Problem 1 (5 points) : Heat Plate

In much the same way that we solved the bug on a bar problem in Team Lab 3, we can solve a linear system to determine the temperature at various locations across a heat plate. We have simplified the problem to allow you to focus on the general concepts used when modeling such a problem as a system of linear equations.

Imagine that you wish to know the temperature at all points on a plate that is insulated except for two positions on the edge where heating elements of known temperature are applied. The plate also has a hole in the center that is insulated. At some point when the temperature at all positions on the plate have stabilized, what will the temperature at each point be?

To solve such a problem, we imagine the plate surface divided into equal sized "tiles" and restate the problem to find the temperature at the center of each tile. We know that the temperature of each tile depends upon its neighboring temperatures. In fact, it is a weighted average of those temperatures based on the proportion of our tile's edge that is shared with its neighbor. Here is a rectangular "plate" with a hole in the center that has been divided into 16 equal area tiles. Assume that each edge of each tile has identical length, so that the temperature of any one tile is a simple average of the temperatures of all neighboring tiles that share a side with the tile in question. This creates a system of 16 equations, one for each "tile" area.

T_A and T_B are the non-insulated positions along the edge where a heat source of known temperature will be applied. The temperatures at T₁through T₁₆ are the unknowns that you wish to find.

1. Define a function named heatPlate that accepts two temperature values TA and TB (representing T_A and T_B, respectively) and returns a single matrix result that is a 36 matrix of temperatures for the unknowns T₁ through T₁₆ (and the hole) in the order labeled above (the spots in the result matrix representing the hole should have the value 0).

   To solve this problem, identify 16 equations. Each equation represents the temperature value for one tile based upon the temperature values of each of its neighboring tiles. For example, the temperature at T₈ is equal to the average of the temperatures: T₂, T₇, and T₁₂ and the temperature at T₁₁ is equal to the average of T₇ and T₁₂.

   How do T_A and T_B factor into the equations? They will be in the equations, but since their values will be known when the function is called, the terms with T_A and T_B should be on the right-hand side of the equations once the equations are put into general form.

   DO NOT SOLVE FOR ANY OF THE TEMPERATURES OF THIS PROBLEM BY HAND. Work out the equations for each tile and then reorder the terms in general form, with the unknown terms on the left-hand side and the known terms on the right-hand side of the equation. Once you have the equations, rewrite the linear system as a matrix equation and put the commands to solve this problem in your heatPlate(TA,TB) function definition.

   Note that if T_A and T_B have the same value, then after the temperatures of the tiles have stabilized, every tile should have the same temperature (i.e., the same value as T_A and T_B). For example, a call to heatPlate(100, 100) should return:

    100.0000 100.0000 100.0000 100.0000 100.0000 100.0000 100.0000 100.0000 0 0 100.0000 100.0000 100.0000 100.0000 100.0000 100.0000 100.0000 100.0000

2. In your homework1.m script,write code to call your heatPlate function to compute the values of T1 through T16 for each of the following conditions. Do not suppress the output of your function calls.

   1. Find T1 through T16 when TA = 87 and TB = 23

   2. Find T1 through T16 when TA = 35 and TB = 64