Hermite Sturm-Liouville Problem

Hermite Sturm-Liouville Problem#

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

\[ u'' - 2 u' + 2 m u = 0, \quad -1 < x < 1 \]

with Dirichlet boundary conditions: $\( u(-1) = H_m(-1), \quad u(1) = H_m(1) \)$

The exact solution to this problem is the Hermite polynomial of degree \(m\), denoted as \(H_m(x)\). For \(m = 4\), the solution is \(H_4(x) = e^{x^2} \frac{d^4}{dx^4}e^{-x^2}\).

Mathematical Background#

Hermite’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 Hermite’s equation, we have:

  • \(p(x) = e^{-x^2}\)

  • \(q(x) = 0\)

  • \(r(x) = e^{-x^2}\)

  • \(\lambda = m^2\)

Discretization#

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

The discrete system is:

\[A u = b\]

where

  • \(A = L - 2 x I_{FC} G + 2 m I\)

  • \(L\) 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 Hermite polynomial \(H_4(x) = e^{x^2} \frac{d^4}{dx^4}e^{-x^2}\).