lec4v2 - 18.152 - Introduction to PDEs , Fall 2004 Prof....

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Unformatted text preview: 18.152 - Introduction to PDEs , Fall 2004 Prof. Gigliola Staffilani Lecture 4 - Types of PDEs and Distributions Equations of second order • Consider the up to second order case 0 = a 11 u xx + 2 a 12 u xy + a 22 u yy + a 1 u x + a 2 u y + a u where we write 2 a 12 because we should have a 12 u xy + a 21 u yx but u xy = u yx . Then set 2 a = a 12 + a 21 (where prime indicates the coefficients in a new equation that is equivalent 12 to the old, then we can drop the primed indices after transforming). The second order part of the differential operator is a 11 u xx +2 a 12 u xy + a 22 u yy = ( a 11 ∂ x 2 + 2 a 12 ∂ xy + a 22 ∂ y 2 ) u we would like to remove this 2 = ( ∂ x + a 12 ∂ y ) 2 + ( − a 12 + a 22 ) ∂ y 2 u (1) by assuming a 11 = 1 (which we can do without loss of generality). There are three cases based upon the sign of a 22 − a 2 : 12 1. a 22 − a 12 2 > 0 Set b 2 = a 22 − a 2 . We want to change variables such that the differential operator looks 12 like L = ∂ x 2 + ∂ y 2 (the equation Lu = 0 is then known as an elliptic equation). We have x ≡ αx + βy y ≡ γx + δy ∂x ∂y ∂ x u = u x + u y ∂x ∂x ∂x ∂u ∂ y u = u x + u y ∂y ∂y ∂ x = α∂ x + γ∂ y ⇒ ∂ y = β∂ x + δ∂ y This implies that in (1) we have α = 1 , β = 0 , γ = a 12 , δ = 2 x = x , y = a 22 − a 12 + ( a 22 − a 2 ) y ⇒ a 12 x 12 2. 2....
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lec4v2 - 18.152 - Introduction to PDEs , Fall 2004 Prof....

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