Heat and Laplace equation problem 3. Solve ut – Uz = 0 with u(1,0) = 1, and u (0,t) = U,(2,t) = 0.
PDE questions. Please show all steps in detail. 2. Consider the initial-boundary value problem 0
Show all steps Problem 1. Consider the solution ur, t) 1-12 -2t of the heat equation ut the location of its maximum and minimum on the closed rectangle {0 x 1,0 Find t 12-2t of the heat equation ut-11x2- Find Problem 1. Consider the solution ur, t) 1-12 -2t of the heat equation ut the location of its maximum and minimum on the closed rectangle {0 x 1,0 Find t 12-2t of the heat equation ut-11x2- Find
1. Solve the boundary value problem ut =-3uzzzz + 5uzz, u(z, 0) = r(z) (-00 < z < oo, t > 0), using direct and inverse Fourier transforms U(w,t)-홅启u(z, t) ei r dr, u(z,t)-二U( ,t) e ur d . You need to explain where you use linearity of Fourier transform and how you transform derivatives in z and in t 2. Find the Fourier transform F() of the following function f(x) and determine whether F() is a continuous function (a)...
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.
3. In class we discussed the heat conduction problem with the boundary conditions a(0, t) 0, t4(1,t)-0, t > 0 and the initial condition u(r,0) f(a) We found the solution to be of the form where (2n-1)n 1,2,3,. TL 20 Now consider the heat conduction problem with the boundary conditions u(0, t) 1,u(T, t)0, t>0 and the initial condition ur,0) 0. Find u(r,t). Hint: First you must find the steady state. 3. In class we discussed the heat conduction problem...
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)
Consider the problem minimize 1[r(-)] = 2 / r,(t)2 dt subject to the conditions r(0) - r(T)0 and the constraint 0 r(t)2 dt 1. = Suppose that r : [0, π] R is a C2 function that! solves the above Let y : [0, π] R be any other C2 function such that y(0) Define problem a(s): (r(t) + sy(t))2 dt and a(s) a. Explain why a(0) 1 and i'(0) 0. b. Show that i'(0)= | z'(t) y' (t) dt-X...
4. Consider the boundary-value problem on the region given by {(r, 0, 6)|1 < r < 2}: vu= 0, 1 <r< 2, u(r = 1)= 1, ur(r = 2) = -u(r = 2). Using our work with the Laplace equation in class, find the solution to this problem. [Hint: it depends only on r, not on 0 or ø. 4. Consider the boundary-value problem on the region given by {(r, 0, 6)|1
Solve heat equation for the following conditions ut = kuxx t > 0, 0 < x < ∞ u|t=0 = g(x) ux|x=0 = h(t) 2. g(x) = 1 if x < 1 and 0 if x ≥ 1 h(t) = 0; for k = 1/2