Question

2. Heat equation Let ult, 2) satisfy the equation 4472(t, 2) +1, 0<r <1, t>0 with initial condition u(0,2) = 0, 0<x<1, and boundary conditions u(t,0) = 0, u(t,1)= 0, t> 0. This equation describes the temperature in a rod. The rod initially has a temperature of 0 (zero degree Celsius), and is then heated at a uniform rate 1. However, its two endpoints are kept at the temperature of 0 at all times. The unknown function u(t, x) describes the temperature in the rod at time t > 0 at the point 2 € [0,1]. (a). Set up the forward-Euler method. (b). Set up the backward-Euler method. Write out the tri-diagonal system one needs to solve at every time step.

Answer 1

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a forward - Guler method И Let i=ih, t= jk where i = 1, 2, 3 4 .. J-1,2,3 h= xi tini :k: tjre ti gu U41t.nl, Ui, j+1-41,5 tock) ot gu Uitli -243,j.tui-lj : ley- toch) n values in substitute these U(4,0): 4044 (t, x) + 1 Lijti-liis 41 4+,; -24j +4;-.;) + k ů ui, iti -Ui,j Ak h L nk (uin,j - 247,5 + U1-13 * By neglecting och"), o(k) and ik we get U j41-45j +a (Li +1,; -247,3 +41-1,;) Ui , j+1= dui 41,j + (1-2«)ui, j + d Uitlej

Backward Cullen methoden du Uj-Unj. st tock) k Uiti, j - 24;, ; tuin,j + och?) del from we get uij - uij- 4 (4141,5 - 247.j thi+1,3)+1 K th Ak uij-U13-1 ( uitij ik 4 -245j +4- Neglecting och!), 0(1) my they are very small, we get 41,5-} = 45,5 - (uitlij - 203+ U14,j) (: a 4k Backward Euler method # U1j4) =(1+22 Juij - Uitlij -44141,j Tvi-cliogonal systemi uijti o dutti,j.+ (1-20) ligj + dU1-13 i=1,69:-). Uitla düz,j' + (1-2d)uj + 440.j U2, jtt = dus, j + (1-2d ) 42, + dulj U3.j+1 = 244,3 + (1-26 ) ugoj, td vej you 1-24.0 الروما 31 (2) us,d Uz j+1 0 1-20 Toi-diogonal inatrix where it z Uk ul