
Solve the PDE (e* + e-*)ut +2ux = 0, and sketch a few of the characteristic...
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).
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)
PDE Question thank you
0 0, t 4. Solve for ρ(x, t) on x Hint: Sketch the characteristics and consider how they relate to the initial data
0 0, t 4. Solve for ρ(x, t) on x Hint: Sketch the characteristics and consider how they relate to the initial data
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) =
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?
PDE
question
Consider the one dimensional wave equation on the half line: Ut(x,0) = g(x) Utt - Uzx= 0 0 < < u(0,t) = 0 u(x,0) = f(x) (a) What is the solution? (b) For the particular initial conditions 12 - 2 25254 f(x) = { 6- 4<r<6 otherwise g(x) = 0 sketch the solution u(x, t) for t = 0, 2, 4, 6.
Assignment 0220 Marks) Solve the following IVBP: PDE : Uxx = (1/25) utt ICs: u (x,0) = x2 (nt - x), ut (x,0) = sin(x) BCs: u(0,t) = 0, u(nt,t) = 0 for 0<x<, t> 0. for 0<x<T. for t>0.
MATH2018 Quiz The PDE ar2 can be solved using D'Alembert method. That is, it has a solution of the form u(x, t) = φ(x + ct) + ψ(x-ct). where c 6 Solve the PDE with the initial conditions u(x, 0) 6 sin (x), ut (x, 0) 3 er Enter the expression for u(x, t) in the box below using Maple syntax. Note: the expression should be in terms of x andt, but not c
MATH2018 Quiz The PDE ar2 can...
Solve the circularly symmetric vibrating membrane PDE given
as
u_tt = ∇^2*u
BC : u(1, θ, 0) = 0, 0 < t < ∞
ICs :
u(r, θ, 0) = J_0*(2.4r) − 0.25*J_0*(14.93r), 0 ≤ r ≤ 1
u_t(r, θ, 0) = 0
Solve the circularly symmetric vibrating membrane PDE given as Utt = Dau BC : u(1,0,0) = 0, 0<t< oo ICs : u(r,0,0) = J.(2.4r) – 0.25J(14.93r), 0 <r <1 Ut(r,0,0) = 0
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?