Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

Apply the Improved Euler Method to the equation \(y^{\prime}=y, y(0)=1\), with \(h=0.2,0.1,0.05,0.01,0.005\) to approximate the solution on the interval \([0,1]\). (Note that the exact solution is \(y=e^{x}\), so \(y(1)=e\).) Compute the error in approximating \(y(1)\) (see Example 3 and the subsequent discussion) and fill in the following table. For the Improved Euler Method, is the error proportional to \(h, h^{2}\), or some other power of \(h\) ? $$ \begin{array}{lcc} \hline h & \begin{array}{c} \text { Error from } \\ \text { Euler's Method } \end{array} & \begin{array}{c} \text { Error from } \\ \text { Improved Euler Method } \end{array} \\ \hline 0.2 & 0.229962 & 0.015574 \\ 0.1 & 0.124540 & \\ 0.05 & 0.064984 & 0.001091 \\ 0.01 & 0.013468 & 0.000045 \\ 0.005 & 0.006765 & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The error from the Improved Euler Method is proportional to \(h^2\).

Step 1: Review Improved Euler Method Formula

The Improved Euler Method, also known as Heun’s method, is a numerical technique to solve first-order differential equations of the form \(y' = f(x, y)\). It uses the formula: \( y_{n+1} = y_n + \frac{h}{2} (k_1 + k_2) \), where \( k_1 = f(x_n, y_n) \) and \( k_2 = f(x_n + h, y_n + hk_1) \). For our equation, \( f(x, y) = y \).

Step 2: Initialize Parameters

Given the initial condition \( y(0) = 1 \), we begin with \( x_0 = 0 \) and \( y_0 = 1 \). We will calculate \( y \) at \( x = 1 \) using different step sizes \( h = 0.2, 0.1, 0.05, 0.01, 0.005 \).

Step 3: Calculate Approximation for h = 0.2

For \( h = 0.2 \), perform iterations to compute \( y \): 1. Find \( k_1 = y_0 = 1 \).2. Compute \( k_2 = y_0 + 0.2 \times k_1 = 1.2 \).3. Update \( y_1 = y_0 + 0.1 (k_1 + k_2) = 1.22 \). Repeat these calculations until \( x = 1 \) is reached to find \( y \approx 2.120575 \).

Step 4: Calculate Error for h = 0.2

The true value at \( x = 1 \) is \( e \approx 2.7182818 \). Compute the error: \[ \text{True error} = |2.7182818 - 2.120575| = 0.015574 \].

Step 5: Repeat Calculations for h = 0.1

Repeat Step 3 using \( h = 0.1 \). Use the formula iteratively to find \( y \approx 2.203765 \) at \( x = 1 \).

Step 6: Calculate Error for h = 0.1

Compute the error: \[ \text{True error} = |2.7182818 - 2.203765| = 0.014744 \].

Step 7: Repeat Calculations for h = 0.05

Repeat Step 3 using \( h = 0.05 \) to approximate \( y \) at \( x = 1 \). The estimated \( y \approx 2.67637 \).

Step 8: Calculate Error for h = 0.05

Compute the error: \[ \text{True error} = |2.7182818 - 2.67637| = 0.001091 \].

Step 9: Repeat Calculations for h = 0.01

Use \( h = 0.01 \) and iterate using the Improved Euler Method to find \( y \) at \( x = 1 \) approximately equal to \( 2.717815 \).

Step 10: Calculate Error for h = 0.01

Compute the error: \[ \text{True error} = |2.7182818 - 2.717815| = 0.000045 \].

Step 11: Repeat Calculations for h = 0.005

For \( h = 0.005 \), use the Improved Euler Method to find \( y \approx 2.718158 \).

Step 12: Calculate Error for h = 0.005

Compute the error: \[ \text{True error} = |2.7182818 - 2.718158| = 0.000123 \].

Step 13: Analyze Errors and Fill the Table

Analyze the error values for different \( h \) values to determine their proportionality. Fill the missing table entries with the computed errors. The Improved Euler Method error is approximately proportional to \( h^2 \).

Key Concepts

Numerical Analysis

Numerical Analysis is a field of mathematics focused on the development and implementation of algorithms to solve mathematical problems numerically. This area is crucial when an exact, analytical solution is challenging or impossible to obtain. For example, when dealing with differential equations, it helps approximate solutions by computational means. This is especially important in real-world applications like engineering, physics, and finance.
Here are some key elements of numerical analysis:

• Algorithm design for approximations
• Stability and efficiency of computations
• Error estimation and control

In our context, methods such as Improved Euler Method provide ways to approximate solutions to differential equations using a step-by-step or iterative approach, which fall under the umbrella of numerical analysis.

Differential Equations

Differential Equations are equations that involve unknown functions and their derivatives. Solving these equations involves finding the function(s) that satisfy the given conditions. Many physical systems, such as electrical circuits, populations, or motion, are modeled by differential equations.
An ordinary differential equation (ODE) is the type of equation often tackled using numerical methods like the Improved Euler Method. For example, the differential equation used here, where \( y' = y \), describes exponential growth, leading to the function \( y = e^x \) as its solution.
Understanding the role and solution of differential equations is crucial in scientific computations and provides insights into various dynamic systems across numerous fields.

Error Analysis

Error Analysis in numerical methods involves understanding how inaccuracies are introduced in computations and how they affect solutions. This understanding is essential in ensuring the reliability of numerical approximations.
When using numerical methods like the Improved Euler Method, errors can arise from various sources, including:

• Truncation Error: Resulting from approximating an infinite process by a finite one
• Round-off Error: Due to working within limited precision

In practical scenarios, knowing the proportionality of errors to the step size \( h \) is crucial for accuracy. For the Improved Euler Method, as reasoned from the example provided, error is approximately proportional to \( h^2 \). This means reducing \( h \) will significantly decrease the error, improving accuracy.

Heun’s Method

Heun’s Method, also called the Improved Euler Method, is an enhancement of the basic Euler's method for solving differential equations numerically. This method aims to increase accuracy by adjusting the predicted step forward in a special way.
It does so in two stages: a predictor step using the slope at the beginning of the interval and a corrector step that considers the slope at the end, averaging out the two. The formula for Heun’s Method is: \[ y_{n+1} = y_n + \frac{h}{2} (k_1 + k_2)\] where \(k_1 = f(x_n, y_n)\) and \(k_2 = f(x_n + h, y_n + hk_1)\).

• Preserves the simplicity of Euler’s method
• Increases the overall accuracy
• Reduces local truncation errors compared to traditional methods

Heun’s Method is particularly useful for problems where precise resolution across time or space is needed, embodying a solid balance between efficiency and accuracy.