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Unformatted text preview: + + + + 18.152  Introduction to PDEs , Fall 2004 Prof. Gigliola Staﬃlani 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 coeﬃcients 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 = κu xx , κ > 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
 Derivative, Partial differential equation, PDEs, VVVV VVVV VVVV

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