Find a 1D Laplacian's Symbolic Eigenfunctions

Specify a 1D Laplacian operator.

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\[ScriptCapitalL] = -Laplacian[u[x], {x}];

Specify homogeneous Dirichlet boundary conditions for the eigenfunctions.

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\[ScriptCapitalB]1 = DirichletCondition[u[x] == 0, True];

Find the five smallest eigenvalues and eigenfunctions.

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{vals, funs} = DEigensystem[{\[ScriptCapitalL], \[ScriptCapitalB]1}, u[x], {x, 0, \[Pi]}, 5];

Inspect the eigenvalues.

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vals

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Inspect the eigenfunctions.

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funs

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Visualize the eigenfunctions.

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Plot[Evaluate[funs + 2 Range[5]], {x, 0, \[Pi]}]

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Specify a homogeneous Neumann boundary condition.

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\[ScriptCapitalB]2 = NeumannValue[0, True];

Find the five smallest eigenvalues and eigenfunctions.

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{vals, funs} = DEigensystem[\[ScriptCapitalL] + \[ScriptCapitalB]2, u[x], {x, 0, \[Pi]}, 5];

Inspect the eigenvalues. Relative to the Dirichlet conditions, a zero mode has been added.

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vals

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Sines have replaced cosines in the eigenfunctions.

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funs

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Visualize the eigenfunctions.

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Plot[Evaluate[funs + 2 Range[5]], {x, 0, \[Pi]}]

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