Classification of Second-Order PDEs: Hyperbolic, Parabolic, Elliptic Hyperbolic PDEs: A hyperbolic PDE is a second-order ordinary differential equation...
Classification of Second-Order PDEs: Hyperbolic, Parabolic, Elliptic
Hyperbolic PDEs:
A hyperbolic PDE is a second-order ordinary differential equation (ODE) that exhibits the following characteristics:
The characteristics of the solution are determined by the sign of the discriminant of the characteristic polynomial.
If the discriminant is positive, the solution has two distinct real roots.
If the discriminant is negative, the solution has two distinct complex roots.
Examples:
Heat equation: ∂u/∂t = α ∂²u/∂x² for positive α (hyperbolic)
Wave equation: ∂²u/∂t² = c² ∂²u/∂x² for c² > 1 (hyperbolic)
Parabolic PDEs:
A parabolic PDE is a second-order ODE that exhibits the following characteristics:
The characteristics of the solution are determined by the sign of the discriminant of the characteristic polynomial.
If the discriminant is negative, the solution is a parabola.
If the discriminant is positive, the solution has two distinct real roots.
Examples:
KdV equation: ∂²u/∂t² = u for k > 0 (parabolic)
Poisson's equation: ∂²u/∂x² = u for k = 1 (parabolic)
Elliptic PDEs:
An elliptic PDE is a second-order ODE that exhibits the following characteristics:
The characteristics of the solution are determined by the sign of the discriminant of the characteristic polynomial.
If the discriminant is positive, the solution is an ellipse.
If the discriminant is negative, the solution has two distinct complex roots.
Examples:
Wave equation with periodic boundary conditions: ∂²u/∂t² = c² ∂²u/∂x² for -∞ < x < ∞ with periodic boundary conditions (elliptic)
Heat equation with insulated boundaries: ∂²u/∂t² = α ∂²u/∂x² for -∞ < x < ∞ with insulated boundaries (elliptic)