Chebyshev Sturm-Liouville Problem

Chebyshev Sturm-Liouville Problem#

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

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

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

The exact solution to this problem is the Chebyshev polynomial of the first kind of degree \(n\), denoted as \(T_n(x)\). For \(n=2\), the solution is \(T_2(x) = 2x^2 - 1\).

Mathematical Background#

Chebyshev’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 Chebyshev’s equation, we have:

  • \(p(x) = 1-x^2\)

  • \(q(x) = 0\)

  • \(r(x) = 1\)

  • \(\lambda = n^2\)

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:

  • \(A = (1-x^2) L - x I G + n^2 I\)

  • \(L\) is the mimetic Laplacian

  • \(G\) is the mimetic gradient

  • \(I\) is the interpolation operator from faces to centers

Boundary conditions are applied using the addScalarBC1D function.

This example is implemented in:

Results#

The numerical solution closely matches the exact solution, which is the Chebyshev polynomial \(T_2(x) = 2x^2 - 1\).

Chebyshev polynomials are important in numerical analysis and approximation theory because they:

  1. Minimize the maximum error in polynomial approximation

  2. Have roots that are optimal interpolation points (Chebyshev nodes)

  3. Are closely related to the Fourier cosine series

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