Problem 4 - Advanced Engineering Mathematics

Question

Consider \(y^{\prime \prime}+\lambda y=0\) subject to the periodic boundary conditions \(y(-L)=y(L), y^{\prime}(-L)=y^{\prime}(L)\). Show that the eigenfunctions are \(\left\\{1, \cos \frac{\pi}{L} x, \cos \frac{2 \pi}{L} x, \ldots, \sin \frac{\pi}{L} x, \sin \frac{2 \pi}{L} x, \sin \frac{3 \pi}{L} x, \ldots\right\\}\) This set, which is orthogonal on \([-L, L]\), is the basis for the Fourier series.

Step-by-Step Solution

Answer

The eigenfunctions are \(\{ 1, \cos(\frac{n\pi}{L}x), \sin(\frac{n\pi}{L}x) \}_{n=1}^{\infty}\), serving as a Fourier basis on \([-L, L]\).

Step 1: Propose a general solution for the differential equation

The given differential equation is \(y'' + \lambda y = 0\). We propose a general solution of the form \(y(x) = A \cos(kx) + B \sin(kx)\), where \(k\) is the wave number that we need to determine.

Step 2: Determine the relation between \(\lambda\) and \(k\)

Substitute \(y(x) = A \cos(kx) + B \sin(kx)\) into the differential equation to obtain \(-Ak^2\cos(kx) - Bk^2\sin(kx) + \lambda A \cos(kx) + \lambda B \sin(kx) = 0\). Simplifying this gives \(-(k^2) + \lambda)(A \cos(kx) + B \sin(kx)) = 0\). Therefore, \(\lambda = k^2\).

Step 3: Apply the periodic boundary conditions

The boundary conditions are \(y(-L) = y(L)\) and \(y'(-L) = y'(L)\). Applying \(y(-L) = y(L)\) is equivalent to \(A \cos(-kL) + B \sin(-kL) = A \cos(kL) + B \sin(kL)\), leading to the condition \(\sin(kL) = 0\).

Step 4: Find the allowed values of \(k\)

The requirement \(\sin(kL) = 0\) implies that \(kL = n\pi\) for \(n = 0, 1, 2, \ldots\). Thus, \(k = \frac{n\pi}{L}\).

Step 5: Determine the eigenfunctions

For each \(n\), we have the eigenfunctions \(y_n(x) = A_n \cos\left(\frac{n\pi}{L}\right) x\) and \(y_n(x) = B_n \sin\left(\frac{n\pi}{L}\right) x\), where \(n = 0, 1, 2, \ldots\). Including the constant solution, the eigenfunctions are \[ \left\{ 1, \cos\left(\frac{\pi}{L} x\right), \cos\left(\frac{2\pi}{L} x\right), \ldots, \sin\left(\frac{\pi}{L} x\right), \sin\left(\frac{2\pi}{L} x\right), \ldots \right\} \].

Step 6: Explain Orthogonality and Fourier Basis

The set of functions \( \left\{ 1, \cos\left(\frac{n\pi}{L} x\right), \sin\left(\frac{n\pi}{L} x\right) \right\} \) is orthogonal over the interval \([-L, L]\). These functions form the basis of the Fourier series because of their orthogonality and completeness over the stated interval.

Key Concepts

Fourier Series

Fourier series are a powerful tool in mathematics for expressing complex periodic functions as sums of simpler sine and cosine waves. By breaking down a function into

• trigonometric components,
• each of which repeats periodically,
• a Fourier series allows for the analysis and manipulation of signals in various domains such as acoustics, electrical engineering, and physics.

In the context of our exercise, we find that the eigenfunctions can be expressed using a Fourier series. The family of eigenfunctions we derive from the solutions of our differential equation can be represented as sums of sine and cosine terms: \[ y(x) = A_0 + \sum_{n=1}^{\infty} \left(A_n \cos\left(\frac{n\pi}{L} x\right) + B_n \sin\left(\frac{n\pi}{L} x\right)\right) \] These components represent the basis functions for Fourier series expansion, effectively recreating any periodic function on the interval r r These eigenfunctions, therefore, demonstrate how any periodic structure can be broken into these intrinsic oscillatory patterns.

Orthogonality

Orthogonality is an essential concept when dealing with eigenfunctions, particularly within Fourier series. Two functions are considered orthogonal over a certain interval if their inner product is zero. For example, in our exercise, we see

• an orthogonal set of sine and cosine functions,
• which serves as the basis functions for Fourier expansion,
• are orthogonal on the interval \([-L, L]\).

Mathematically, for two functions \( f(x) \) and \( g(x) \) to be orthogonal over the interval \([-L, L]\), their inner product is expressed as: \[ \int_{-L}^{L} f(x)g(x) \, dx = 0 \] The concept of orthogonality ensures that each component of the Fourier series does not interfere with the others, allowing for independent control of each frequency component within the function. r r This independence is critical for accurately reconstructing the original function through its series components.

Boundary Conditions

Boundary conditions are constraints necessary for solutions of differential equations to be determined uniquely. In our exercise, the boundary conditions specified are periodic, meaning they require that

• the function and its derivative must coincide at the start and end of a chosen interval.

Concretely, for our differential equation, the boundary conditions are \[ y(-L) = y(L) \quad \text{and} \quad y'(-L) = y'(L) \] This constraint leads directly to the requirement that \( \sin(kL) = 0 \) which determines necessary conditions for the periodicity, ensuring that the eigenfunctions satisfy these periodic properties. r r By applying such boundary conditions, we ensure that the mathematical model correctly describes physical phenomena that are periodic over the interval. This condition ensures synchronized behavior of the function over its defined domain, leading to a repeatable, predictable pattern in its solutions.

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. They describe various physical phenomena, such as heat, sound, and light propagation, making them vital for modeling real-world problems. In our exercise, we look into the differential equation: \[ y'' + \lambda y = 0 \] The task is to determine functions (eigenfunctions) that satisfy this equation under given conditions. For this, we use a generic form \( y(x) = A \cos(kx) + B \sin(kx) \), proposing a solution format that accommodates the potential behaviors, oscillations, or decays in the system. r r The solving process involves

• finding the relationship between the parameters \( \lambda \) and \( k \),
• and applying boundary conditions to find specific forms of \( k \) which satisfy the system.

This equation and its solutions highlight the role differential equations play in understanding the dynamic behaviors of systems they represent, predicting how certain properties evolve under given initial or boundary conditions.