7 Laplace’ s Equation 7.1 Green’ s Function SLIDE 10 血n2D If u=log(V(-Io)2+(y-yo)2 hen岩+=0(x,y)≠(xo,) In 3D If hen+分+=0(x,y,2)≠(x0,,20) Proof: Just differentiate and see! Note 7 Greens function for Laplaces equation In the next few slides, we will use an informal semi-numerical approach to deriving the integral form of Laplace's equation. We do this inpart because such a derivation lends insight to the subsequent numerical procedures To start, recall from basic physics that the potential due to a point charge is related only to the distance between the point charge and the evaluation point In 2-D the potential is given by the log of the distance, and in 3-D the potential is inversely proportion to the distance. The precise formulas are given on the slide. A little more formally, direct differentiation reveals that satisfies the 2-D Laplace's equation everywhere except = To, y= yo and u(, y, 2) (x-x0)2+(y-y0)2+(2-20)2 satisfies the 3-D Laplace's equation everywhere except I=Io, y= yo, 2=20 These functions are sometimes referred to as Greens functions for Laplace's equation b Exercise 1 Show by direct differentiation that the functions in(4)and(5) satisfy V-u=0, in the appropriate dimension almost everywhere7 Laplace’s Equation 7.1 Green’s Function Slide 10 In 2D If u = log (x − x0)2 + (y − y0)2  then ∂2u ∂x2 + ∂2u ∂y2 = 0 ∀ (x, y) = (x0, y0) In 3D If u = √ 1 (x−x0)2+(y−y0)2+(z−z0)2 then ∂2u ∂x2 + ∂2u ∂y2 + ∂2u ∂z2 = 0 ∀ (x, y, z) = (x0, y0, z0) Proof: Just differentiate and see! Note 7 Green’s function for Laplace’s equation In the next few slides, we will use an informal semi-numerical approach to deriving the integral form of Laplace’s equation. We do this inpart because such a derivation lends insight to the subsequent numerical procedures. To start, recall from basic physics that the potential due to a point charge is related only to the distance between the point charge and the evaluation point. In 2-D the potential is given by the log of the distance, and in 3-D the potential is inversely proportion to the distance. The precise formulas are given on the slide. A little more formally, direct differentiation reveals that u(x, y) = log (x − x0)2 + (y − y0)2 (4) satisfies the 2-D Laplace’s equation everywhere except x = x0, y = y0 and u(x, y, z) = 1 (x − x0)2 + (y − y0)2 + (z − z0)2 (5) satisfies the 3-D Laplace’s equation everywhere except x = x0, y = y0, z = z0. These functions are sometimes referred to as Green’s functions for Laplace’s equation.  Exercise 1 Show by direct differentiation that the functions in (4) and (5) satisfy ∇2u = 0, in the appropriate dimension almost everywhere. 7