91Ó°ÊÓ

Eigenvalues are a fundamental part of solving differential equations in Sturm-Liouville theory. They act like special numbers that allow differential equations to have non-trivial solutions. In the context of the exercise, we explore eigenvalues for the Sturm-Liouville problem

• Given by the equation: \( y^{\prime\prime} + \lambda y = 0 \)
• Subject to boundary conditions: \( y^{\prime}(0) = 0 \) and \( y^{\prime}(m) = 0 \)

To solve for eigenvalues, assume a trial solution of the form \( y(x) = A \cos(\sqrt{\lambda}x) + B \sin(\sqrt{\lambda}x) \). By applying the boundary conditions:

• The condition \( y^{\prime}(0) = 0 \) simplifies to \( B \sqrt{\lambda} = 0 \), implying \( B = 0 \).
• The condition \( y^{\prime}(m) = 0 \) gives \( -A \sqrt{\lambda} \sin(\sqrt{\lambda}m) = 0 \).

Since \( A eq 0 \) (as \( y(x) \) must not be null), we require \( \sin(\sqrt{\lambda}m) = 0 \). Therefore, \( \sqrt{\lambda}m = n\pi \) for \( n = 1,2,3, \ldots \), giving eigenvalues \( \lambda_n = \frac{n^2\pi^2}{m^2} \).Productively identifying these eigenvalues helps find specific solution "shapes" or modes that satisfy the Sturm-Liouville problem.

Eigenfunctions represent the backbone solutions to the differential equations in the Sturm-Liouville problem once the eigenvalues are known. For each eigenvalue \( \lambda_n \), there corresponds an eigenfunction \( y_n(x) \). These eigenfunctions develop as:

• With the general form: \( y(x) = A \cos(\sqrt{\lambda}x) + B \sin(\sqrt{\lambda}x) \)
• With \( B = 0 \) due to boundary conditions, leading to \( y(x) = A \cos(\sqrt{\lambda}x) \)

For the eigenvalues \( \lambda_n = \frac{n^2\pi^2}{m^2} \), the corresponding eigenfunctions become \( y_n(x) = A_n \cos\left(\frac{n\pi x}{m}\right) \). Eigenfunctions are essential as they provide the fundamental "building blocks" for constructing solutions to differential equations over given domains. For practical applications, one can view them as waves or modes that satisfy the constraints given by the boundary conditions and the differential equation.

The orthogonality of eigenfunctions is a crucial property in Sturm-Liouville theory. It means that different eigenfunctions, corresponding to different eigenvalues, are mathematically orthogonal over a given interval (like a dot product being zero). For our eigenfunctions \( y_n(x) \) and \( y_k(x) \) associated with different eigenvalues \( \lambda_n eq \lambda_k \), they satisfy:

• The integral of their product over the interval \((0, m)\) is zero, \( \int_0^m y_n(x) y_k(x) \, dx = 0 \).

This orthogonality greatly aids in determining the coefficients for expanding functions in terms of these eigenfunctions. Essentially, it means that any function can be decomposed or expressed as a series of these eigenfunctions without interference from others. When expanding a function using a series of eigenfunctions, the orthogonality ensures that we can independently calculate each coefficient without mixes from different terms. This property is analogous to how perpendicular vectors interact in geometry.

Series expansion in the context of Sturm-Liouville problems enables us to express a function as a sum of scaled eigenfunctions over the defined interval. Given a function \( f(x) = x \) and the system’s eigenfunctions \( y_n(x) \), we approximate the function:

• As \( f(x) \approx \sum_{n=1}^{\infty} c_n y_n(x) \)

To determine the coefficients \( c_n \), use the orthogonality property:

• Integrate the product of \( f(x) \) and each eigenfunction \( y_k(x) \) over the interval \((0, m)\)

This gives \( c_k = \frac{\int_0^m f(x) y_k(x) \, dx}{\int_0^m y_k^2(x) \, dx} \). The series expansion thus systematically reconstructs the function \( f(x) \) by summing each eigenfunction properly weighted by its respective coefficient. This process provides a powerful method for solving and approximating solutions to various boundary value problems within physics and engineering. It transforms complex interactions into manageable series form, capturing essential features of the original function.