lec1v2 - + + + + 18.152 - Introduction to PDEs , Fall 2004...

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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 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 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 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 } 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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lec1v2 - + + + + 18.152 - Introduction to PDEs , Fall 2004...

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