A Partial Differential Equation is a Differential Equation which has more than one independent variable: $u(x,y), u(t,x,y), …$

For instance:

\begin{equation} \pdv{U}{t} = \alpha \pdv[2]{U}{x} \end{equation}

Key Intuition

• PDEs may have no solutions (unlike Uniqueness and Existance for ODEs)
• yet, usually, there are too many solutions—so… how do you describe all solutions?
• usually, there are no explicit formulas

Laplacian of \(u(x,y)\)

Laplacian of \(u(x,y)\)

Examples

Heat Equation

See Heat Equation

Wave Equation

see Wave Equation

Transport Equation

\begin{equation} \pdv{u}{t} = \pdv{u}{x} \end{equation}

Boundary Value Problem

A BVP for an ODE is defined at two different points \(x_0\) and \(x_1\) at two different values of \(l\), whereby we are given:

\begin{equation} X_0 = a, X(L) = b \end{equation}

which we use to further specify a PDE. BVPs can either have no or lots of solutions.

To aid in the discovery of solutions, for:

\begin{equation} X’’ = \lambda X \end{equation}

we have:

\begin{equation} X = \begin{cases} c_1 e^{\sqrt{\lambda}x} + c_2 e^{-\sqrt{\lambda}x}, \lambda > 0 \\ c_1 x + c_2, \lambda =0 \\ c_1 \cos \qty(\sqrt{|\lambda|}x) +c_2 \sin \qty(\sqrt{|\lambda|}x), \lambda < 0 \end{cases} \end{equation}

Fourier Decomposition

Main idea, any induction \(f(x)\) on an interval \([0, L]\) can be written as a sum:

\begin{equation} f(x) = a_0 + \sum_{k=1}^{\infty} a_{k} \cos \qty( \frac{2\pi k}{L} x) + \sum_{k=1}^{\infty} b_{k} \sin \qty( \frac{2\pi k}{L} x) \end{equation}

L-periodicity

A function is $L$-periodic if \(f(x+L) = f(x)\) for nonzero \(L\) for all \(x\). The smallest \(L > 0\) which satisfies this property is called the period of the function.

$L$-periodicity is preserved across…

more on Fourier Series.

decomposition of functions to even and odd

Suppose we have any function with period \(L\) over \([-\frac{L}{2}, \frac{L}{2}]\), we can write this as a sum of even and odd functions:

\begin{equation} f(x) = \frac{1}{2} (f(x) - f(-x)) + \frac{1}{2} (f(x) + f(-x)) \end{equation}

And because of this fact, we can actually take each part and break it down individually as a Fourier Series because sin and cos are even and odd parts.