5. For the transport equation PDE Uz-ut + u = 0 IC u(z,0) cos z (a) What is the associated ODE af...
1. Solve the vibrating string problem PDE BC BC IC IC utt T.T Uz(0,t) = 0 u(1,t0 u(x,0cos(3T) 14(2.0) = x
PDE: Ut = Uxx, -00 < x < 0, t> 0 IC: u(x,0) = 38(x) + 28(x – 6) where is the Dirac delta function (impulse). u(x, t) =
2. Let u(z,t) be a differentiable function on R x [0, 0o). a) Show that the directional derivative of u at (x, t) = (zo, to) along v is Dvu(x, t) = ▽u(ro, to) , v b) Solve the following homogeneous linear transport equation ul + uz = 0, u(x,0) =-2 cosx c) Solve the following non-homogeneous equation ut-2uz--2 cos (x-t), u(x, 0) = sin x d) Solve the following second-order homogeneous linear euqation u(z,0) = sin x, ut (z,...
PDE questions. Please show all steps in detail. 2. Consider the initial-boundary value problem 0
Heat and Laplace equation problem 3. Solve ut – Uz = 0 with u(1,0) = 1, and u (0,t) = U,(2,t) = 0.
Find solution to the IBVP PDE BCs Ic u(0, t)-0, 0<oo l u(1, t) 0, 0<t< oo u(z,0)=x-x2ババ1
b) The transport equation ut +r3t3uz-0 has the characteristics if to >0 E(t) = 0 if ro < 0, where zo = E(0) is the starting point. Given the initial values u(zo, 0) = uo(20) =婧+sin(ro), find the solution u(r, t) to the transport equation b) The transport equation ut +r3t3uz-0 has the characteristics if to >0 E(t) = 0 if ro
Problem # 3 [20 Points] Solve PDE: ut = uxx - u, 0 < x < 1, 0 < t < ∞ BCs: u(0, t)=0 u(1, t)=0 0 < t < ∞ IC: u(x, 0) = sin(πx), 0 ≤ x ≤ 1 directly by separation of variables without making any preliminary trans- formation. Does your solution agree with the solution you would obtain if transformation u(x, t)= e(caret)(-t) w(x, t) were made in advance?
clear writing please and thank you Problem 2. Given the PDE ut + uy = u? u(3,0) = g(x) for IER, >0, for ER. (a) Sketch the characteristic curve that passes through P(2,3) in the xt-plane. Find u(2,3) without using the exact solution ulit, t). (b) Use the method of characteristics to find the solution u(, t).
Problem # 1 [15 Points] Consider the following PDE which describes a typical heat-flow problem PDE: ut = ↵2uxx, 0 < x < 1, 0 < t < 1 BCs: ux(0, t)=0 ux(1, t)=0 0 < t < 1 IC: u(x, 0) = sin(⇡x), 0 x 1 (a) What is your physical interpretation of the above problem? (b) Can you draw rough sketches of the solution for various values of time? (c) What about the steady-state temperature?