📂Partial Differential Equations

Parabolic Partial Differential Equation

Definition^{1 2}

Consider the following second-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 discriminant of $0$ is referred to as a parabolic PDE.

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

Explanation

In fact, it is rare to explicitly call it a parabolic partial differential equation, and it is commonly referred to simply as [parabolic PDE]. Of course, the origin of the name is naturally from parabolas.

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

In a narrow sense, it specifically refers to the heat equation.

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