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Unformatted text preview: + + + + 18.152  Introduction to PDEs , Fall 2004 Prof. Gigliola Stalani Lecture 1  Introduction and Basic Facts about PDEs The Content of the Course Definition of Partial Differential Equation (PDE) Linear PDEs V V V V V V V V V V V V V V V V V V V V homomgeneous nonhomogeneous (no forcing term) (with forcing term) V V V V V V V V V V V V V V V V V V V V V with variable and constant coecients T T T T T T T T T T T T T T T T T T T T T T parabolic example * hyperbolic example elliptic example (diffusion equation) (wave equation) (Laplace equation) (heat equation) 2 u t = u xx , > 0 T T T T T T T T T T T T T T T T T T T T T u tt = c u xx u xx = 0 ) and nonhomogeneous case In studying these examples of PDEs we will learn how to impose conditions to make the problem wellposed, we will introduce fundamental mathematical concepts like distribu tions, Fourier Transform, and Fourier Series. These tools are by now classical, but still heavily used in the study of more complex PDEs, in particular, the nonlinear ones. What is a partial differential equation? This is an equation involving a function u ( x 1 , . . . , x n ) of n variables and its partial derivatives up to order m : F ( u, u x 1 , . . . , u x n , . . . , u x i 1 x i 2 , . . . , u x i 1 x i 2 ...x i m ) = 0 , i j { 1 , . . . , n } This functional defines the equation by involving u and its partial derivatives. In this case m is known as the order of the equation....
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 Fall '04
 GigliolaStaffilani

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