Partial Differential Equations (PDEs): Classification, Analytical Solutions, and Numerical Discretization

Partial Differential Equations (PDEs) are the mathematical foundation of continuum physics, fluid dynamics, electromagnetism, quantitative finance, and structural mechanics. Unlike Ordinary Differential Equations (ODEs) which involve functions of a single independent variable, PDEs describe multivariable dynamic systems where unknown functions depend on multiple spatial and temporal dimensions ( $x, y, z, t$ ).

This guide details the classification of second-order linear PDEs (Elliptic, Parabolic, and Hyperbolic), classical analytical solution methods (Separation of Variables, Fourier Transforms, and Green's Functions), and modern numerical discretization frameworks (Finite Difference Method FDM, Finite Element Method FEM, and Physics-Informed Neural Networks PINNs).

1. Quick-Reference: Second-Order PDE Classification Matrix

A general second-order linear PDE in two independent variables has the form:

A \frac{\partial^2 u}{\partial x^2} + B \frac{\partial^2 u}{\partial x \partial y} + C \frac{\partial^2 u}{\partial y^2} + D \frac{\partial u}{\partial x} + E \frac{\partial u}{\partial y} + F u = G

The mathematical behavior and appropriate numerical solver depend strictly on the Discriminant $\Delta = B^2 - 4AC$ :

+-----------------------------------------------------------------------------------------+
|                               SECOND-ORDER PDE CLASSIFICATION                           |
+-----------------------------------------------------------------------------------------+
| Type        | Discriminant (B² - 4AC) | Canonical Archetype       | Physical Phenomenon | Solver Stability |
+-------------+-------------------------+---------------------------+---------------------+------------------+
| Elliptic    | Δ < 0                   | Laplace / Poisson Eq      | Steady-State Equil. | Global Boundary  |
|             |                         | ∇²u = f(x, y)             | (Electrostatics,    | (Elliptic matrix |
|             |                         |                           |  steady heat/flow)  |  solvers / PCG)  |
+-------------+-------------------------+---------------------------+---------------------+------------------+
| Parabolic   | Δ = 0                   | Heat / Diffusion Eq       | Dissipative Systems | Time-Marching    |
|             |                         | ∂u/∂t = α ∇²u             | (Thermal conduction,| (Crank-Nicolson, |
|             |                         |                           |  Black-Scholes)     |  implicit A-stab)|
+-------------+-------------------------+---------------------------+---------------------+------------------+
| Hyperbolic  | Δ > 0                   | Wave Equation             | Propagating Waves   | Explicit / CFL   |
|             |                         | ∂²u/∂t² = c² ∇²u          | (Acoustics, seismic,| (Courant limit   |
|             |                         |                           |  electromagnetics)  |  c · Δt/Δx ≤ 1)  |
+-----------------------------------------------------------------------------------------+

2. Classical Analytical Methods

Analytical Solution Frameworks:
[ Boundary & Initial Value Problem (PDE + BCs + ICs) ]
                          |
        +-----------------+-----------------+
        |                                   |
[ Separation of Variables ]       [ Integral Transforms (Fourier / Laplace) ]
- Assumes u(x,t) = X(x) · T(t)    - Converts PDE into algebraic ODE in frequency domain
- Produces Sturm-Liouville ODEs   - Inverts via contour integration / IFFT
        |                                   |
        +-----------------+-----------------+
                          |
                          v
         [ Orthogonal Series Expansion (Fourier / Bessel) ]

The Heat Equation via Separation of Variables

For 1D heat conduction along a rod of length $L$ with Dirichlet boundary conditions $u(0, t) = u(L, t) = 0$ :

\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}

Setting $u(x, t) = X(x) T(t)$ yields:

\frac{T'(t)}{\alpha T(t)} = \frac{X''(x)}{X(x)} = -\lambda^2

Solving the spatial eigenvalue problem gives eigenfunctions $X_n(x) = \sin\left(\frac{n \pi x}{L}\right)$ with eigenvalues $\lambda_n = \frac{n \pi}{L}$ , producing the general solution:

u(x, t) = \sum_{n=1}^\infty c_n \exp\left(-\alpha \left(\frac{n \pi}{L}\right)^2 t\right) \sin\left(\frac{n \pi x}{L}\right)

where Fourier coefficients $c_n = \frac{2}{L} \int_0^L f(x) \sin\left(\frac{n \pi x}{L}\right) dx$ match the initial temperature profile $f(x) = u(x, 0)$ .

3. Numerical Discretization Paradigms

When geometry, non-linear coefficients, or complex boundary conditions render analytical solutions impossible, computational algorithms discretize continuous domain fields:

+---------------------------+-----------------------------------+------------------------+
| Discretization Method     | Mathematical Foundation           | Best Suited For        |
+---------------------------+-----------------------------------+------------------------+
| Finite Difference (FDM)   | Taylor series expansion on grids  | Regular grid geometries|
| Finite Element (FEM)      | Variational weak form over meshes | Complex CAD geometries |
| Finite Volume (FVM)       | Conservative flux integral balance| Fluid dynamics (CFD)   |
| Physics-Informed NNs      | Neural network loss penalized     | High-dimensional &     |
| (PINNs)                   | by PDE residual $\| \mathcal{N}(u) - f \|^2$ | inverse parameter est. |
+---------------------------+-----------------------------------+------------------------+

Crank-Nicolson Implicit Time-Marching Scheme

To avoid the severe Courant-Friedrichs-Lewy (CFL) time-step restriction of explicit forward Euler ( $\Delta t \le \frac{\Delta x^2}{2\alpha}$ ), the Crank-Nicolson method averages spatial derivatives at time steps $n$ and $n+1$ :

\frac{u_i^{n+1} - u_i^n}{\Delta t} = \frac{\alpha}{2} \left[ \frac{u_{i+1}^{n+1} - 2u_i^{n+1} + u_{i-1}^{n+1}}{\Delta x^2} + \frac{u_{i+1}^n - 2u_i^n + u_{i-1}^n}{\Delta x^2} \right]

Crank-Nicolson is unconditionally stable ( $A$ -stable) and second-order accurate in both space and time ( $\mathcal{O}(\Delta t^2 + \Delta x^2)$ ), requiring only a fast tridiagonal Thomas algorithm solve per time-step.

References

Evans, L. C. (2010). Partial Differential Equations (2nd ed.). American Mathematical Society (Graduate Studies in Mathematics, Vol. 19).
Strauss, W. A. (2007). Partial Differential Equations: An Introduction (2nd ed.). John Wiley & Sons.
LeVeque, R. J. (2007). Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM.
Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686–707.