Generate an Eigenfunction Expansion

Compute the eigenfunction expansion of the function with respect to the basis provided by a Laplacian operator with Dirichlet boundary conditions on the interval .

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basis = DEigensystem[{-Laplacian[u[x], {x}], DirichletCondition[u[x] == 0, True]}, u[x], {x, 0, \[Pi]}, 6, Method -> "Normalize"][[2]]

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Compute the Fourier coefficients for the function .

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f[x_] := E^(-x) x^2 (\[Pi] - x) Sin[4 x]

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coeffs = (Table[Integrate[f[x] basis[[i]], {x, 0, Pi}], {i, 6}] // FullSimplify);

Define as the partial sum of the expansion.

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eigexp[x_, n_] := Sum[coeffs[[i]] basis[[i]], {i, n}]

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eigexp[x, 3] // N

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Compare the function with its eigenfunction expansion for different values of .

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Table[Plot[{f[x], eigexp[x, i]} // Evaluate, {x, 0, Pi}], {i, 3, 6}]

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