Problem 3 - A First Course in the Numerical Analysis of Differential Equations

Question

Construct the Gaussian quadrature formulae for the weight function \(\omega(t) \equiv\) \(1,0 \leq t \leq 1\), of orders two, four and six.

Step-by-Step Solution

Answer

For Gaussian quadrature: order 2 points at \( \frac{1}{2}\pm\frac{1}{2\sqrt{3}} \); order 4 points at \( \frac{1}{2}\pm\frac{1}{2}\sqrt{\frac{3}{5}} \); order 6 generally requires numerical methods for points and weights.

Step 1: Understanding Gaussian Quadrature

Gaussian quadrature is a method of approximating integrals of the form \( \int_{a}^{b} f(x) \, dx \) by a weighted sum of function values at specified points within the domain \([a, b]\). The weights and points are chosen such that the formula is exact for polynomials of degree up to \(2n-1\), where \(n\) is the number of points used.

Step 2: Identifying the Weight Function

The given weight function is \( \omega(t) = 1 \) for \(0 \leq t \leq 1\). This is a uniform weight function across the integration domain.

Step 3: Constructing 2-Point Gaussian Quadrature

For a 2-point Gaussian quadrature, we need to find two points \( t_1 \) and \( t_2 \), and corresponding weights \( w_1 \) and \( w_2 \), such that the integral approximates polynomials of degree up to 3. Solving the system of equations based on orthogonality and weight distribution gives \( t_1 = \frac{1}{2} - \frac{1}{2\sqrt{3}}, t_2 = \frac{1}{2} + \frac{1}{2\sqrt{3}} \), and \( w_1 = w_2 = \frac{1}{2} \).

Step 4: Constructing 4-Point Gaussian Quadrature

For a 4-point Gaussian quadrature, we find four points \( t_1, t_2, t_3, t_4 \) with corresponding weights \( w_1, w_2, w_3, w_4 \) to approximate polynomials up to degree 7. Solving the equations yields points \( t_1 = \frac{1}{2} - \frac{1}{2} \sqrt{\frac{3}{5}}, t_2 = \frac{1}{2} + \frac{1}{2} \sqrt{\frac{3}{5}}, t_3 = \frac{1}{2} - 0, t_4 = \frac{1}{2} + 0 \), and weights \( w_1 = w_4 = \frac{5}{18}, w_2 = w_3 = \frac{4}{9} \).

Step 5: Constructing 6-Point Gaussian Quadrature

For a 6-point Gaussian quadrature, we need six points and weights that approximate polynomials up to degree 11. The system equates integrals with derived polynomials to solve for these parameters, typically utilizing Hermite polynomials and numerical solutions for symmetry and roots. The solutions tend to be non-trivial and solved using specialized computational methods.

Key Concepts

Weight Function

In numerical integration, a weight function is crucial. It influences how the integrand is treated across the integration domain. For Gaussian quadrature, the weight function determines the importance of each point within the interval of integration. In this exercise, the weight function is simple:

• \( \omega(t) = 1 \) for \( 0 \leq t \leq 1 \)

This means each part of the interval is equally significant. This is known as a uniform weight function. The role of the weight function can drastically change the integral's computation. It can either simplify or complicate the process, depending on its form.If the weight function is more complex, it can require special techniques or transformations for effective integration.

Numerical Integration

Numerical integration helps calculate definite integrals, especially when an analytical solution is complex. Gaussian quadrature is a precise numerical integration method. It estimates the integral of a function over an interval by summing its weighted values at specific points.This approach is optimal because:

• The chosen points and weights result in the exact computation for polynomials of degree up to \(2n-1\), where \(n\) is the number of chosen points.
• It maximizes efficiency and accuracy using fewer points than other methods.

Gaussian quadrature is ideal for problems where the integrand is well-approximated by a polynomial. It reduces computational work while maintaining high precision.

Polynomial Approximation

Polynomial approximation is at the core of many numerical methods. It involves representing a function as a polynomial to make calculations more manageable. In Gaussian quadrature, the goal is to approximate the integrand as a polynomial, which allows precise integration. Why polynomials? Because:

• Polynomials are easy to evaluate and differentiate.
• They provide a systematic approach to achieve desired accuracy.

In practice, solving integrals involves matching the function's essence with the best approximate polynomial. As a result, Gaussian quadrature bases its efficiency on how well polynomials approximate the given function.

Hermite Polynomials

Hermite polynomials play a significant role in constructing Gaussian quadrature rules, especially when no simple algebraic solution exists. They are orthogonal polynomials over a certain interval with a specific weight function. Why Hermite polynomials?

• They simplify finding the roots and weights needed for higher-order Gaussian quadrature.
• Their orthogonality property makes computations more straightforward.

In more complex problems, Hermite polynomials allow a systematic approach to solve for points and weights. They ensure higher accuracy, addressing non-trivial solutions that arise in complex integrals. Solving such systems frequently involves computational tools designed to handle polynomial equations efficiently.