(0) 0 and f (3) 0 and with boundary conditions u(t,0)-0, r 30 Using separation of variables, the solution of this problem is 4X with the normalization conditions un(m3ī)-. n@) : ї, a. (5/10) Find the functions wn with index n1. Wnlz) b. (5/10) Find the functions vn with index n 1. n(t)...
please dont copy other answer, it is not correct model from before: Continue with the same network availability model from the previous problem. Engineers decided to pool two types of failures together considering the two-state Markov chain with the state space. S-(0,1), and infinitesimal transition probabilities defined (under the assumption that A 0) as follows. They set λ-λι + λ2 and μ-μι + μ2 and then considered the infinitesimal transition probabilities as P [x(t + Δ) = 1 lx(t) =...
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z 382u(t,z), tE (0,oo), E (0,3); with initial condition u(0,x)-f(x)- and with boundary conditions Find the solution u using the expansion u(t,x) n (t) wn(x), with the normalization conditions vn (0)1, Wn (2n -1) a. (3/10) Find the functionswn with index n 1. b. (3/10) Find the functions vn, with index n 1 C. (4/10) Find the coefficients cn , with index n 1. Let...
Let u be the solution to the initial boundary value problem for the Heat Equation, Otu(t, x) = 2 &n(t, x), ț e (0,00), x e (0,5); with initial condition u(0,xf(x)- and with boundary condition:s Find the solution u using the expansion with the normalization conditions (2n - 1) a. (3/10) Find the functions w, with indexn>1. Wnsin(2n-1)pix/10) b. (3/10) Find the functions v, with indexn > 1. Vnexp(-2(2n-1)pi/10)(2)t) 1. C. (4/10) Find the coefficients cn , with index n...
Let u be the solution to the initial boundary value problem for the Heat Equation, dụı(t, x)-20 11(t, x), IE(0, oo), XE(0,3); with initial condition u(0,x)-f (x), where f(0) 0 andf'(3)0, and with boundary conditions Using separation of variables, the solution of this problem is with the normalization conditions 3 a. (5/10) Find the functions wn, with index n 1. wn(x) = 1 . b. (5/10) Find the functions vn, with index n Let u be the solution to the...
Find the solution u(a, t) to the initial boundary value problem for the heat equation 4urx te (0, 00), a e (0,5), with initial condition e [0,) e ,5 | 3, u(0, ar) f(ar) = 4, and with boundary conditions ug (t, 0) = 0, un (t, 5) = 0. Find the solution u(a, t) to the initial boundary value problem for the heat equation 4urx te (0, 00), a e (0,5), with initial condition e [0,) e ,5 |...
Let u be the solution to the initial boundary value problem for the Heat Equation au(t,) -48Fu(t,), te (0,oo), z (0,5); with boundary conditions u(t,0) 0, u(t,5) 0, and with initial condition 5 15 15 The solution u of the problem above, with the conventions given in class, has the form with the normalization conditions vn(0)-1, u Find the functions vnwn and the constants cn n(t) wnr) Let u be the solution to the initial boundary value problem for the...
Let u be the solution to the initial boundary value problem for the Heat Equation an(t,r)-301a(t, z), te(0,00), z E (0,3); with initial condition 3 0 and with boundary conditions 6xu(t,0)-0, u(t, 3) 0 Find the solution u using the expansion with the normalization conditions vn (0)-1, wn(0) 1 a. (3/10) Find the functionsw with index n1 b. (3/10) Find the functions vn with index n1 Un c. (4/10) Find the coefficients cn, with index n 1 Let u be...