We’ve gone over Heat Equation, Wave Equation, and let’s talk about some more stuff.

• damped heat equation
• damped wave equation
• two-dimensional heat equation

What if, Fourier Series, but exponential?

This also motivates Discrete Fourier Transform.

Also Complex Exponential.

Review

Recall again that if we have a periodic function, we’ve got:

\begin{equation} f(x) = \sum_{k=0}^{\infty} a_{k} \sin \qty( \frac{2\pi k}{l} x) + b_{n} \cos \qty( \frac{2\pi k x}{L}) \end{equation}

We note that this breaks individually into the sign and cosine series depending of the function’s oddness.

Complex Fourier Series

This will begin by feeling like a notation rewrite:

Fourier Transform

heat equation on the entire line

\begin{equation} \pdv{u}{t} = \frac{1}{2} \pdv[2]{u}{x} \end{equation}

We can try to find a:

\begin{equation} U(0,x) = f(x) \end{equation}

if we write:

\begin{equation} \hat{U}(t,\lambda) = \int e^{-i x \lambda} U(t,x) \dd{x} \end{equation}

which means we can write, with initial condtions:

\begin{equation} \hat{U} (t, \lambda) = \hat{f}(\lambda) e^{- t \frac{\lambda^{2}}{2}} \end{equation}

We want to reach a close form:

\begin{equation} U (t, x) = \frac{1}{\sqrt{2\pi} t} \int_{-\infty}^{\infty} f(y) e^{-\frac{(x-y)^{2}}{2t}} \dd{y} \end{equation}