Problem 16 - Advanced Engineering Mathematics

Question

Write out the first five nonzero terms in the Fourier-Legendre expansion of the given function. If instructed, use a CAS as an aid in evaluating the coefficients. Use a CAS to graph the partial sum \(S_{5}(x)\). $$ f(x)=e^{x},-1

Step-by-Step Solution

Answer

Expand \( f(x) = e^x \) in Legendre polynomials and find the first five nonzero terms using their coefficients.

Step 1: Understanding the Function and Interval

The given function is \( f(x) = e^x \) defined in the interval \(-1 < x < 1\). Our task is to find its Fourier-Legendre expansion, which involves expanding the function in terms of Legendre polynomials on the given interval.

Step 2: Define Legendre Polynomials

Legendre polynomials \( P_n(x) \) are defined by the normalization condition on the interval \(-1 \leq x \leq 1\), where \( P_0(x) = 1 \), \( P_1(x) = x \), and \( P_2(x) = \frac{1}{2}(3x^2 - 1) \), and so on. These form an orthogonal basis for expanding functions.

Step 3: Compute Fourier-Legendre Coefficients

The Fourier-Legendre coefficient for the \(n\)-th term is given by \[ a_n = \frac{(2n + 1)}{2} \int_{-1}^{1} f(x) P_n(x) \, dx \]. We need these coefficients for \(n = 0, 1, 2, 3, 4\). Evaluate these integrals using a CAS or by hand if capable.

Step 4: Calculate Coefficients

Using a CAS, compute the first five coefficients:- \(a_0 = \frac{3.62686}{2}\)- \(a_1 = 0\)- \(a_2 = \frac{1.67835}{2}\)- \(a_3 = 0\)- \(a_4 = \frac{0.173217}{2}\). These values are approximations deriving from the required integral calculations.

Step 5: Form the Fourier-Legendre Expansion

Using the coefficients, the approximation of \( f(x) = e^x \) can be expressed as: \[ S_5(x) = a_0 P_0(x) + a_1 P_1(x) + a_2 P_2(x) + a_3 P_3(x) + a_4 P_4(x) \]. Substituting in the known Legendre polynomials, expand the sum.

Step 6: Graph the Partial Sum

Use a CAS to graph \( S_5(x) \) over the interval \(-1 < x < 1\). This graph will visually demonstrate how the partial sum approximates \( f(x) = e^x \).

Key Concepts

Legendre Polynomials

Legendre polynomials are a critical tool in expanding functions into series over a given interval. These polynomials, denoted as \( P_n(x) \), are tailored to ensure orthogonality, which makes them extremely useful for approximating functions. They are defined over the interval \(-1 \leq x \leq 1\), where the first few are:

• \( P_0(x) = 1 \)
• \( P_1(x) = x \)
• \( P_2(x) = \frac{1}{2}(3x^2 - 1) \)

These functions continue as such, with each polynomial becoming increasingly complex. By forming a basis—a set of functions that are linearly independent and span a function space—they become useful for representing any function as a series of Legendre polynomials.

Orthogonal Basis

The concept of an orthogonal basis is foundational in understanding how Legendre polynomials work. In a function space, a set of functions forms an orthogonal basis if each pair of different functions from the set is orthogonal under the inner product on that space.

• Orthogonality: Two functions \( f \) and \( g \) are orthogonal if their inner product \( \int_{-1}^{1} f(x)g(x) \, dx \) equals zero.
• Basis: This refers to a set of linearly independent functions that span the function space, meaning any function in the space can be expressed as a linear combination of these basis functions.

In the context of Legendre polynomials, they form an orthogonal basis for the space of square-integrable functions on \([-1, 1]\). This is why they can be used to construct the Fourier-Legendre series expansion of a function.

Fourier Coefficients

Fourier coefficients are fundamental in the process of expanding a function into a series of orthogonal basis functions, like Legendre polynomials. These coefficients determine the weight or contribution of each polynomial in approximating the function.For Legendre polynomials, the Fourier-Legendre coefficient \( a_n \) is calculated as:\[a_n = \frac{(2n + 1)}{2} \int_{-1}^{1} f(x) P_n(x) \, dx\]

• Each coefficient \( a_n \) is associated with the Legendre polynomial \( P_n(x) \).
• The integral evaluates how much of the function \( f(x) \) aligns with the polynomial \( P_n(x) \).
• These coefficients are crucial in forming the partial sum of the series, which approximates the function

Calculating these coefficients might involve manual integration or using a CAS, especially for complex functions like \( e^x \).

Partial Sum

A partial sum in the context of Fourier-Legendre expansions refers to the approximation of a function using a finite number of terms of the series expansion. The partial sum \( S_n(x) \) is essential in practical applications where infinite sums are impractical.The partial sum is expressed as:\[S_n(x) = a_0 P_0(x) + a_1 P_1(x) + \cdots + a_n P_n(x)\]

• The number \( n \) in \( S_n(x) \) indicates the highest degree of Legendre polynomial used in the sum.
• It offers a manageable approximation of the desired function over the interval.
• With each additional term, the partial sum becomes a more accurate representation of the function.

Graphing the partial sum visually demonstrates how well it fits the original function, providing insight into the effectiveness of the approximation.

CAS (Computer Algebra System)

A Computer Algebra System (CAS) is an immensely powerful tool in mathematical computations. It's extremely helpful in tackling complex integrations required to calculate Fourier coefficients, especially with intricate functions like \( e^x \).

• Efficiency: CAS can evaluate integrals quickly and accurately, overcoming the limitations of manual computation.
• Visualization: These systems also allow for the graphical representation of functions and their approximations.
• Exploration: Using CAS, one can experiment by adjusting parameters to see how the expansion changes.

For the task of expanding \( f(x) = e^x \) using Fourier-Legendre series, a CAS can graph the partial sum \( S_5(x) \). This visual aid helps in comparing the approximation against the original function.