Assignment-
Problem 1) Solve the following system of non-linear equations using Newton's Method and fixed point iteration:
x2 + 2x + 2y2 - 26 = 0
2x3 - y2 + 4y - 19 = 0
Start at x= 1.0, y = 1.0. Carry out the first five iterations for both cases.
Problem 2) A coating on the panel of a surface is cured by radiant energy from a heater. The temperature of the coating is determined by radiative and convective (Non-linear) processes. Using the following governing equations solve the system of simultaneous equations to determine Jh, Jc, Tc, Th. Follow the steps (parts a and b) in the text below-
5.67 x 10-8Tc4 + 17.41Tc - Jc = 5188.18
Jc - 0.71Jh + 7.46Tc = 2352.71
5.67 x 10-8Th4 + 1.865Th - Jh = 2250
Jh - 0.71Jc + 7.46Th = 11093
where Jh and Jc are the radiosities of the heater and coating surfaces, respectively, and Th and Tc are the respective temperatures.
(a) Show that the following iteration function can be used for solving the nonlinear system of equations with the fixed-point iteration method:
Tc = [Jc - 17.41Tc + 5188.18/5.67 x 10-8]1/4
Th = [2250 + Jh - 1.865Th/5.67 x 10-8]1/4
Jc = 2352.71 + 0.71Jh - 7.46Tc
Jh = 11093 + 0.71Jc - 7.46Th
(b) Solve the nonlinear system of equations with the fixed-point iteration method using the iteration functions from part (a). Use the following initial values: Th = Tc = 298 K, Jc = 3000 W/m2, and Jh = 5000 W/m2. Carry out 100 iterations, and plot the respective values to observe their convergence. The final answers should be: Tc = 481 K, Jc = 6222 W/m2, Th = 671 K, Jh = 10504 W/m2.