Chapter 9: Separation of Variables

### Boundary and Initial Conditions for PDEs

#### The Main Idea

In physics situations, the classification and types of boundary conditions are typically straightforward: if there are two time derivatives, the equation is hyperbolic and we will need two initial conditions on the entire spatial region to make the solution unique; if there is only a single time derivative, the equation is parabolic and we will need only a single initial condition; if the equation has no time derivatives, the equation is elliptic and the solutions are qualitatively different from the previous two cases. It is easiest to understand the elliptic case from an explicit example.

#### Types of Boundary Conditions

In addition to initial conditions, we will need boundary conditions on the spatial variables. The two main type of boundary conditions encountered in physics are Dirichlet, when the value of the solution of the PDE goes to zero on a continuous portion of the boundary, and Neumann, when the normal (to the boundary) derivative of the solution goes to zero on a continuous portion of the boundary. Most of the theorems below are true when the (spatial) boundary conditions are either Dirichlet or Neumann on each piecewise smooth piece of the boundary.

#### Optional Mathematical Details

##### Elliptic Equations

Example: Poisson's Equation

$$\nabla^2 \psi(x_k) = f(x_k)$$

Theorem: If $\psi(x_k)$ satisfies Poisson's equation throughout a closed, bounded region $R$ and satisfies Dirichlet conditions on the the boundary $\partial R$ of $R$, then $\psi$ is unique.

Theorem: If $\psi(x_k)$ satisfies Poisson's equation throughout a closed, bounded region $R$ and satisfies Neumann conditions on the the boundary $\partial R$ of $R$, then $\psi$ is unique up to an additive constant.

Corollary: If the boundary is piecewise smooth, you can specify either Dirichlet or Neumann conditions on each piece. If Dirichlet conditions are satisfied on at least one piece then $\psi$ is unique.

Corollary: If the region $R$ is unbounded (in some or all directions) but $\psi = o(r^{-{1/2}})$ as $r\rightarrow\infty$ (i.e. $\psi$ falls off faster than $r^{-1/2}$) in the unbounded directions, then $\psi$ is unique.

##### Parabolic Equations

Example: Inhomogeneous Diffusion Equation

$$\left({\partial \over \partial t} -k\nabla^2\right) \psi(t,x_k) = f(x_k)$$

Theorem: If $\psi(t, x_k)$ satisfies the inhomogeneous diffusion equation throughout a closed, bounded region $R$ and satisfies either Dirichlet or Neumann conditions on the the boundary $\partial R$ of $R$, and $\psi$ satisfies the initial condition

$$\psi(t=0, x_k) = g(x_k)$$

then $\psi$ is unique.

Corollary: If the spatial boundary is piecewise smooth, you can specify either Dirichlet or Neumann conditions on each piece.

Corollary: If the region $R$ is unbounded (in some or all spatial directions) but $\psi = o(r^{-{1/2}})$ as $r\rightarrow\infty$ (i.e. $\psi$ falls off faster than $r^{-1/2}$) in the unbounded directions, then $\psi$ is unique.

##### Hyperbolic Equations

Example: Inhomogeneous Wave Equation

$$\left({-1 \over v^2}{\partial^2 \over \partial t^2} +\nabla^2\right) \psi(t,x_k) = f(x_k)$$

Theorem: If $\psi(t, x_k)$ satisfies the inhomogeneous wave equation throughout a closed, bounded region $R$ and satisfies either Dirichlet or Neumann conditions on the the boundary $\partial R$ of $R$, and $\psi$ satisfies the two initial conditions $$\psi(t=0, x_k) = g(x_k)$$ $${\partial\psi \over \partial t}(t=0, x_k) = h(x_k)$$ then $\psi$ is unique.

Corollary: If the spatial boundary is piecewise smooth, you can specify either Dirichlet or Neumann conditions on each piece.

Corollary: If the region $R$ is unbounded (in some or all spatial directions) but $\psi = o(r^{-{1/2}})$ as $r\rightarrow\infty$ (i.e. $\psi$ falls off faster than $r^{-1/2}$) in the unbounded directions, then $\psi$ is unique.