Compute Symbolic Eigenvalues

Specify a 1D Laplacian operator.

In[1]:=

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

Specify a homogeneous Dirichlet boundary condition.

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

Find expressions for the 5 smallest eigenvalues on the interval .

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DEigenvalues[{\[ScriptCapitalL], \[ScriptCapitalB]}, u[x], {x, a, b}, 5]

Out[3]=

Specify an Airy operator.

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

Find the 5 smallest eigenvalues and corresponding eigenfunctions.

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

The eigenvalues are roots of a transcendental equation.

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vals[[1]] // TraditionalForm

Out[6]//TraditionalForm=

Compute a transcendental eigenvalue with high precision.

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N[vals[[1]], 500] // TraditionalForm

Out[7]//TraditionalForm=

Visualize the eigenfunctions.

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Plot[Evaluate[funs + Range[5]], {x, 0, 1}, ImageSize -> Medium, PlotTheme -> {"Business", "Bare"}, AspectRatio -> 1]

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