Lecture02

# Lecture02 - Note Thursday office hours changed to 2:00 –...

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Unformatted text preview: Note: Thursday office hours changed to 2:00 – 3:30 PM. 12 Partial Differential Equations (PDEs) As mentioned in the overview, PDEs often come up in the mathematical modeling of physical phenomena in continuous media, such as the propaga- tion of sound, heat conduction, electrostatics, electrodynamics, fluid flow, and elasticity. There are several mathematical textbooks on the subject; I would recommend Lawrence C. Evans: Partial Differential Equations. AMS, 1998 to students who like to learn more about the theory of PDEs. 12.1 Basic Concepts In the Kreyszig textbook, this section is in words only. We shall follow a more formal approach here to describe the general concept. The special PDEs treated later in the lecture will not take advantage of the notation introduced here, but it is useful to be acquainted with it. We consider a function u : Ω → R , where Ω ⊂ R d is an open subset. As mentioned in the overview, one of the variables is often time, t . For the partial derivatives of u , we use the multiindex notation: α = ( α 1 , . . . , α d ) ∈ N d is a multiindex of order | α | = α 1 + ··· + α d . (42) Given a multiindex α ∈ N d , we define the partial derivative D α u := ∂ | α | u ∂x α 1 1 ··· ∂x α d d ≡ ∂ α 1 x 1 ··· ∂ α d x d u. (43) Example: d = 3, α = (2 , , 1), | α | = 3. We obtain the partial derivative D (2 , , 1) u = ∂ 3 u ∂x 2 1 ∂x 3 . (44) If the partial derivative exists, D α u : Ω → R is again a function. Partial derivatives of order k ∈ N (if they exist) are collected in a k-th order tensor, D k u := { D α u | α ∈ N d , | α | = k } , k ∈ N . (45) 10 The tensor D k u may be represented by a k-dimensional array. Note: For d > 1, there should be no confusion between D k u , k ∈ N , and D α u , α ∈ N d , because α is a multiindex and k is a scalar. Examples: D 2 u is a second-order tensor, which may be represented by a d × d matrix: D 2 u = ∂ 2 u ∂x 2 1 ··· ∂ 2 u ∂x 1 ∂x d ....
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Lecture02 - Note Thursday office hours changed to 2:00 –...

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