16 Chapter in Review Answers to selected odd-numbered problems begin on page ANS-44.

  1. Consider the boundary-value problem

    Approximate the solution of the differential equation at the interior points of the region with mesh size h = . Use Gaussian elimination or Gauss–Seidel iteration.

  2. Solve Problem 1 using mesh size h = . Use Gauss–Seidel iteration.
  3. Consider the boundary-value problem
    1. 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.
    2. 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.
    3. Are Tables 16.R.1 and 16.R.2 related in any way? Use a larger time interval if necessary.
TABLE 16.R.1  
A table has seven columns and six rows. The column headings are as follows: Column 1, Time. Column 2, x = 0.00. Column 3, x = 0.20. Column 4, x = 0.40. Column 5, x = 0.60. Column 6, x = 0.80. Column 7, x = 1.00. The row entries are as follows: Row 1. Time, 0.00. x = 0.00, 0.0000. x = 0.20, 0.2000. x = 0.40, 0.4000. x = 0.60, 0.6000. x = 0.80, 0.8000.x = 1.00, 0.0000. Row 2. Time, 0.01. x = 0.00, 0.0000. x = 1.00, 0.0000. Row 3. Time, 0.02. x = 0.00, 0.0000. x = 1.00, 0.0000. Row 4. Time, 0.03. x = 0.00, 0.0000. x = 1.00, 0.0000. Row 5. Time, 0.04. x = 0.00, 0.0000. x = 1.00, 0.0000. Row 6. Time, 0.05. x = 0.00, 0.0000. x = 1.00, 0.0000.
TABLE 16.R.2  
A table has seven columns and six rows. The column headings are as follows: Column 1, Time. Column 2, x = 0.00. Column 3, x = 0.20. Column 4, x = 0.40. Column 5, x = 0.60. Column 6, x = 0.80. Column 7, x = 1.00. The row entries are as follows: Row 1. Time, 0.00. x = 0.00, 0.0000. x = 0.20, 0.2000. x = 0.40, 0.4000. x = 0.60, 0.6000. x = 0.80, 0.8000.x = 1.00, 0.0000. Row 2. Time, 0.01. x = 0.00, 0.0000. x = 1.00, 0.0000. Row 3. Time, 0.02. x = 0.00, 0.0000. x = 1.00, 0.0000. Row 4. Time, 0.03. x = 0.00, 0.0000. x = 1.00, 0.0000. Row 5. Time, 0.04. x = 0.00, 0.0000. x = 1.00, 0.0000. Row 6. Time, 0.05. x = 0.00, 0.0000. x = 1.00, 0.0000.