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lec1v2 - 18.152 Introduction to PDEs Fall 2004 Prof...

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+ + 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 non-homogeneous (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 non-homogeneous case In studying these examples of PDEs we will learn how to “impose conditions” to make the problem “well-posed”, 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 }
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