Y y q y in this case 2 pxq y sin mxdx

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Unformatted text preview: aplace’s equation ∂2u ∂2u + 2 = 0. ∂x2 ∂y (5.4) 2. u(x, y ) satisfies the homogeneous boundary conditions u(0, y ) = u(a, y ) = u(x, 0) = 0. 3. u(x, y ) satisfies the nonhomogeneous boundary condition u(x, b) = f (x), where f (x) is a given function. Note that the Laplace equation itself and the homogeneous boundary conditions satisfy the superposition principle—this means that if u1 and u2 satisfy these conditions, so does c1 u1 + c2 u2 , for any choice of constants c1 and c2 . Our method for solving the Dirichlet problem consists of two steps: Step I. We find all of the solutions to Laplace’s equation together with the homogeneous boundary conditions which are of the special form u(x, y ) = X (x)Y (y ). By the superposition principle, an arbitrary linear superposition of these solutions will still be a solution. Step II. We find the particular solution which satisfies the nonhomogeneous boundary condition by Fourier analysis. To carry out Step I, we substitute u(x, y ) = X (x)Y (y ) into Laplaces equation (5.3) and obtain X (x)Y (y ) + X (x)Y (y ) = 0. Next we separate variables, puttin...
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This document was uploaded on 01/12/2014.

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