We have solved the given PDE using the method of separation of variables.
3. For the equation 24 = r, in 0 <<1,0<t<1, (1,0) = sin(x), on 0 SEST...
1. Consider the insulated heat equation up = cum, 0 <r<L, t > 0 u (0,t) = u (L, t) = 0, t > 0 u(x,0) = f(2). What is the steady-state solution? 2. Solve the two-dimensional wave equation (with c=1/) on the unit square (i.e., [0, 1] x [0,1) with homogeneous Dirichlet boundary conditions and initial conditions: (2, y,0) = sin(x) sin(y) (,y,0) = sin(x). 3. Solve the following PDE: Uzr + Uyy = 0, 0<<1,0 <y < 2...
1. For pdf f (r, y) = 1.22, 0 < x < 1,0 < y < 2, z +y > 1, calculate: EY) and () E (X2)
2. Use separation of variables to solve the IBVP: utt (z, y, t) u(0, y, t) u (x, y,0) uzz(z, y, t) + un, (x, y, t) = 0, 0 < x < 1, 0 < y < 1, 0, u(1,y,t)=0, u(z,0,t)=0, u(z, l,t) = 0 sin(r) sin (2my), ue (r, y,02 sin(2mx) sin(ry) t > 0, = =
a) Use the d'Alembert solution to solve au au - <r< ,t> 0, at2 48,2 ux,0) = cos 3x, u(,0) = 21 b) Consider the heat equation диди 0<x<1, t > 0, at ax? with boundary conditions uz (0,t) = 0, uz(1,t) = 0, > 0, and initial conditions u(x,0) = { 0, 2.0, 0<r < 0.5, 0.5 <<1. Use the method of separation of variables to solve the equation.
Solve heat equation in a rectangle du = k ( ou + dou), 0<x<t, 0<y< 1, t> 0 u(x, 0, 1) = 0, uy(x,1,1) = 0, with boundary conditions u(O, y,t) = 0, u(r, y, t) = 0, and initial condition u(x, y,0) = (y – į v?) sin(2x).
4. (10 points) Find the solution to the wave problem Ut = c+421 +COSI, <0, t>0, with initial conditions u(1,0) = sin r, 4(1,0) = 1+I.
Question. Consider () - ( cos(t), sin(t)) for 0 +< 2. Parameterine this curve by are length. Chat
2. Solve the linear homogeneous IVP U+ rtuz = 0, u.1,0) = sinr, -o0<< 0, t> 0.
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
8. Solve V?u=0, 2<r<4,0<O<21, (u(2,0) = sin 0, u(4,0) = cos 0,0 5 0 5 21.