Question

Initial temp of wood is: T0 which is heated by Tinf over boundaries at x=0 and x=L

Temperature of wood model:

Use the crank-Nicholson method (Θ1 = Θ2 = ½)to develop a finite volume method for the diffusion equation

0<<L, t>O subject to at x = 0, at r = L, +ht = hlm, >0 +ht = ht«, t> 0 and T(3,0) = To, 0<x<L. The analytical solution to this problem is given by: T(2, t) = 1 + (T. -T.) (ET) where BO sin - COS le- and (in, ) = Hn cos(un) + B, sin(pin), N (1:1= } [~ + (1 +371 BP) +B with { = 1, B; = T = prezt. The eigenvalues An are the roots of the equation tanjin = 26, and are given in Roots AnalyticalSolution.m file on Black Board.

Answer 1

using crank nicholson methods of equations we derive the finite volume method for the diffusion equation.
Page no. 1 Solutions desire the Cue Con earutos Tan the gives Cerditons, 01 = 02 = Y₂ lusoy Canka Nicholson method, we can desire ScxoT = kotoLaL, to K.Theroneel Cendeativity at azo, kot tht = hla, toff- Temperature SC=L, KOT thT=htw, too h-heat and M T (2,0) = To, oLXLL, The Chaily ficel Sorter to this roaublem is given by T(2+) = Tat (To-Ta ) o (8,2) where (1) (A ) (sinh, B, C, A, B, 4 (un, 2) = Mn Cos (una) + B; 30 (Me) the alcom-[(**o*)(****)

Puge nov 2 2 With &= a BishL, 2= 1 Scir the roots of the The eigen values Ma con ecuadas tanda 2 Bicha and are given in the Roots Anely freeil Solutions in File on Bluk boood. Those is She Finike volume method Tes diffusi cauatus the