2. Let u(z,t) be a differentiable function on R x [0, 0o). a) Show that the directional derivativ...
9. Use a suitable Fourier Transform to find the solution of the IVP utt (x, t) Uz(0, t) u(x, t) , uz (z, t) 4uzz (x, t) + q (x, t), 0, t> 0, 0as x → 00, x > 0, t > 0, = = t>0. → = 0, ut (2,0)-( = { t, 0 0-x-2, -1, 0, > 2, u(x, 0) q(a, t) Leave your answer in the form of an integral. 9. Use a suitable Fourier Transform...
Exercise 9. Solve the BVP a(0, t) = 0 u(r, t)-Uz(n, t) = 0 u(z, 0) = sin z 0<x<π, t>0, t >0 t > 0 (z,0) = 0 2
5. For the transport equation PDE Uz-ut + u = 0 IC u(z,0) cos z (a) What is the associated ODE after applying the method of characteristics? (b) Solve the associated ODE to find u(s,T) c)Find u(x, t) 5. For the transport equation PDE Uz-ut + u = 0 IC u(z,0) cos z (a) What is the associated ODE after applying the method of characteristics? (b) Solve the associated ODE to find u(s,T) c)Find u(x, t)
Problem #2115 Points) Solve the following initial-boundary-value prob- lem: u(0,t) = 1 uz(1,1) + ßu(1,1) = 1 BCs: 0 < t < oo 0 IC: u(z,0)= sin(nx)+x, 1 x by transforming it into homogeneous BCs and then solving the transformed problem Problem #2115 Points) Solve the following initial-boundary-value prob- lem: u(0,t) = 1 uz(1,1) + ßu(1,1) = 1 BCs: 0
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, = =
If you were to solve the variant of wave equation utt=uxx+u for 0<x<6 and t>0 with u(0,t)=u(2 ,t)=0, u(x,0)=2x, ut(x,0)=0 using separation of variables, what would be the correct form of Xn (x)? Xn (x)=cosh( nπ 4 Xn (x)=sin( nπ 2 Xn (x)=sin( n2 π2 4 Xn (x)=cos nπ 2 None of these
nonhomogeneous vibrating string problem for u(x with homogeneous boundary conditions t > 0 u(0, t) u(r,t) = 0, 0, = and the initial conditions 0stst tr(z,0)=0, u(z, 0) sin(2x), = Find the solution u(x,t) to the IBVP using an eigenfunction expansion: u(z, t) = Σ an(t) sin(nz) n-1 nonhomogeneous vibrating string problem for u(x with homogeneous boundary conditions t > 0 u(0, t) u(r,t) = 0, 0, = and the initial conditions 0stst tr(z,0)=0, u(z, 0) sin(2x), = Find the...
Problem 1. Consider the nonhomogeneous heat equation for u,t) ut = uzz + sin(2x), 0<x<π, t>0 subject to the nonhomogeneous boundary conditions u(0, t) t > 0 u(n, t) = 0, 1, - and the initial condition Lee) Find the solution u(z, t) by completing each of the following steps: (a) Find the equilibrium temperature distribution ue(x). (b) Denote v(x, t) u(a, t) - e(). Derive the IBVP for the function v(x,t). (c) Find v(x, t) (d) Find u(, t)...
Solve the BVP for the wave equation (∂^2u)/(∂t^2)(x,t)=(∂^2u)/(∂x^2)(x,t), 0<x<5pi u(0,t)=0, u(5π,t)=0, t>0, u(x,0)=sin(2x), ut(x,0)=4sin(5x), 0<x<5pi. u(x,t)=
(2) Solve the following BVP for the Wave Equation using the Fourier Serics solution formulac (42- u(r, t)= 0 u(0, t) 0 u(5, t) 0 u(r, 0) sin(T) 12sin(T) ut(r, 0)0 (r, t) E (0,5) x (0, oo) t > 0 t > 0 (2) Solve the following BVP for the Wave Equation using the Fourier Serics solution formulac (42- u(r, t)= 0 u(0, t) 0 u(5, t) 0 u(r, 0) sin(T) 12sin(T) ut(r, 0)0 (r, t) E (0,5) x...