Classifying Second Order Linear Partial Differential Equations in Two Variables
Second order partial differential equations in two variables – sayand– take the formwhereare all functions of andand
Ifare constant then the equation is constant coefficient.
If G=0 then the equation is homogeneous and if G neq 0 the equation is non – constant coefficient.
If non ofare functions of u or any partial derivative of u, then the equation is linear.
Equations of the form (1) may also be classified as parabolic, hyperbolic or elliptic.
Parabolic equations describe heat flow and diffusion processes and satisfy
Hyperbolic equations describe vibrating systems and wave phenomena and satisfye.g.
Elliptic equations describe steady state phenomena and satisfye.g.
A function may be parabolic, hyperbolic or elliptic in different parts of theplane. For examplehasso is elliptic forparabolic forand parabolic forOn the other handis hyperbolic everywhere since