lec13-1 - Laplace equation in Cartesian coordinates The...

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Unformatted text preview: Laplace equation in Cartesian coordinates The Laplace equation is written 2 = 0 For example, let us work in two dimensions so we have to find ( x , y ) from, 2 x 2 + 2 y 2 = 0 We use the method of separation of variables and write ( x , y ) = X ( x ) Y ( y ) X 00 X + Y 00 Y = 0 Laplace equation in Cartesian coordinates, continued Again we have two terms that only depend on one independent variable, so Y 00 Y =- k 2 This is called a Helmholtz equation (weve seen in before), and we can write it Y 00 + k 2 Y = 0 The we have another equation to solve, X 00- k 2 X = 0 Here k is real and k Could we have done this a different way? Yes! Laplace equation in Cartesian coordiates, continued We could have a different sign for the constant, and then Y 00- k 2 Y = 0 The we have another equation to solve, X 00 + k 2 X = 0 We will see that the choice will determine the nature of the solutions, which in turn will depend on the boundary conditions Steady-state temperature in a semi-infinite plate Imagine a metal plate bounded at y = 0 but extending to infinity...
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lec13-1 - Laplace equation in Cartesian coordinates The...

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