Classification and Separation of Variables

Partial differential equations in physics are classified by their type -- elliptic, parabolic, or hyperbolic -- each corresponding to fundamentally different physical phenomena. Separation of variables reduces PDEs to systems of ODEs, leveraging the symmetry of the domain.

Classification of Second-Order PDEs

Definition6.1Elliptic, Parabolic, and Hyperbolic PDEs

A second-order linear PDE $Au_{xx} + 2Bu_{xy} + Cu_{yy} + \cdots = 0$ is classified by the discriminant $\Delta = B^2 - AC$ : elliptic if $\Delta < 0$ (e.g., Laplace equation $\nabla^2 u = 0$ ), parabolic if $\Delta = 0$ (e.g., heat equation $u_t = \alpha u_{xx}$ ), hyperbolic if $\Delta > 0$ (e.g., wave equation $u_{tt} = c^2 u_{xx}$ ). Each type has distinct well-posedness requirements: elliptic needs boundary conditions, parabolic needs initial + boundary, hyperbolic needs initial conditions on a spacelike surface.

ExampleSeparation of Variables for Laplace's Equation

In spherical coordinates, $\nabla^2 u = 0$ separates as $u(r,\theta,\phi) = R(r)\Theta(\theta)\Phi(\phi)$ . This yields: $\Phi'' + m^2\Phi = 0$ (azimuthal), $\frac{1}{\sin\theta}\frac{d}{d\theta}(\sin\theta\Theta') + [\ell(\ell+1) - m^2/\sin^2\theta]\Theta = 0$ (associated Legendre), and $r^2R'' + 2rR' - \ell(\ell+1)R = 0$ (Euler-type). Solutions: $R = Ar^\ell + Br^{-(\ell+1)}$ , $\Theta\Phi = Y_\ell^m(\theta,\phi)$ (spherical harmonics).

Green's Functions for PDEs

Definition6.2Green's Function for the Laplacian

The Green's function $G(\mathbf{r}, \mathbf{r}')$ for $-\nabla^2$ on domain $\Omega$ with Dirichlet boundary conditions satisfies $-\nabla^2 G = \delta(\mathbf{r} - \mathbf{r}')$ with $G|_{\partial\Omega} = 0$ . The solution to $-\nabla^2 u = f$ with $u|_{\partial\Omega} = g$ is $u(\mathbf{r}) = \int_\Omega G(\mathbf{r},\mathbf{r}')f(\mathbf{r}')\,d^3r' + \oint_{\partial\Omega} g(\mathbf{r}')\frac{\partial G}{\partial n'}\,dS'$ . In free space ( $\mathbb{R}^3$ ): $G_0 = 1/(4\pi|\mathbf{r}-\mathbf{r}'|)$ .

RemarkMethod of Images

For simple geometries, the Green's function is constructed by the method of images: $G = G_0 + H$ where $H$ is harmonic and chosen so $G|_{\partial\Omega} = 0$ . For a half-space $z > 0$ : $G(\mathbf{r},\mathbf{r}') = \frac{1}{4\pi|\mathbf{r}-\mathbf{r}'|} - \frac{1}{4\pi|\mathbf{r}-\mathbf{r}''|}$ where $\mathbf{r}''$ is the mirror image of $\mathbf{r}'$ across $z = 0$ .