### Description

Hello,

You will find attached the files with the project decription and other files attached with it to help with the project. You will need to use MATLAB for some part of the analysis and the plotting.

Deliverables:

– Analytical solution graph

– 100×100 grid Finite difference method graph (suggested to start with a 4×4 grid first before the 100×100)

– Comparison of the two solutions.

– Build plots with “Contour” and add elevations labels. The elevation will correspond to the temperatures.

Also, attached is an analytical solution slide show in pdf format, and here is the description for each slide:

Slide 1

The first slide shows the infinite series solution. Note that when you code this you will not be able to specify an upper bound of infinity. Start with a small upper bound like 5 or 10 and increase it as needed. There will be a point when the upper bound is “too” large and you’ll get an error in Matlab. That’s fine just use a smaller upper bound. I give the example of sine in an infinite series form. We use this because we can easily graph the exact solution to use as comparison. Note that for sin(x) we need to solve the infinite series for each x location. In your project you will need to solve this solution for each x and y location. So you should expect to need a 3 nested for loop. If you do not remember how to use a for loop brush up on it, it will be used in both parts of the project.

Slide 2

The second slide simply specifies the notation to setup a m-file in Matlab and plot the exact sine solution. Remember to use a semicolon after your equations in order to prevent your matrices/vectors from plotting. This can severely slow down your code if the for loop or matrix is large.

Slide 3

In the third slide we begin thinking about the structure of the infinite series. I start with defining the x vector outside of the for loop so I can reference it in the for loop. Since I know I need to solve the infinite series for each x location I need a for loop for x that is “for i=1:length(x)”. “Length(x)” tells me the length of x and this makes sure if I change my vector x, the for loop changes along with it. Then I need a for loop to go through the infinite series itself. I define the upper bound as J=10. Hence “j=0:10”.

Slide 4

In slide 4 we fill in the missing parts of the for loops. At the top I specify “Summation=zeros(1,length(x))” this is not a required step but creates an empty vector which my solution which I call “Summation” can be filled. In larger projects this speeds up the computation time.

The line under the outer for loop is “counter=0” this is resets the counter which is used for the infinite series. In Matlab the “%” indicates that anything after it on that line is just a comment. Using comments is good to keep track of your code.

The line under the inner for loop is “counter=counter+(the infinite series equation)”. The reason we have it as counter = counter + … is so we can build the series. Take a step back and make sure this process makes sense to you, you can use the same concept for your project. Note that instead of x we write x(i). If you write x Matlab thinks you are calling the entire vector. Instead writing x(i) calls that specific location x. You will do the same for your project but just use x(i) and y(j). The line “Summation(i)=counter” takes the final counter solution for a given x value and stores it in the vector at the given index location Summation(i).

The bottom 3 lines ares just the plotting terms. Try doing this in Matlab yourself. Slides 5 and 6 just show the plots when we change the upper limit.

**University of New Hampshire Heat Transfer Project Using MATLAB for Analysis**

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**University of New Hampshire Heat Transfer Project Using MATLAB for Analysis**