A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena in, for example, engineering science, quantum mechanics and financial mathematics. Examples include the heat equation, time-dependent Schrödinger equation and the Black–Scholes equation.
Definition To define the simplest kind of parabolic PDE, consider a real-valued function u ( x , y ) {\displaystyle u(x,y)} of two independent real variables, x {\displaystyle x} and y {\displaystyle y} . A second-order, linear, constant-coefficient PDE for u {\displaystyle u} takes the form
A u x x + 2 B u x y + C u y y + D u x + E u y + F = 0 , {\displaystyle Au_{xx}+2Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}+F=0,}
where the subscripts denote the first- and second-order partial derivatives with respect to x {\displaystyle x} and y {\displaystyle y} . The PDE is classified as parabolic if the coefficients of the principal part (i.e. the terms containing the second derivatives of u {\displaystyle u} ) satisfy the condition
B 2 − A C = 0. {\displaystyle B^{2}-AC=0.}
Usually x {\displaystyle x} represents one-dimensional position and y {\displaystyle y} represents time, and the PDE is solved subject to prescribed initial and boundary conditions. Equations with B 2 − A C < 0 {\displaystyle B^{2}-AC<0} are termed elliptic while those with B 2 − A C > 0 {\displaystyle B^{2}-AC>0} are hyperbolic. The name "parabolic" is used because the assumption on the coefficients is the same as the condition for the analytic geometry equation A x 2 + 2 B x y + C y 2 + D x + E y + F = 0 {\displaystyle Ax^{2}+2Bxy+Cy^{2}+Dx+Ey+F=0} to define a planar parabola. The basic example of a parabolic PDE is the one-dimensional heat equation
u t = α u x x , {\displaystyle u_{t}=\alpha \,u_{xx},}
where u ( x , t ) {\displaystyle u(x,t)} is the temperature at position x {\displaystyle x} along a thin rod at time t {\displaystyle t} and α {\displaystyle \alpha } is a positive constant called the thermal diffusivity. The heat equation says, roughly, that temperature at a given time and point rises or falls at a rate proportional to the difference between the temperature at that point and the average temperature near that point. The quantity u x x {\displaystyle u_{xx}} measures how far off the temperature is from satisfying the mean value property of harmonic functions. The concept of a parabolic PDE can be generalized in several ways. For instance, the flow of heat through a material body is governed by the three-dimensional heat equation
u t = α Δ u , {\displaystyle u_{t}=\alpha \,\Delta u,}
where
Δ u := ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 + ∂ 2 u ∂ z 2 , {\displaystyle \Delta u:={\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}},}
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