# Me and My Friend Found Where the Coolest Part in My Room is Using the Heat Equation Because it is REALLY REALLY HOT OUTSIDE!

Jack Lee1, Aaron Wiener1, Superman Action Figure2

1Ms Makema 2nd Grade Maths, Department of Young Academic Misfits, Southeastern Northwest Primary School, Oceania

2My (second) favourite superhero, he’s also here for moral support and in case we need more time to write the paper he can turn back time by making it go backwards by flying really really really really fast and reversing the spin of the earth. I saw it once!

## Abstract

In this paper me and my best friend Aaron try to figure out where the coolest part in my room is during summer so that we don’t have to sit in front of the fan all day. The summer is almost over but I still think this will be useful to know for next summer and I don’t think it’s going to stop being hot for a WHILE! I’m finally not grounded anymore but now my mommy says it’s too hot to go outside. I think she is right! It IS HOT! Finding the coldest spot in my room would be really awesome! Not only would my mommy stop shouting at us to stop using the fan and driving up the power bill, but it means that me and Aaron could use that electricity to instead make progress in our hardcore Minecraft survival world. It’s REALLY Tough. We have gotten REALLY good at Minecraft. We are going to try and use the heat equation to get to find a cold spot in our room! The fan doesn’t work that well anyway, it just blows hot air around! I think we need a bigger fan. I like it when Aaron talks through the fan and makes his voice all funny though.

## Introduction

After me and Aaron finished our previous paper[1], we showed it to our maths teacher. Ms Makema was not impressed though and after telling us that we were “wasting our time”, “not going to solve the Collatz conjecture”, and “not sleeping during nap time like she told us to” she made us sit inside and write down lines while everyone else played outside. While writing “We will not write any more pointless papers during nap time” for the fiftieth time I noticed that the room was starting to get quite warm. The outside was as high as 40 $^\circ$C! Aaron was sitting next to me, so when I mentioned that it was getting hot he said that we should move, then Ms Makema said to keep quiet! I thought that this was a great idea, so we immediately started looking for a different part of the room to sit at.

This was when we found the problem. We weren’t sure where the coolest part of the room was, and we didn’t really want to run around with a thermometer. We only had ONE thermometer and it was HARD TO READ! PLUS, we didn’t want Ms Makema to think that we were goofing around and extend our punishment. PLUS, running around made us even hotter! We decided to solve this with MATH! We like Math and are VERY good at it! In math there is a thing called the heat equation! We are VERY good at equations as we tried to tell Ms Makema even though she tells us we don’t need to learn about equations yet. I think we might be smarter than her and she doesn’t know how to use equations. By doing it with equations and not a thermometer, not only would we be in the coolest part in the classroom during a 40 $^\circ$C day, we would not need the fan at home! We’re killing two birds with one stone, although I like birds, so I probably wouldn’t kill them! One time I was throwing rocks at birds with Aaron and Michael and John and some other kids in class and I was actually trying to not hit the birds, but don’t tell them that!

## Background

My brother Oscar, who is going to university now, told me that one of the things he learned about, besides the $3n+1$ problem we solved earlier, was the heat equation[2]. I miss Oscar, I can’t wait for him to come back at the next break. He told me that the equation is a partial differential equation, and that it will tell me how temperature diffuses over time. Diffusing is a big word for people like Geoff who can’t understand big words. I can explain it, because I understand it and read about it online. For a temperature to diffuse, it has to spread out, like when they finally let all of us outside we can spread out into the playground. If we have been inside longer we will run outside faster, this is like temperature, it can spread out faster or slower. I think things that spread out must have something to do with the Mess On particle which is a particle me and Aaron discovered to explain why my room is so messy in another paper [3]. We didn’t even need to buy a particle accelerator to find it.

I still don’t know what a differential equation is or what a partial differential equation is. It has something to do with things changing and because the part that changes is only a part of the equation I don’t know if it’s worth learning about it if it’s going to change all the time. Differential means change because it stands for difference which is like subtraction but it’s not subtraction. I tried learning the rules to not get in trouble in Ms. Makema’s class but she keep CHANGING THEM when she doesn’t like how we follow her rules. She tells us to do our math homework but gets mad when we are drawing more differential equations to try and figure out what they are instead of learning how to do more arithmetic. I think my teacher is a partial differential equation because she changes so much and I don’t understand her. Arithmetic is boring, I’m already perfect at it. I make NO mistakes! I’m not perfect at differential equations, they keep changing like Ms. Makema. The heat equation is written with partial differential equations like:

$\displaystyle \frac{\partial u}{\partial t} = \dot{u} = \alpha \Delta u = \alpha \nabla^2 u \ \ \ \ (1)$

Apparently the $\frac{\partial u}{\partial t}$ is the part that changes. I think $\partial$ is a silly letter, or maybe it’s a partial because it’s an incomplete $8$. I asked Oscar what $\alpha$ is and he said it was actually made up of a lot of things. It’s something called a thermal diffusivity which has something to do with the types of atoms and particles and how they touch each other. Me and Aaron are really smart at particles. We discovered a particle once called the Mess On [3] even though we didn’t have a particle accelerator because my dad wouldn’t let me spend the million-billion-trillion dollars on it. We still found it. Thermal dissusivity is like, when, heat travels through things, and how fast? I think. Water is slow. That’s why it’s so cold in the summer, but concrete and metal is high, because I burn my hand when I touch it, or the things in the kitchen can heat up REAL fast. I think that’s because the particles are closer together which makes it easier for heat to go through. Me and Aaron decided to rate the thermal diffusivity of the different objects in my room based on how hot they were when we got home from school. We put the alphas in a table below based on what we felt. I couldn’t touch the window for very long.

Me and Aaron agree on the numbers so they should be correct. Apparently, we also needed the $\alpha$ of the air, so we decided to make it $\alpha_{air} = 1$ because air can’t be hot and it can’t be cold like an object except when it’s from something like a refrigerator, oven or by the window. Maybe it can have a temperature but then I wouldn’t know what number to give it. The right side of the equation is now defined as:

$\displaystyle \Delta u = \nabla^2 u = \alpha(\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} + \frac{\partial^{2} u}{\partial z^{2}}) \ \ (2)$

Because an upside-down triangle means Laplacian. A Laplacian is a three dimensions derivative and means “The Place”-ian in Spanish where “ian” means change except I don’t know the language “ian” is from. It is very special because it can find the change of heat at any three dimensional point in my room! Did you know that you are three dimensions? So am I! Some cartoons are two dimensional but I think most are three dimensional now.

While we are talking about the heat equation in three dimensions, there is nothing stopping us from using it with more than three spatial dimensions. I think we could keep adding on dimensions if we wanted to. We are smart enough I think we can. We could even apply it to four dimensions or even A THOUSAND DIMENSIONS! Even though I can’t picture that, someone has come up with a method to make Tesserectane[4] which I guess is something more than 3d! We will now use the heat equation to answer our question and find out where we should sit so that we can stay cool!

## Analysis

I wish my room was one dimensional space, because if it were in one dimensional space, the heat equation can be solved by something called separation of variables. In separation of variables, the variables get separated like when I get peas and carrots and don’t want to mix them so that I can eat the carrots first because they taste better than the peas. I can do that because peas and carrots are one dimensional which is like when something is a line or one thing. I think it would be kind of boring if my room was one dimensional, I couldn’t do anything except sleep I think because I’m three dimensions. Apparently Oscar knows how to solve through separation of variables but he refused to show us how it was done, even when we offered to buy him give him one of my toys. I think if he showed me, we could figure out how to do it in three dimensional space which is my room.

It won’t be hard to solve because we already figured out the $\alpha$ for all the objects in my room, except the air conditioning vent I can’t reach. And we set $\alpha$ to one for air because it doesn’t do anything. Unfortunately, I found out I can’t just use my partial differential equations with just my $\alpha$ values, I need initial conditions and boundary conditions. I did not know how to look this up or what they were so we’ll figure that out later. I think it means the walls, or the doors, but I’m not sure which. Me and Oscar tried to do the analytical solution, but we ran out of paper to make our calculations on and we couldn’t write on toilet paper.

On our family computer, we didn’t run out of paper. BUT it was hard to write on in math form. We decided to try and solve it numerically like Oscar told us we would have to do. I still think we can solve it on paper, but we need more paper and pencils to do that. We tried erasing and re-writing, but it took too long. We were getting frustrated and hot, and we were doing this to not be hot! Solving something numerically is like solving it with numbers instead of with letters. BUT the numbers are close numbers, they’re not exact numbers. It’s like when I try and guess how long we have left at school, I am normally not correct but I am close. It always is longer than I think it is. Solving with numbers would be A LOT! Since we had even less clue how to do SO MANY calculations, we first asked Oscar. He said that we would need a really REALY big computer and that we could not do it on our iPad or my dad’s computer.

I think we could still do it because we are REALLY GOOD at using computers! I once put a frog on the desktop and my dad was mad because I guess he doesn’t like frogs. He doesn’t like it when we use his computer either, but we don’t know how to solve something numerically on an iPad. It got put into a parental lock mode last time we tried to watch videos on it. Computers now are REALLY SMALL compared to how they used to be. I guess people have to do less math on computers because old computers used to be the SIZE OF A ROOM! My dad’s computer is only as big as my arm!

My dad sometimes doesn’t turn off his computer so that’s how we could get on sometimes. It has all of the applications. That’s why he was so mad at me for putting a frog on his desktop background I think. I don’t think he doesn’t like frogs. It was OUR FROG WE CAUGHT! Jerry. I miss that frog. He could jump THREE FEET! While we were trying to get enough ender pearls to drop we thought about asking Chat GPT how we would use my dad’s computer to solve the heat equation next time he leaves his computer on. Chat GPT is a person on my dads computer he talks to a lot. He’s VERY SMART and answers ALL my questions. Even Oscar doesn’t answer all my questions.

While we didn’t have access to GPT4 because mom stopped letting me use the credit card and I can’t remember that many numbers, we could still ask GPT 3.5[5] how to do it. I think Chat GPT4 must be smarter because four is a bigger number. At first Chat GPT 3.5 said it was a bad idea to use my dad’s computer but then I asked it without telling it we were using my dad’s computer. It said to use something called python! I don’t know how because a python is a snake my mom won’t buy me from the Pet store and snakes don’t have fingers and you need fingers to use computers. We eventually realized that python was a language like English or Spanish. Except instead of using the language to talk to my friend Juan, I use it to talk to a computer. I know thirty words in Spanish. Hola, Adios, Por favor, Gracias, Si, No, Amigo, Familia, Gato, Perro, Casa, Comido, Jugar, Escuela, Libro, Pelota, Feliz, Triste, Ayuda, Sol, Luna, Estrella, Agua, Arbol, Flor, Color, Numero, Tren, Coche, Avion

## Results

We started writing our code in the python language. Oh! I forgot that I actually know fourty words in Spanish because I can count to ten! Uno, Dos, Tres, Cuatro, Cinco, Seis, Siete, Ocho Nueve, Diez! Thankfully Juan knows English because it’s hard to talk with only forty words. We didn’t know any words in Python so we asked Chat GPT how to talk to the computer and we typed exactly what it told us to type. I thought it didn’t work but then we realized we forgot to type a “}” after a letter “+” and before another line. Python is really hard even when you’re typing directly from Chat GPT! My dad is only away from his computer for thirty minutes so we had to type FAST!

We used the code from Chat GPT, we were able to use it to find out the temperature of a 1 metre cube in our room! It made a whirring sound which I was afraid my dad would hear while he was in the kitchen and not at his computer. After running the script, we did get a very funny looking picture we put in Figure 4. We think that the purple means cold and the orange-yellow means hot so we want to stay away from the orange-yellow part. I think Chat GPT said that my room started out colder than it actually was because that part of my room was not that much hotter than the rest. I think that might have been a mistake with the initial condition.

The python code provided by ChatGPT worked I think, it wasn’t exactly right but I think that’s why it is a numerical and not analytical solution. I said that earlier! If you were paying attention. Numerical is not exactly the same as it is in real life so I think we did it good enough. We had finally found a way to find the cold spot in our room that worked! We thought about solving it numerically on paper when we found another four pieces of paper then we gave up almost immediately! It was A LOT of calculations and it would have taken us a VERY long time to do them. I know we would have done them PERFECTLY because we don’t make mistakes.

The python code provided by ChatGPT worked I think, it wasn’t exactly right but I think that’s why it is a numerical and not analytical solution. I said that earlier! If you were paying attention. Numerical is not exactly the same as it is in real life so I think we did it good enough. We had finally found a way to find the cold spot in our room that worked! We thought about solving it numerically on paper when we found another four pieces of paper then we gave up almost immediately! It was A LOT of calculations and it would have taken us a VERY long time to do them. I know we would have done them PERFECTLY because we don’t make mistakes.

Figure 4 shows the solution to the heat equation for a 1m sized room after a very short amount of time with uniform initial conditions. Our room is larger than 1m squared but I think if we stretch out the picture on our computer it can be big enough to cover our ENTIRE room! We wanted to stretch out the picture on the BIG SCREEN TV, but couldn’t figure it out and had to use our imaginations. My room I think is at least eight times bigger than 1m.

We were very happy to find a solution to such a hard problem; we were now able to extend this to my bedroom to answer the original question and stay cool. It still wasn’t that much cooler. But it WAS COLDER! We almost IMMEDIATELY encountered a problem with the answer. The computers answer was for only a region where the boundary conditions were zero! My room’s walls aren’t zero degrees! AND the end time of the program was ONE SECOND! I think there are at least THOUSANDS of seconds in a day! Because sixty seconds in sixty minutes in one hour with twenty four hours in a day adds up to A LOT! Except you don’t add, you multiply those numbers!

## Conclusion

We did solve the heat equation to find where the coolest part in my room was during summer, even IF it was only for one meter, a second and an approximation instead of a solution. We did realized that I wasn’t even really in my room most of the time, and that trying to figure out where it was hot made more heat which should be in our heat equation! Most of the time I was either at school, playing with my friends outside or over at Aaron’s house because he has a pool WITH A SLIDE! And when I was at home, I would usually be in the living room. And at night, it was cool enough that I didn’t really have to worry about the heat. There were mosquitoes which I think are VERY ANNOYING, but I’ll have to solve the mosquito’s problem in another paper! There wasn’t much reason in solving the heat equation for my room since I wouldn’t even be in it half the time. I still think that it was REALLY fun solving a hard math problem like a heat equation! I like math!

## Acknowledgements

Me and Aaron came up with most of what to say for this paper, we still had Oscar actually write and typeset all of this up for us. We only talked to Chat GPT to talk to the computer for us because he speaks computer. I don’t know if Chat GPT was a boy or girl because he I mean she I mean I don’t know didn’t tell me. Oscar was even more annoyed than when we got him to write up our previous paper.

## References

1. Lee, Wiener – Me and My Best Friend Aaron Prove the 3n+1 Problem even though it is a REALLY Hard Problem, Journal of Astrological Big Data Ecology, Chapter 16 of Et al.: Because not all research Deserves a Nobel Prize
2. Heat Equation – https://en.wikipedia.org/wiki/Heat_equation, Wikipedia
3. Lee, Wiener – A Particle Physics Model of Why My Room is Never Clean and Why I Shouldn’t have to Clean it Up Everyday Like Mommy Tells Me To, Journal of Astrological Big Data Ecology, Chapter 15 of Et al.: Be- cause not all research Deserves a Nobel Prize
4. Freely, Jackson, Cupid, Cedric the Fourth – A Total Synthesis of Tesseractane: The First Preparation of a 4-Dimensional Natural Product, Journal of Immaterial Science
5. ChatGPT – https://chat.openai.com/, OpenAI

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