F(x1, x2, …, xn, u, ∂u/∂xi, ∂²u/∂xi∂xj, …)=0

Describes how a quantity changes with respect to multiple variables simultaneously

Space variables (e.g., x,y,z)

Time variable (e.g., t)

Other parameters (e.g., age, frequency, velocity coordinates)

Field/State variable (e.g., temperature T(x,t), pressure p(x,y), wave height u(x,t))

First-order (gradients, rates of change)

Second-order (curvature, diffusion-like effects)

Higher-order (beam bending, dispersive phenomena)

Constant vs variable coefficients

Forcing/input sources (e.g., heat generation, external loads)

ut = α∇²u

Models spreading/smoothing over space with time

utt = c²∇²u

Models propagation of disturbances

∇²u = 0

Equilibrium fields (electrostatics, steady heat)

∇²u = f

Equilibrium with sources

ut + v·∇u = 0

Pure movement of a quantity with a flow

ut + uux = νuxx

Combines advection + diffusion; shocks possible

Momentum + incompressibility for fluids

Strongly coupled, nonlinear multivariable behavior

First-order (e.g., advection, Hamilton–Jacobi)

Second-order (e.g., heat, wave, Laplace/Poisson)

Higher-order (e.g., biharmonic equation)

Linear

Semilinear

Quasilinear

Fully nonlinear

Elliptic

Parabolic

Hyperbolic

Unknown depends on one independent variable

Unknown depends on multiple independent variables

Requires additional information across boundaries/initial states in more than one dimension

Specify state at initial time (e.g., u(x,0)=u0(x))

Dirichlet

Neumann

Robin (mixed)

Periodic

Geometry/region where the PDE is defined (1D/2D/3D, complex shapes)

Existence, uniqueness, and continuous dependence on data

Sufficiently smooth; satisfies PDE pointwise

Satisfies an integrated/variational form; allows less regularity

Uses generalized derivatives (distributions)

Handles nonsmoothness in first-order nonlinear PDEs

Separation of variables (common in simple geometries)

Fourier series/transform methods

Method of characteristics (first-order hyperbolic)

Green’s functions and fundamental solutions

Eigenfunction expansions

Symmetry and transform techniques

Finite difference methods (FDM)

Finite element methods (FEM)

Finite volume methods (FVM)

Spectral methods

Time-stepping schemes (explicit/implicit; stability considerations)

Energy estimates, maximum principles, conservation laws

Stability and long-time behavior

Unknown represents a field over space/time (temperature, displacement, concentration)

Derived from conservation + constitutive relations (e.g., flux laws)

Multiple PDEs for multiple unknowns (e.g., velocity + pressure)

Direction-dependent or location-dependent behavior encoded in coefficients

Heat conduction, waves, electromagnetism, quantum mechanics

Fluid flow, structural mechanics, acoustics, materials

Weather/climate, groundwater flow, ocean dynamics

Reaction–diffusion patterns, tumor growth, bioelectric fields

Option pricing PDEs (e.g., Black–Scholes)