GT MATH 4581 - CHAPTER PoEq (29 pages)

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CHAPTER PoEq



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CHAPTER PoEq

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Lecture Notes


Pages:
29
School:
Georgia Institute of Technology
Course:
Math 4581 - Math Methods in Engr
Math Methods in Engr Documents

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CHAPTER PoEq P 1 The Potential Equation In principle little will change when eigenfunction expansions are applied to potential problems In practice however the mechanics of solving such problems will become more complicated because the expansion coefficients are found as solutions of boundary value problems which often are more difficult to solve than the initial value problems arising in diffusion and wave propagation problems P 1 1 The Solution Technique A standard model for potential problems solvable with a separation of variables technique is the following formulation P 1 1 Lu uxx uyy F x y x y D x y 0 x a 0 y b u g x y x y D where D denotes the boundary of the rectangle D The operator Lu uxx uyy is known as the two dimensional Laplacian and is commonly denoted by Lu u 2 u u u is the preferred notation for the Laplacian in the mathematics literature and we shall adopt it here u 0 is known as Laplace s equation and u F for non zero F is called Poisson s equation Hence problem P 1 1 calls for the solution of Poisson s equation subject to Dirichlet data which implies that the function u is given on the boundary of the domain D 1 In order to obtain zero boundary conditions it is customary to decompose P 1 1 into two problems of the form P 1 2 u1 F x y u1 0 y 0 u1 a y 0 u1 x 0 g x 0 P 1 3 u1 x b g x b u2 0 u2 x 0 0 u2 x b 0 u2 0 y g 0 y u2 a y g a y since u x y u1 x y u2 x y then will solve P 1 1 If the boundary values of P 1 1 are continuous but not zero at the corners of D then such crude splitting of P 1 1 will introduce artificial discontinuities into problems P 1 2 and P 1 3 at the corners of D Any eigenfunction expansion will invariably show a Gibbs phenomenon at such points Such discontinuities can be avoided if the problem is so reformulated that the boundary data are zero at the corner points of D before the splitting is carried out This is easy to achieve for the Dirichlet problem if we subtract from u a function v x y which takes on the values of g at the



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