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Unformatted text preview: Math 124A October 12, 2010 Viktor Grigoryan 5 Classification of second order linear PDEs Last time we derived the wave and heat equations from physical principles. We also saw that Laplaces equation describes the steady physical state of the wave and heat conduction phenomena. Today we will consider the general second order linear PDE and will reduce it to one of three distinct types of equations that have the wave, heat and Laplaces equations as their canonical forms. Knowing the type of the equation allows one to use relevant methods for studying it, which are quite different depending on the type of the equation. One should compare this to the conic sections, which arise as different types of second order algebraic equations (quadrics). Since the hyperbola, given by the equation x 2- y 2 = 1, has very different properties from the parabola x 2- y = 0, it is expected that the same holds true for the wave and heat equations as well. For conic sections, one uses change of variables to reduce the general second order equation to a simpler form, which are then classified according to the form of the reduced equation. We will see that a similar procedure works for second order PDEs as well. The general second order linear PDE has the following form Au xx + Bu xy + Cu yy + Du x + Eu y + Fu = G, where the coefficients A,B,C,D,F and the free term G are in general functions of the independent vari- ables x,y , but do not depend on the unknown function u . The classification of second order equations depends on the form of the leading part of the equations consisting of the second order terms. So, for sim- plicity of notation, we combine the lower order terms and rewrite the above equation in the following form Au xx + Bu xy + Cu yy + I ( x,y,u,u x ,u y ) = 0 . (1) As we will see, the type of the above equation depends on the sign of the quantity ( x,y ) = B 2 ( x,y )- 4 A ( x,y ) C ( x,y ) , (2) which is called the discriminant for (1). The classification of second order linear PDEs is given by the following. Definition 5.1. At the point ( x ,y ) the second order linear PDE (1) is called i) hyperbolic , if ( x ,y ) > ii) parabolic , if ( x ,y ) = 0 ii) elliptic , if ( x ,y ) < Notice that in general a second order equation may be of one type at a specific point, and of another type at some other point. In order to illustrate the significance of the discriminant = B 2- 4 AC , we next describe how one reduces equation (1) to a canonical form. Similar to the second order algebraic equations, we use change of coordinates to reduce the equation to a simpler form. Define the new variables as = ( x,y ) , = ( x,y ) , with J = det x y x y 6 = 0 . (3) We then use the chain rule to compute the terms of the equation (1) in these new variables....
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