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Course: ME 303, Fall 2009
School: W. Alabama
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OF CLASSIFICATION LINEAR PDEs OF SECOND ORDER Second order linear PDEs with two independent variables x; y have the general form: Au + Bu + Cu + Du + Eu + Fu = G xx xy yy x y where the coe cients A; B; C; D; E; F; and G are functions of x and y or they could be constants. The PDEs can be classi ed as either: 1: Hyperbolic if B 2 , 4AC 3: Elliptic if 0 2: Parabolic if B 2 , 4AC = 0 B 2 , 4AC 0 The PDEs are de...

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OF CLASSIFICATION LINEAR PDEs OF SECOND ORDER Second order linear PDEs with two independent variables x; y have the general form: Au + Bu + Cu + Du + Eu + Fu = G xx xy yy x y where the coe cients A; B; C; D; E; F; and G are functions of x and y or they could be constants. The PDEs can be classi ed as either: 1: Hyperbolic if B 2 , 4AC 3: Elliptic if 0 2: Parabolic if B 2 , 4AC = 0 B 2 , 4AC 0 The PDEs are de ned to be homogeneous if G = 0, otherwise they are de ned to be nonhomogeneous. The PDEs are de ned to be nonlinear if they contain terms like 2 u @u = 1 @u @x 2 @x EXAMPLES OF HYPERBOLIC, PARABOLIC, AND ELLIPTIC The PDEs 1. Di usion Heat Equation u =u xx t 1 is a second-order linear PDE with coe cients: A=1 Therefore B=0 C=0 F =0 D=0 G=0 E = ,1 B 2 , 4AC = 0 for all x and t. The Di usion Equation is Parabolic. 2. The Wave Equation u =u xx tt is a second-order linear PDE with coe cients: A=1 Therefore B=0 E =0 C = ,1 F =0 G=0 D=0 B 2 , 4AC = 4 0 for al...

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