Bessel Sturm-Liouville Problem

Bessel Sturm-Liouville Problem#

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

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

with Dirichlet boundary conditions: $\( u(0) = 0, \quad u(1) = J_\nu(1) \)$

The exact solution to this problem is the Bessel function of the first kind of order \(\nu\), denoted as \(J_\nu(x)\). For \(\nu = 3\), the solution is \( J_3(x) = \frac{1}{\pi}\int^{\pi}_0 \cos(3\tau) - x\sin(\tau)\,d\tau \).

Mathematical Background#

Bessel’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 Bessel’s equation, we have:

  • \(p(x) = x^2\)

  • \(q(x) = x\)

  • \(r(x) = \frac{1}{x}\)

  • \(\lambda = -\nu^2\)

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:

  • \(A = x^2 L + x I_{FC} G + (x^2 - \nu^2) I\)

  • \(L\) 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 Bessel function of the first kind of order \(\nu\):

\(J_\nu(x) = \frac{1}{\pi}\int^{\pi}_0 \cos(\nu\tau) - x\sin(\tau)\,d\tau\).