Advanced Engineering Mathematics, Lecture 6.1: The Heat and Wave Equations on the Real Line

Advanced Engineering Mathematics, Lecture 6.1: The Heat and Wave Equations on the Real Line

We learn how to solve an initial value problem for the heat equation on the real line. Our approach involves two steps: (1) solve an "easier" IVP where u(x,0)=H(x), the Heavyside (or unit step) function, and (2) use that solution and superposition to solve the original problem. The solution for (1) involves the "error function" erf(x) from statistics, which is the area under a bell curve from [-x,x]. The so-called "fundamental solution" or "heat kernel" involves a decaying Gaussian distribution. We conclude with solving the wave equation on the real line, which leads us to D'Alebert's formula.

[thomasjefferson]

Well i guess I can say I'm better than you with one thing!

I love this series so much! Thank you for letting me learn this for free from you!