📂Partial Differential Equations

Hyperbolic Partial Differential Equations

Definition^{1 2}

Consider the following 2nd order linear partial differential equation for $u(t,x)$.

$$Au_{tt} + Bu_{tx} + Cu_{xx} + Du_{t} + Eu_{x} + Fu + G = 0\qquad (ABC \ne 0) \tag{1}$$

Here, the coefficients $A, \dots, G$ are functions of $(t,x)$. $\Delta = B^{2} - 4AC$ is called the discriminant. A partial differential equation $(1)$ with a positive discriminant is called a hyperbolic PDE.

$$(1) \text{ is called hyperbolic, if } \Delta (t,x) \gt 0.$$

Explanation

In fact, it is rarely called a hyperbolic partial differential equation, and is commonly referred to simply by its phonetic translation, hyperbolic PDE. The origin of the name, of course, comes from the hyperbola.

A conic section $Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0$ is a hyperbola if it satisfies $B^{2} - 4AC \gt 0$.

In a narrow sense, it refers to the wave equation.

$$u_{tt} - \Delta u = 0 \qquad (\Delta = 0^{2} - 4(1)(-1) = 4)$$