Parabolic partial differential equation

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A parabolic partial differential equation is a second-order partial differential equation of the form

Au_{xx} + 2B_{xy} + Cu_{yy} + Du_{x} + Eu_{y} + F \quad

in which the matrix Z=\begin{bmatrix}A&B\\B&C\end{bmatrix} has the determinant equal to 0.

Some examples of parabolic partial differential equations are Schrödinger's equation and the heat equation.

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