The family of second-order linear equations are, in general, classified into three distinct types: hyperbolic, parabolic, and elliptic equations. In spite of very basic definition, we would like to concentrate on classification of linear equations for functions in two independent variables . Such an equation has the form

where are given functions of and is the unknown function. Suppose that the coefficients satisfy . Then, the operator

that consists of the highest-order terms of the operator is called the *principal part* of . We are going to classify the equation according to the sign of . More precisely, it turns out the following definition.

**Definition 1.** Let be an open connected set. Equation (1) is said to be *hyperbolic* in if for all , it is said to be *parabolic* if , and it is said to be *elliptic* if .

From this definition, we can verify that the wave equation is hyperbolic, the heat equation is parabolic, and the Laplace equation is elliptic.

**Example 2.** Three following equations are hyperbolic, parabolic and elliptic, respectively.

**References**

[1] S. Larsson, V. Thomée, *Partial Differential Equations with Numerical Methods*, Springer, 2003.

[2] Y. Pinchover, J. Rubinstein, *An Introduction to Partial Differential Equations*, Cambridge, 2005.

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