1. An iron bar 10m long as shown in the picture, with both ends given a certain temperature. With the conditions shown in the figure, the iron bars flow heat following the usual differential equations d²T +h (TT)=0 dx² where T = temperature distribution in bars, T₁ = temperature around bars, and h = heat dispersion coefficient, 1/m². T₁ = 40°C T₁ = 20°C 10 m T₂ = 200°C Determine the heat distribution of the bars by dividing the domain into 5 segments a. Write down the difference equation finite to approximate the equation b. Write the SPL equation in matrix form for all interior points c. Complete the SPL at c. 2. Solve Laplace's Equation- =0 with the domain as shown in the following image b C h g d 113 3 115 10 x where u₁ = u3 = f(x, y) = 0, uzu₁ = us = 100. Note that the domain of symmetry is at y = 4. 3. Solve Laplace's Equation for the domain with the boundary of the following Neumann boundary condition =0 g Ju Ju りょ on ox V²=0 Note: Use a backward difference to calculate u, (1, y) = 9y at points c and i. 4. A disc-shaped domain as shown below In this domain, heat propagation occurs following the diffusion equation, with the outer side temperature of the Trout circle and the inner side temperature of the Trin circle = 100°C. The heat distribution in the disks is the same for the same distance from the center point (radial symmetry), so the problem can be viewed as a 1-D heat equation in polar coordinates. a. Show that the thermal equation in polar coordinates can be written as = α Hint use transform. x = r cost, dan y = r sin t T b. Use the FTCS scheme to solve the heat propagation problem by using the heat equation in polar coordinates ba Computing domain image bb Write a different equation for this problem b.c Divide the domain in 5 segments, then calculate T in 2 time levels (i = 1,2) where j = 1,2,3,4,5 5. A domain in the form of a rectangular slab with length = width = 1 m. In this domain, heat spreads following the 2-D heat equation. When t = 0 the initial temperature at u(x,t) = 0°C. If the temperatures on the sides are given as follows. u East 150, u_west -300, u_north 50, u_south=100. Solve the problem equation with the program