Classification of PDEs

PDEs can be classified in different ways. The classification is important because the solution methods often apply only to a specified class of PDEs. To start with, PDEs can be classified by the number of variables, e.g. Uf = which contain two independent variables, t and x. The order of a PDE is the order of the highest order derivative that appears in the PDE. For example, Ut = is a first-order PDE, whereas Ut = Uxx is a second-order PDE. In addition, it is important to make a distinction between nonlinear and linear PDEs. An example of a well-known non-linear PDE is the Navier-Stokes equation, which describes the motion of fluids. In a linear PDE, the dependent variable and its derivatives appear in a linear fashion. The linear second-order PDE in Equation (6.69) can be classified as 

The B 4AC term is referred to as the discriminant of the solution, and the behavior of the solution of Equation (6.69) depends on its sign. The smoothness of a solution is affected 

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