Classification Of Linear Second Order Partial Differetiation Equation

Partial Differential Equation :Consider the general form of a second-order partial differential equation in two independent variables x and y which is linear with respect to its second-order derivatives (more precisely, we consider a semi-linear equation)


where A,B,C and F are given differentiable functions and subscripts denote partial derivatives.

The equation can be reduced to a normal form using a transformation of independent variables.


Then Eq. (1.1) takes the form



is independent of the second derivatives of u.

It can be directly verified that

so that the sign of

Definition:  Equation (1) is said to be


1.Elliptic if B2 − AC < 0 , 

2.parabolic if B2 − AC = 0 ,

3.hyperbolic if B2 − AC > 0 .


If A,B and C are functions of position, the type of a second-order PDE can be different in different regions . In three and more dimensions, second-order PDEs can be of one type in one pair of variables and of another type in the other variables, e.g., there can occur elliptic-hyperbolic equations, ultra-hyperbolic equations.