Repeat Prob. 12.3 but use Jacobi iteration.

PROBLEMS 301 Solve this problem with the Gauss-Seidel method to ε s = 5.

12.6 Use the Gauss-Seidel method a without relaxation

and b with relaxation λ = 1.2 to solve the following sys- tem to a tolerance of ε s = 5. If necessary, rearrange the equations to achieve convergence. 2x 1 − 6x 2 − x 3 = − 38 − 3x 1 − x 2 + 7x 3 = − 34 − 8x 1 + x 2 − 2x 3 = − 20

12.7 Of the following three sets of linear equations, identify

the sets that you could not solve using an iterative method such as Gauss-Seidel. Show using any number of iterations that is necessary that your solution does not converge. Clearly state your convergence criteria how you know it is not converging. Set One Set Two Set Three 8x + 3y + z = 12 x + y + 5z = 7 −x + 3y + 5z = 7 − 6x + 7z = 1 x + 4y − z = 4 −2x + 4y − 5z = −3 2x + 4y −z = 5 3x + y − z = 3 2y −z = 1

12.8 Determine the solution of the simultaneous nonlinear

equations y = −x 2 + x + 0.5 y + 5x y = x 2 Use the Newton-Raphson method and employ initial guesses of x = y = 1.2.

12.9 Determine the solution of the simultaneous nonlinear

equations: x 2 = 5 − y 2 y + 1 = x 2 a Graphically. b Successive substitution using initial guesses of x = y = 1.5. c Newton-Raphson using initial guesses of x = y = 1.5. 12.10 Figure P12.10 depicts a chemical exchange process consisting of a series of reactors in which a gas flowing from left to right is passed over a liquid flowing from right to left. The transfer of a chemical from the gas into the liquid occurs at a rate that is proportional to the difference between the gas and liquid concentrations in each reactor. At steady state, a mass balance for the first reactor can be written for the gas as Q G c G − Q G c G 1 + Dc L 1 − c G 1 = and for the liquid as Q L c L 2 − Q L c L 1 + Dc G 1 − c L 1 = where Q G and Q L are the gas and liquid flow rates, respec- tively, and D = the gas-liquid exchange rate. Similar balances can be written for the other reactors. Use Gauss-Seidel with- out relaxation to solve for the concentrations given the fol- lowing values: Q G = 2, Q L = 1, D = 0.8, c G = 100, c L 6 = 10.

12.11 The steady-state distribution of temperature on a

heated plate can be modeled by the Laplace equation: 0 = ∂ 2 T ∂ x 2 + ∂ 2 T ∂ y 2 If the plate is represented by a series of nodes Fig. P12.11, centered finite differences can be substituted for the second derivatives, which result in a system of linear algebraic equations. Use the Gauss-Seidel method to solve for the temperatures of the nodes in Fig. P12.11. 12.12 Develop your own M-file function for the Gauss- Seidel method without relaxation based on Fig. 12.2, but c G 1 c G c G 2 c G 3 c G 4 Q G Q G Q L c G 5 Q L D c L 1 c L 2 c L 3 c L 4 c L 5 c L 6 FIGURE P12.10