SecondOrder.tex

\section{PDEs of Second Order}

Linear PDEs of second order have the form
\begin{align*}
	\sum_{i,j=1}^n a_{ij}\frac{\partial ^2}{\partial x_i\partial x_j}u+\sum_{i=1}^nb_i\frac{\partial}{\partial x_i}u + cu = f
\end{align*}

The equations fall into these categories:
\emph{Elliptic} (potential problem), \emph{parabolic} (heat equation, diffusion)
and \emph{hyperbolic} (wave equation, linearised supersonic flow).

\symbolicsubsection{Splitting the Solution}

Given a second order linear differential operator $L$,
we have the PDE $Lu = f \text{ in }\Omega$ with $u = g \text{ on }\partial\Omega$.

\begin{enumerate}
	\item We try to find a particular solution {\color{blue}$Lu_p = f$ in $\Omega$}, satisfying only the PDE and neglecting boundary conditions
	\item{
		To solve the original problem,
		we need an additional summand $u_r$ taking care of boundary values to receive the solution $u_p + u_r$.
		However, $u_r$ still needs to solve the PDE but due to linearity, this reduces to a homogeneous problem:
		\begin{align*}
			L(u_p + u_r) = f + Lu_r = f \Rightarrow{\color{blue} Lu_r = 0\text{ in }\Omega}
		\end{align*}
	}
	\item{
		Ensure that the boundary conditions are satisfied:
		\begin{align*}
			u_p+u_r = g \Rightarrow {\color{blue}u_r = g - u_p\text{ on }\partial\Omega}
		\end{align*}
	}
	\item{
		If the solution is not unique, we're able to find other solutions using an additional term $u_h$:
		\begin{align*}
			L(u_p + u_r + u_h) = f + Lu_h &\ \Rightarrow Lu_h = 0\text{ in }\Omega \\
			u_p + u_r + u_h = g + u_h &\ \Rightarrow u_h = 0\text{ on }\partial\Omega
		\end{align*}
	}
\end{enumerate}

For example, consider the PDE $\nabla^2 u = 4$ in $\Omega = \{(x,y)\ |\ x^2 + y^2 < 1\}$ and $u = 0.5x+0.5$ on $\partial\Omega$.
This has the particular solution $u_p(x,y)=x^2+y^2$.
To fix boundary conditions, we find a solution $u_r$ of the homogeneous problem with boundary values
$u_r = 0.5x + 0.5 - u_p(x,y) = 0.5x + 0.5 - \underbrace{1}_{\text{on }\partial\Omega} = 0.5x - 0.5$ giving the complete solution
$u(x,y) = u_p(x,y) + u_r(x,y) = x^2 + y^2 + 0.5x - 0.5$.