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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 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 } This functional “defines” the equation by involving u and its partial derivatives. In this case m is known as the order of the equation....
View Full Document

Page1 / 6

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online