Parabolic and Hyperbolic Equations - Applications

Parabolic Applications: Diffusion processes (mass, heat, momentum), financial mathematics (Black-Scholes: $V_t + \frac{1}{2}\sigma^2 S^2V_{SS} + rSV_S - rV = 0$ ), population dynamics (Fisher-KPP: $u_t = Du_{xx} + ru(1-u)$ ), image processing (anisotropic diffusion), mean curvature flow in geometry.

Hyperbolic Applications: Acoustics, electromagnetics (Maxwell equations), elastodynamics, seismology, general relativity (Einstein equations are hyperbolic), traffic flow models, shallow water waves, gas dynamics (Euler equations).

ExampleSystems of Conservation Laws

Hyperbolic systems $\mathbf{u}_t + \mathbf{F}(\mathbf{u})_x = 0$ model:

Gas dynamics: $\rho_t + (\rho v)_x = 0$ , $(\rho v)_t + (\rho v^2 + p)_x = 0$
Shallow water: $h_t + (hv)_x = 0$ , $(hv)_t + (hv^2 + gh^2/2)_x = 0$

Shocks form generically, requiring entropy conditions and weak solutions.

Coupled Systems: Reaction-diffusion (parabolic) models pattern formation (Turing instability), climate models couple parabolic diffusion with hyperbolic advection, population dynamics involves both local diffusion and long-range dispersal (integro-differential parabolic equations).

Remark

Modern applications: Machine learning uses parabolic equations for generative models (diffusion models), hyperbolic PDEs appear in meta-learning and neural ODEs. Hamilton-Jacobi equations (related to hyperbolic conservation laws) connect optimal control to PDEs.