Question

Use the Euler method and a calculator to approximate the values of the exact solution at \(t=0,0.1, \ldots, 0.5\) by using \(h=0.1 .\) For Exercises \(1-8\) find the exact solutions \(x\) and \(y ;\) for Exercises \(9-13\) find the exact solution \(x .\) Evaluate the exact solutions at \(t=0,0.1, \ldots, 0.5 .\) Compare the approximations to the exact values by calculating the errors and percentage relative errors. Repeat the exercise using the Runge-Kutta method in place of the Euler method. How much better are the approximations obtained using the Runge-Kutta method? \(x^{\prime}=6 x-3 y+e^{2 t}\) \(y^{\prime}=2 x+y-e^{2 t}\) \(x(0)=4, y(0)=4\)

Short answer

To approximate solutions to the given system of ODEs using the Euler and Runge-Kutta methods, we perform several iterations for each method, using the update rules presented in the step-by-step solution. After finding the approximations at each step, we compare these with the exact values by calculating the errors and percentage relative errors. We should observe that the Runge-Kutta method provides better approximations compared to the Euler method.

Step-by-step solution

Apply the Euler method

First, we need to initialize our variables x and y with their given initial values, which are x(0)=4 and y(0)=4, and set the step size to h=0.1. Let's use the given ODEs to apply the Euler method: Euler method update rule: \(x_{i+1} = x_i + h \cdot x^{\prime}_i\) \(y_{i+1} = y_i + h \cdot y^{\prime}_i\) Using the given ODEs: \(x^{\prime} = 6x - 3y + e^{2t}\) \(y^{\prime} = 2x + y - e^{2t}\) We perform the following iterations for each value of t=0, 0.1, 0.2, 0.3, 0.4: t=0: x(0)=4 y(0)=4 t=0.1: x(0.1) = x(0) + h * (6 * x(0) - 3 * y(0) + e^(2 * 0)) x(0.1) = 4 + 0.1 * (6 * 4 - 3 * 4 + 1) y(0.1) = y(0) + h * (2 * x(0) + y(0) - e^(2 * 0)) y(0.1) = 4 + 0.1 * (2 * 4 + 4 - 1) Now, do the same calculations for t=0.2, 0.3, 0.4, and 0.5.

Apply the Runge-Kutta method

For the Runge-Kutta method, we use the following 2nd order Runge-Kutta method (improved Euler method): 1. k1x = h * (6 * x(0) - 3 * y(0) + e^(2*t)) k1y = h * (2 * x(0) + y(0) - e^(2*t)) 2. k2x = h * (6 * (x(0) + k1x/2) - 3 * (y(0) + k1y/2) + e^(2*(t+h/2))) k2y = h * (2 * (x(0) + k1x/2) + (y(0) + k1y/2) - e^(2*(t+h/2))) 3. x(0.1) = x(0) + k2x y(0.1) = y(0) + k2y Repeat the same process for t=0.2, 0.3, 0.4, and 0.5.

Compare the approximations

Now that we have the approximated values for x and y using the Euler and Runge-Kutta methods, please use the exact solutions provided in your textbook or by your instructor to calculate the errors and percentage relative errors. Once you have calculated the errors, compare them to decide which method provides more accurate results. You should see that the Runge-Kutta method gives better approximations compared to the Euler method.

Key concepts

Euler Method

The Euler Method is one of the simplest numerical techniques for solving ordinary differential equations (ODEs). It is particularly useful for approximating the solution of initial value problems. Here’s how it works:

• Start with an initial value, such as given for functions of time, e.g., \( x(t_0) = x_0 \) and \( y(t_0) = y_0 \).
• Select a step size, \( h \), which determines how frequently the function is sampled. For our exercise, \( h = 0.1 \).
• Use the derivative of the function, derived from the ODE, to estimate the next value: this is done by adding the product of the step size and the derivative at the present value to the current value.

In the example from the exercise, the Euler update rules are:

\[ x_{i+1} = x_i + h \cdot (6x - 3y + e^{2t}) \]
\[ y_{i+1} = y_i + h \cdot (2x + y - e^{2t}) \]

It is important to note that, while simple, the Euler Method can accumulate significant errors, especially with larger step sizes or more complex equations. Thus, it is often the first step before introducing more accurate methods.

Runge-Kutta Method

The Runge-Kutta Methods are a group of iterative methods used to improve the accuracy of solutions to ordinary differential equations. Among them, the second-order Runge-Kutta or improved Euler method offers a balance between complexity and accuracy.

This method employs an approach that uses intermediate calculations to estimate the slope of the solution over the interval, making it significantly more accurate than the straightforward Euler method. Here is a brief overview:

• Calculate "slopes" at the beginning of the interval and at an intermediate point.
• The intermediate point is calculated using the average of the initial slope and the slope at half the interval.
• The new estimate is then made by adding these weighted slopes.

For the exercise, this involved:

• Calculating \( k_1 \), the initial slope:
• \[ k1_x = h \cdot (6x - 3y + e^{2t}) \]
• \[ k1_y = h \cdot (2x + y - e^{2t}) \]
• Then \( k_2 \) for the midpoint:
• \[ k2_x = h \cdot (6(x + \frac{k1_x}{2}) - 3(y + \frac{k1_y}{2}) + e^{2(t + \frac{h}{2})}) \]
• \[ k2_y = h \cdot (2(x + \frac{k1_x}{2}) + (y + \frac{k1_y}{2}) - e^{2(t + \frac{h}{2})}) \]
• Finally, compute the new values:
• \[ x_{i+1} = x_i + k2_x \]
• \[ y_{i+1} = y_i + k2_y \]

This approach reduces error significantly and is preferred for more complex systems where greater precision is required.

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are equations that involve functions and their derivatives. They are "ordinary" because they involve functions of a single variable, typically time. An ODE is essential in describing the behavior of a dynamic system. In our exercise, we deal with the following ODEs:

- \( x' = 6x - 3y + e^{2t} \)
- \( y' = 2x + y - e^{2t} \)

These equations describe the rate of change of \( x \) and \( y \), which are functions dependent on time \( t \). For these types of problems, we must determine the initial conditions, such as \( x(0) = 4 \) and \( y(0) = 4 \), to find specific solutions.

Methods like the Euler and Runge-Kutta discussed earlier are established to solve these equations numerically when exact analytical solutions are either unavailable or difficult to find. ODEs capture a wide range of phenomena from physical sciences to economics, making them a core area of study in numerical analysis.

Error Analysis

Error analysis in numerical methods is crucial to understanding and improving the accuracy of approximate solutions to differential equations like our exercise. Every numerical approximation carries with it a certain level of error, which can come from different sources such as:

• Truncation error: arises from using a finite number of terms in a sequence or series to estimate an entire function.
• Round-off error: occurs because computers use finite precision to represent numbers.

For our exercise, one goal is to analyze these errors by comparing the numerical solutions (obtained by Euler and Runge-Kutta methods) to the exact solutions, calculating the absolute and percentage relative errors for each method.

This involves:

• Calculating the exact solution values using the given functions.
• Comparing them against the approximations.
• Computing the difference to find the magnitude of errors.

Running a relative error analysis highlights the superiority of one method—often, the Runge-Kutta provides closer approximations than Euler, indicating less error. Error analysis not only pinpoints discrepancies but also guides method selection and adjustment for more complex systems.