Laguerre Sturm-Liouville Problem

Laguerre Sturm-Liouville Problem#

This example solves the Laguerre differential equation, which is a classic Sturm-Liouville problem:

\[x u'' + (1 - x) u' + n u = 0, \quad 0 < x < 2\]

with Dirichlet boundary conditions: $\( u(0) = L_n(2), \quad u(2) = L_n(2) \)$

The exact solution to this problem is the Laguerre polynomial of degree \(n\), denoted as \(L_n(x)\). For \(n = 4\), the solution is \(L_4(x) = \frac{e^x}{4!} \frac{d^4}{dx^4} (e^{-x}x^4) \).

Mathematical Background#

Laguerre’s differential equation is a special case of the Sturm-Liouville problem, which has the general form:

\[\frac{d}{dx}\left(p(x)\frac{du}{dx}\right) + q(x)u + \lambda r(x)u = 0\]

For Laguerre’s equation, we have:

  • \(p(x) = xe^{-x}\)

  • \(q(x) = 0\)

  • \(r(x) = e^{-x}\)

  • \(\lambda = n\)

Discretization#

The equation is discretized using mimetic finite difference operators. The spatial derivative operators are constructed with a specified order of accuracy \(k\).

The discrete system is:

\[A u = b\]

where:

  • \(A = x L + (1 - x) I_{FC} G + n I\)

  • \(L\) is the mimetic Laplacian

  • \(G\) is the mimetic gradient

  • \(I_{FC}\) is the interpolation operator from faces to centers

  • \(I\) is the identity matrix

Boundary conditions are applied using RobinBC.

This example is implemented in:

Results#

The numerical solution closely matches the exact solution, which is the Laguerre polynomial \(L_4(x) = \frac{e^x}{4!} \frac{d^4}{dx^4} (e^{-x}x^4) \).