Partial Differential Equations

Partial differential equations are the language in which most of mathematical physics is written. Quantum mechanics, fluid dynamics, electromagnetism, elasticity, finance, biology — wherever a quantity depends on more than one independent variable and the local rate of change relates to the local value, you get a PDE. The mathematics is dramatically richer than the ODE case: TYPE matters (elliptic, parabolic, hyperbolic), boundary conditions matter, function-space subtleties matter, and even the existence and uniqueness of solutions can be delicate.

This section covers the analytical foundations. Numerical methods get their own home in the finite elements section (which already has 1D and 2D Poisson, heat, elasticity, and unstructured-mesh content), and the related differential equations section covers ODE-specific machinery.

Topics

• The three classical equations — heat, wave, and Laplace. The taxonomy elliptic / parabolic / hyperbolic, why their solutions behave so differently (smoothing vs propagation vs maximum principle), and side-by-side numerical demonstrations on a common domain.
• Method of characteristics — first-order PDEs reduce to ODEs along characteristic curves. The linear transport equation, the quasilinear Burgers equation, and the geometric reason characteristics intersect to produce shocks. The Rankine-Hugoniot jump condition and entropy selection of physical shocks.
• Separation of variables — the workhorse analytical technique: assume $u(x,t)=X(x)T(t)$, separate, solve eigenvalue problems for $X$, ODEs for $T$, and sum. Sturm-Liouville structure, completeness, and how Fourier series fall out as a special case.
• Green's functions — fundamental solutions for linear PDEs: $\hat{L}G=\delta$ and then $u=\int Gf$. The Green's functions for Poisson, Helmholtz, heat, and wave equations; the method of images for bounded domains; the bridge to scattering and response theory.

Numerical methods

• Finite difference methods — discretizing derivatives; convergence order; the von Neumann stability analysis; the $r\le 1/2$ bound for FTCS heat and the CFL bound for upwind advection.
• Spectral methods — Chebyshev collocation and Fourier methods; exponential convergence for smooth solutions; the Boyd reference. Hits machine precision with O(20) degrees of freedom on smooth problems.
• Method of lines — discretize space first, integrate in time as a large ODE system; the bridge between PDE and ODE methods that unlocks adaptive time stepping and stiff solvers.
• Operator splitting — Lie-Trotter and Strang splittings; trade multiplicative complexity for additive structure; the textbook 2×2 test gives clean 1st- and 2nd-order convergence.

Specific equations

• Time-dependent Schrödinger — split-step Fourier method and Crank-Nicolson; norm-conserving unitary evolution; coherent-state oscillation on a harmonic potential as the canonical test.
• Conservation laws and Riemann problems — weak solutions, Rankine-Hugoniot, entropy conditions; Godunov's method as Riemann-solver-at-cell-interfaces; the path to modern shock-capturing CFD.

The unifying frame is that PDEs are studied at three layers: ANALYTICAL (Green's functions, separation of variables, transforms; tells you what the solution should be), QUALITATIVE (maximum principles, energy methods, regularity; tells you how the solution behaves without computing it), and NUMERICAL (finite differences, finite elements, spectral methods; tells you what the solution actually is for a specific problem). A solid understanding of the analytical layer makes the numerical choice obvious; without it, numerical methods are recipes without intuition.