Question

u(x,t) is solution to heat equation, ,with following parameters for numerical approximation:

0 < x < 2,    0 < t < 0.1, n = 20, m = 100, c =1. Boundary conditions: u(0,t) =0, and u(2,0) = 0.
Initial conditions: u (x,0) =30^o for 0<x<=1
                                            0^o for 1<x<2
Set the approximate difference equation for this equation.

Do you think this equation converges to a numerical solution.

Continuing with problem 1, calculate u(0.1,0.001) by iteration

Continuing with problem 1, calculate u(0.2,0.001) by iteration

Answer 1

Forward time central difference scheme: 4, +1,5 – 24,, +U:-1,5 – 4,;+1 –Uj (14 ,1,–21., +4,-43] =1:38 U Here, h=2-0 = 0.1 20 _ 0.1-0 =0.001 100 Thus, 0.001 091 [1,1,–21, , +1,4., ] = 14, – 11, 0.1[14, 41, -201, +0,-.,]=1,341 – 11,1 Ujj+z = 0.14, 41, -0.24; ; +1,+0.130:-1, U1, 5+1 = 0. 10, 41, +0.84, +0.10,44 (a) U0.1,0.001 = 4,1 = U10+1 (:: i=1, j =0) = 0. lu2.0 +0.814.0 +0.lu. = 0.1% 30+0.8x30+0.1x0 = 27 (b) 40.2,0.001 = U2,1 = U2,0+1 (:.i = 2, j =0) = 0. luz,0 +0.8u2,0 +0.1u1.0 = 0.1% 30+0.8% 30+0.1% 30 = 30