Classification of Partial Differential Equations

Partial differential equations are classified according to their order, linearity, and boundary conditions. 

The order of a partial differential equation is determined by the highest-order partial derivative present in that equation. Examples of first-, second-, and third-order partial differential equations are  

Partial differential equations are categorized into linear, quasilinear, and nonlinear equations. Consider, for example, the following second-order equation  

If the coefficients are constants or functions of the independent variables only [(.)=(x, y)l, then Eq. (6.14) is linear. If the coefficients are functions of the dependent variable and/or any of its derivatives of lower order than that of the differential equation [(.)=(.r, y, u, diddx, d/i/5y)], then the equation is quasilinear. Finally, if the coefficients are functions of derivatives of the same order as that of the equation f(.) = (x, y, u, d u/dx, du ldy, d iddxdy)], then the equation is nonlinear. In accordance with these definitions, Eq. (6.11) is linear, (6.12) is quasilinear, and (6.13) is nonlinear. 

Linear second-order partial differential equations in two independent variables are further classified into three canonical forms elliptic, parabolic, and hyperbolic. The general form of this class of equations is 

---