Classification of second-order linear equations


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 x, y. Such an equation has the form

\mathcal{L}[u](x,y) = a u_{xx} + 2b u_{xy} + c u_{yy} + d u_{x} + e u_{y} + fu = g,\quad (1)

where a, b,c,d,e,f,g are given functions of x,y and u(x,y) is the unknown function. Suppose that the coefficients a,b,c satisfy a^2 + b^2 + c^2 \ne 0. Then, the operator

\mathcal{L}_0[u](x,y) = a u_{xx} + 2b u_{xy} + c u_{yy},\quad (2)

that consists of the highest-order terms of the operator \mathcal{L} is called the principal part of \mathcal{L}. We are going to classify the equation according to the sign of \delta(\mathcal{L}) := b^2 - ac. More precisely, it turns out the following definition.

Definition 1. Let \Omega \subset \mathbb{R}^2 be an open connected set. Equation (1) is said to be hyperbolic in \Omega if \delta(\mathcal{L}) > 0 for all (x,y) \in \Omega, it is said to be parabolic if \delta(\mathcal{L}) = 0, and it is said to be elliptic if \delta(\mathcal{L}) < 0.

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.

u_{xx} - 2\sin{x} u_{xy} - \cos^2 {x} u_{yy} - \cos{x} u_y = 0,

x^2 u_{xx} - 2xy u_{xy} + y^2 u_{yy} + x u_{x} + y u_{y} = 0,

4y^2 u_{xx} + u_{yy} - \dfrac{2y}{1+y^2} (2 u_{x} - u_{y}) = 0.

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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