1. Solve fully the heat equation problem: ut = 5u: u(0,t) = u(1,t) = 0 (3,0)...
1. Solve fully the heat equation problem: Ut = 5ucx u(0,t) = u(1, t) = 0 u(x,0) = x – 23 (Provide all the details of separation of variables as well as the needed Fourier expansions.)
Heat and Laplace equation problem 3. Solve ut – Uz = 0 with u(1,0) = 1, and u (0,t) = U,(2,t) = 0.
Solve the heat equation by the method of separation of variables 1(1, t) = 0 Эт u,(0, t) = 0, u(x,0) =-2cos( 12. Solve the heat equation by the method of separation of variables 1(1, t) = 0 Эт u,(0, t) = 0, u(x,0) =-2cos( 12.
1 & 5 Solve the following heat equations using Fourier series ux Ut, 0 <x <1,t>0, u (0,t) = 0 = u(1,t), u(x,0) = x/2 1/ 2/ Ux=Ut, 0<x< m ,t>0 ,u(0,t) = 0 = u( 1, t), u(x, O) = sinx- sin3x 3/ usxut, O <x < 1 ,t>0, u(0,t) = 0 = u,(1, t), u(x,0) = 1 -x2 Ux=Ut,O<x <m ,t>0, u(0, t) = 0 = u,( rt , t) , u(x, 0) = (sinxcosx)2 4/ 5/Solve the...
2. Solve the heat problem: (Trench: Sec 12.1, 17) 9Uxx = ut, 0 < x < 4, t > 0 u(0, t) = 0, u(4,t) = 0, t> 0 u(3,0) = x2, 0 < x < 4
Solve the heat equation 4,0 < x < 3,1 > 0 kou det u(0, 1) = 0, u(3,t) = 0,1 > 0 S2, 0<x< } u(x,0) = { 10, { <x<3 are the eigenfunctions You will need to apply separation of variables to obtain a family of product solutions un(x, t) = x (x)Ty(t) where X of a Sturm-Liouville problem with eigenvalues an (as in Section 12.1). Using the explicit expressions for un(x, t) gives (8,0) = ŠA, n=0 Then...
(1 point) Solve the nonhomogeneous heat problem Ut = uzz + 4 sin(5x), 0< I<T, u(0, t) = 0, u(T, t) = 0 u(x,0) = sin(3.c) u(x, t) = Steady State Solution lim, , u(x, t)
Find the general solution of jutt + 2 ut + 2 u 3 u(0,t)ut)-0for all t s o ater for all x E (0, π), t > 0 Be sure to clearly indicate the following steps in your solution: 1. 2. 3. How to use separation of variables How to solve the resulting elgenfuiction/eigenvalue problem How the superposition principle is used.
(1 point) Solve the nonhomogeneous heat problem Ut Uzz + 3 sin(3.c), 0<x<1, u(0,t) = 0, u(T,t) = 0 u(2,0) sin(52) u(x, t) = Steady State Solution lim oo u(a,t) =
Problem 4: Consider the following problem for the heat equation (1) (2) (3) ut= Uxa + s(t), xE (0,1), t > 0 u(0, t) 2, u(1, t) = 4 и (х, 0) — 2(1 — х). where s(t) describes the source term (a) Find a series solution for u(x, t) with s(t) = e"1. (b) What is the convergence criteria for the transient extension function if s(t) = 0. Problem 4: Consider the following problem for the heat equation (1)...