Consider the boundary-value problem 02 u 0x 2 0u 0t , 0 , x , 1, 0 , t , 0.05 u(0, t) 0

Chapter 16, Problem 3

(choose chapter or problem)

Consider the boundary-value problem

\(\frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial u}{\partial t}\),  0<x<1,  0<t<0.05

u(0, t)=0,  u(1, t)=0,  t>0

u(x, 0)=x,  0<x<1

(a) Note that the initial temperature u(x, 0) = x indicates that the temperature at the right boundary x = 1 should be u(1, 0) = 1, whereas the boundary conditions imply that u(1, 0) = 0. Write a computer problem for the explicit finite difference method so that the boundary conditions prevail for all times considered, including t = 0. Use the program to complete Table 16.R.1.

(b) Modify your computer program so that the initial condition prevails at the boundaries at t = 0. Use this program to complete Table 16.R.2.

(c) Are Tables 16.R.1 and 16.R.2 related in any way? Use a larger time interval if necessary.