SturmLiouvilleEquations

The Sturm-Liouville equations are posed as follows:

Given $p(x) > 0$, $p(x)$ continuous on $[a, b]$, and $w(x) > 0$, $w(x)$ continuous on $[a, b]$, consider the differential equation:

$$\frac{d}{dx}\left(p(x) \frac{dy}{dx}\right) - q(x)y + \lambda w(x) y = 0$$

Subject to boundary conditions of the form:

$$\alpha_1 y(a) + \alpha_2 y'(a) = 0$$ $$\beta_1 y(b) + \beta_2 y'(b) = 0$$

Where:

• $\alpha_1, \alpha_2, \beta_1, \beta_2$ are constants, not all zero.
• $\lambda$ is the eigenvalue parameter.
• $y$ is the unknown function, or the eigenfunction associated with the eigenvalue $\lambda$.

The boundary conditions given here are more general than the ones previously mentioned, and they encompass a wider class of problems. Such boundary conditions can represent various physical constraints, from being held fixed, to being free, to some combination of these.

Additionally, when discussing the Sturm-Liouville problem, there are the following important properties to consider:

1. Orthogonality: If $\lambda_n$ and $\lambda_m$ are two different eigenvalues of the problem, then their corresponding eigenfunctions $y_n(x)$ and $y_m(x)$ are orthogonal with respect to the weight function $w(x)$:

$$\int_a^b y_m(x) y_n(x) w(x) dx = 0$$

2. Completeness: The set of eigenfunctions forms a complete set, which means any piecewise continuous function can be expanded as a series of these eigenfunctions.

3. Regular and Singular Points: Points where $p, q,$ and $w$ are all continuous are termed regular. If any of these are not continuous at a point, the point is termed singular.

4. Self-Adjoint Form: The Sturm-Liouville equation can be written in a self-adjoint or self-adjoint form which has special mathematical properties and is given by:

$$\frac{d}{dx}\left(p(x) \frac{dy}{dx}\right) - q(x)y = \lambda w(x) y$$