Question

NUMERICAL QUADRATURE (see Exercise 3.1.23). (a) Derive the quadrature formula $$ \int_{a}^{b} f(x) d x \approx \frac{h}{6}(f(a)+f(b))+\frac{h}{3} \sum_{i=1}^{n-1} f(a+i h)+\frac{2 h}{3} \sum_{i=1}^{n} f(a+(2 i-1) h / 2) $$ (where \(h=(b-a) / n)\) by applying the Runge-Kutta method to the initial value problem $$ y^{\prime}=f(x), \quad y(a)=0 $$ This quadrature formula is called Simpson's Rule. (b) \(L\) For several choices of \(a, b, A, B, C,\) and \(D\) apply (A) to \(f(x)=A+B x+C x+D x^{3}\), with \(n=10,20,40,80,160,320\). Compare your results with the exact answers and explain what you find. (c) \(\mathrm{L}\) For several choices of \(a, b, A, B, C, D,\) and \(E\) apply (A) to \(f(x)=A+B x+C x^{2}+\) \(\overline{D x}^{3}+E x^{4}\), with \(n=10,20,40,80,160,320\). Compare your results with the exact answers and explain what you find.

Short answer

Based on the step-by-step solution provided, explain how you derived and applied Simpson's Rule using the Runge-Kutta method and discuss the accuracy of the rule. We derived Simpson's Rule using the Runge-Kutta method by considering an initial value problem with the given ODE \(y^{\prime} = f(x)\) and \(y(a)=0\). We divided the interval \([a, b]\) into n equal subintervals, each of length \(h = (b-a)/n\). By applying the second-order Runge-Kutta method, we derived Simpson's Rule to approximate the integral: $$ \int_{a}^{b} f(x) d x \approx \frac{h}{6}(f(a)+f(b))+\frac{h}{3} \sum_{i=1}^{n-1} f(a+i h)+\frac{2 h}{3} \sum_{i=1}^{n} f(a+(2 i-1) h / 2) $$ We then applied Simpson's Rule to the given functions in parts (b) and (c) for various values of n, such as \(n=10,20,40,80,160,320\). We compared the results with the exact answers and observed that the approximation error decreased as n increased, indicating that Simpson's Rule is an effective and accurate technique for numerical integration.

Step-by-step solution

Understand the Runge-Kutta method

The Runge-Kutta method is a popular technique for solving ordinary differential equations (ODEs). In this case, we will apply the method to derive Simpson's Rule for numerical integration.

State the given ODE and formula

We're given the following ODE: $$ y^{\prime} = f(x), \quad y(a)=0 $$ We need to find the quadratic approximating the integral: $$ \int_{a}^{b} f(x) d x $$ using Simpson's Rule.

Apply the Runge-Kutta method and derive Simpson's Rule

To derive Simpson's Rule using the Runge-Kutta method, we will consider the given initial value problem and divide the interval \([a, b]\) into n equal subintervals, each of the length \(h = (b-a)/n\). Applying the second-order Runge-Kutta method with \(y(a)=0\), the integral can be approximated by: $$ \int_{a}^{b} f(x) d x \approx \frac{h}{6}(f(a)+f(b))+\frac{h}{3} \sum_{i=1}^{n-1} f(a+i h)+\frac{2 h}{3} \sum_{i=1}^{n} f(a+(2 i-1) h / 2) $$ #Phase 2: Applying Simpson's Rule to the given functions#

Applying Simpson's Rule for functions in parts (b) and (c)

For parts (b) and (c), we will use the derived Simpson's Rule to approximate the integral of the given functions for various values of n, such as \(n=10,20,40,80,160,320\). Then, we will compare the results with the exact answers and explain our findings.

Comparing the results and explaining the findings

After applying Simpson's Rule for the functions in parts (b) and (c), we can compare the results with the exact answers. As we increase the value of n, the approximation should be getting closer to the exact answer, indicating that Simpson's Rule is an effective numerical integration technique. When comparing the results, we should observe that the error decreases as n increases, which shows the accuracy of Simpson's Rule.

Key concepts

Numerical Quadrature

Numerical quadrature is a mathematical technique used to approximate the definite integral of a function. It's especially useful when the integral cannot be evaluated analytically. Numerical quadrature works by summing up areas of simple geometric figures, such as rectangles, trapezoids, or parabolas, that approximate the area under the curve described by the function.

One of the most famous quadrature rules is Simpson's Rule, which falls under this category. Simpson's Rule approximates the integral by using parabolic arcs. It works by subdividing the total area into multiple segments, computing the area under each segment with a parabolic approximation, and summing them up to obtain an overall approximation of the integral.

• Simpson's rule combines the results from the midpoint and trapezoidal rules into a single, more accurate expression.
• The effectiveness of Simpson's rule depends on the number of segments, denoted by 'n.' The more segments used, the better the approximation tends to be.
• As seen in the exercise, increasing the number of segments for a function can lead to results that closely align with the exact integral.

This rule is widely utilized because it offers a good balance between simplicity and accuracy for many practical applications.

Runge-Kutta Method

The Runge-Kutta method is among the most popular approaches for solving ordinary differential equations (ODEs), especially when a high level of accuracy is required. This method is a part of a family of iterative techniques, which incrementally build better approximations to the solution.

Here's a simple breakdown of the method:

• It starts at an initial point and takes small steps along the curve of the solution, using the derivative information to progress.
• Each step is calculated using several intermediate computations, often involving weighted averages, which significantly improve accuracy over simpler methods like Euler's.
• There are various forms of the Runge-Kutta methods, classified by their order. The higher the order, the more precise each step can be, but also the more complex the calculations.

In the exercise, a specific version of the Runge-Kutta method was applied to derive Simpson's Rule. By segmenting the interval and evaluating the midpoint and endpoints of each segment, this numerical technique can be used to approximate the integral of a function effectively.

Students must clearly understand the steps involved in the Runge-Kutta method to apply it to derive numerical integration rules like Simpson's Rule.

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are equations involving a function and its derivatives. They describe the relationship between a varying quantity and its rates of change and arise naturally in many fields such as physics, engineering, and economics.

Understanding ODEs involves several steps:

• Identifying the order of the equation based on the highest derivative present.
• Looking for an initial condition that helps determine a particular solution to the differential equation.
• Applying appropriate analytical or numerical methods to find the solution.

In the provided exercise, the ODE is relatively simple, with the first derivative of the unknown function being equal to another known function. Even so, this form neatly illustrates the process of solving an ODE using a numerical approach like the Runge-Kutta method.

Students should practice solving a wide range of ODEs to gain a solid foundation in recognizing patterns and applying the correct methods, both analytical and numerical, to find solutions.

Numerical Integration

Numerical integration is concerned with finding approximate solutions to the definite integrals of functions. These techniques become invaluable when the antiderivative of a function is difficult or impossible to find analytically.

Numerical integration has several key features:

• Using geometric shapes to approximate the area under a curve.
• Improving accuracy by increasing the number of subdivisions.
• Incorporating the function's values at different points to refine the approximation.

The exercise demonstrates numerical integration through Simpson's Rule—an application of numerical quadrature which makes use of parabolic interpolation for approximation. Students should recognize how increasing the number of intervals ('n') leads to a more accurate value, an echo of the concept that in numerical methods, refinement of inputs often leads to refinement in results.

It's essential to master different numerical integration methods to tackle various types of integrals effectively, especially in science and engineering problems where precision is paramount.