Click to go through the slides. Select the graph features that you would like to use. Click to graph curves as time increases. Learn about the wonderful properties of heat! Marvel at the beauty of Fourier series!
WARNING: This project has math in it! Lately I've been self-studying a field of mathematics known as "partial differential equations". One of the most important equations is the heat equation, which models the changing of heat in a medium over time. I wanted to graph solutions to this heat equation to see how it changed over time, but my favorite online graphing calculator, Desmos, was too slow! I then wondered if Scratch could run this faster (and if the advanced math could be handled in Scratch). This project showcases that result. All solutions of the heat equation are sums of exponential functions and sine functions, so I used Scratch to calculate a sufficiently large sum of these in a "Fourier series". A Fourier series is a marvelous sum of functions that closely approximates another function. In fact, any function on a bounded interval can be approximated by just a sum of sine waves! Fourier coefficients require an integral to calculate, so I used a trapezoidal approximation with 75 subintervals to do the job. The lines you graph in this project model the heat in a metal bar over time. Any initial temperature distribution in the bar will eventually converge to a linear monotonic temperature distribution as time gets large. This project was loved by @LeiIani and @Rosyda. Comment "Fourier" if you read all of this ;) This might seem boring or complicated, but it's really fun, trust me! ===== CREDITS ===== INSPIRATION Partial Differential Equations Fourier Series The Heat Equation EDUCATION Joseph Fourier (for discovering Fourier series) Martin Braun (for his book on differential equations) Peter Olver (for his book on partial differential equations) All my teachers (for helping discover my love for math) #Math
