4.5.6 Let g(x,t) = cos(π2/1) in Exercise 4.5.4 and obtain the solution ofthe initial and boundary...
1. Let u be a solution of the wave equation u 0. Let the points A, B, C, D be the vertices of the paralleogram formed by the two pairs of characteristic lines r-ctC1,x- ct-2,+ ct- di,r +ct- d2 Show that u (A)+u (C)-u (B) + u (D Use this to find u satisfying For which (x, t) can you determine u (x, t) uniquely this way? 2. Suppose u satisfies the wave equation utt -curr0 in the strip 0...
Let u be the solution to the initial boundary value problem for the Heat Equation 202u(t, ) te (0, o0) (0,3); дли(t, 2) хе _ with boundary conditions ut, 0) 0 u(t, 3) 0 and with initial condition 3 9 u(0, ar) f(x){ 5, | 4' 4 0, Те The solution u of the problem above, with the conventions given in class, has the form ()n "(2)"п (г)"а "," n-1 with the normalization conditions 3 Wn 2n vn (0) 1,...
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, 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, 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...
correct only please. Let u be the solution to the initial boundary value problem for the Heat Equation, du(t, x) = 2 ult, x), te (0,0), X€ (0,3); with boundary conditions u(t,0) = 0, ut, 3) = 0, and with initial condition [0, xe (p.3). b(0, x) = f(x) ={5, xe,), 10, x€ 12,3]. The solution u of the problem above, with the conventions given in class, has the form u(t, x) = cnun(t) wy(x), n=1 with the normalization conditions...
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...
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)...
1. (10 points, part I) Consider the following initial boundary value problem lU (la) (1b) (1c) 0L, t> 0 3 cos ( a(x, 0) (a) Classify the partial differential equation (1a) (b) What do the equations (la)-(1c) model? (Hint: Give an interpretation for the PDE, boundary conditions and intial condition.) c) Use the method of separation of variables to separate the above problem into two sub- problems (one that depends on space and the other only on time) (d) What...
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...