Parabolic and Hyperbolic Equations - Examples and Constructions

Parabolic Examples: Separation of variables for heat equation on bounded domains gives $u(x,t) = \sum a_n\phi_n(x)e^{-\lambda_n t}$ showing exponential decay of each mode. Fundamental solution (heat kernel) provides explicit Cauchy problem solutions via convolution.

Hyperbolic Examples: D'Alembert's formula $u(x,t) = \frac{1}{2}[f(x+ct) + f(x-ct)] + \frac{1}{2c}\int_{x-ct}^{x+ct}g$ solves 1D wave equation. Method of characteristics transforms first-order hyperbolic systems to ODEs along characteristic curves.

ExampleFinite Difference Methods

Parabolic: Explicit scheme $u_j^{n+1} = u_j^n + r(u_{j+1}^n - 2u_j^n + u_{j-1}^n)$ requires $r = k\Delta t/(\Delta x)^2 \leq 1/2$ for stability (CFL-like condition).

Hyperbolic: Leap frog $u_j^{n+1} = u_j^{n-1} + \frac{c\Delta t}{\Delta x}(u_{j+1}^n - u_{j-1}^n)$ requires $c\Delta t/\Delta x \leq 1$ (Courant-Friedrichs-Lewy condition).

Weak Solutions: For hyperbolic conservation laws $u_t + f(u)_x = 0$ , weak solutions allow shocks (discontinuities) satisfying Rankine-Hugoniot jump conditions. Entropy conditions select physical shocks among multiple weak solutions.

Remark

Semigroup methods unify parabolic theory: $u(t) = e^{tA}u_0$ where $A$ generates analytic semigroup. Hyperbolic equations generate $C_0$ semigroups but not analytic (reflecting finite propagation vs infinite smoothing).