Hyperbolic Equations: Definition & Explanation

[verdigris]

(d2^u/dt^2) - (delta u) = 0 is called a hyperbolic equation.

Why is this? What makes an equation a hyperbolic equation?

 

[HallsofIvy]

The partial differential equation
$$\frac{\partial^2u}{\partial t^2}= \frac{\partial^2 u}{\partial x^2}$$
and its extension to 2 or 3 space dimensions, is called "hyperbolic" in an obvious analogy with the hyperbolic equation
$$x^2- y^2= 1$$
In addition, just as the hyperbola has two asymptotes, so the hyperbolic differential equation has two "characteristic" lines that can be used to solve the equation.

The heat (or diffusion) equation,
$$\frac{\partial u}{\partial t}= \frac{\partial u^2}{\partial x^2}$$
has only a single characteristic line and is a "parabolic" equation.

Laplace's equation,
$$\frac{\partial^2 u}{\partial x^2}+ \frac{\partial^2 u}{\partial y^2}= 0$$
has no characteristic lines and is an "elliptic" equation.

 

[tacman]

A hyperbolic equation is a type of partial differential equation that involves second-order derivatives with respect to both time and space variables. In other words, it is an equation that describes the behavior of a physical quantity that changes over time and space. The equation (d2^u/dt^2) - (delta u) = 0 is an example of a hyperbolic equation because it contains both a second-order time derivative and a second-order spatial derivative.

The name "hyperbolic" comes from the shape of the solutions to these types of equations, which are typically hyperbolic curves. These equations are often used to model wave-like phenomena, such as sound or electromagnetic waves, which exhibit a characteristic shape known as a hyperbolic curve.

One of the key features that distinguishes hyperbolic equations from other types of partial differential equations is their characteristic curves. These curves represent the paths along which information travels through the system described by the equation. In hyperbolic equations, these curves are typically smooth and well-behaved, allowing for the accurate prediction of future behavior based on initial conditions.

Additionally, hyperbolic equations are known for their ability to exhibit both oscillatory and shock-like behavior, making them useful in a wide range of applications. This is due to the fact that they can have both real and complex solutions, giving them a greater range of possible behaviors compared to other types of equations.

In summary, an equation is considered a hyperbolic equation if it involves second-order derivatives with respect to both time and space variables and has characteristic curves that are smooth and well-behaved. These equations are commonly used to model wave-like phenomena and exhibit a range of behaviors that make them useful in various applications.