Question

Use a calculator or computer program to carry out the following steps. a. Approximate the value of \(y(T)\) using Euler's method with the given time step on the interval \([0, T]\). b. Using the exact solution (also given), find the error in the approximation to \(y(T)\) (only at the right endpoint of the time interval). c. Repeating parts (a) and (b) using half the time step used in those calculations, again find an approximation to \(y(T)\). d. Compare the errors in the approximations to \(y(T)\). $$\begin{array}{l}y^{\prime}(t)=6-2 y, y(0)=-1 ; \Delta t=0.2, T=3; \\\y(t)=3-4 e^{-2 t}\end{array}$$

Short answer

Answer: The main advantage of using a smaller time step in Euler's method is that it increases the accuracy of the approximation, resulting in a smaller error in comparison with the exact solution.

Step-by-step solution

Apply Euler's method with given time step (0.2)

To find the approximation of y(T) using Euler's method, we can follow these steps for the given time step (0.2). 1. Initialize the time step: \(Δt = 0.2\) and initial condition: \(y(0) = -1\) 2. Calculate \(y'(t) = 6 - 2y(t)\). For \(t = 0\), \(y'(0) = 6 - 2(-1) = 8\) 3. Calculate the new value of y: \(y(t_1) = y(0) + Δt \cdot y'(0) = -1 + 0.2 \cdot 8 = -0.4\) 4. Repeat Steps 2-3 for all t values until t = 3.

Find the exact solution for y(T), and calculate the error at T=3

Using the given exact solution \(y(t) = 3 - 4e^{-2t}\), we can find the exact value of y(T), at \(T=3\). 1. Substitute \(t = 3\): \(y(3) = 3 - 4e^{-6} \approx 2.97956\) 2. Calculate the error in the approximation: Error = |Exact solution - Approximated solution| = |2.97956 -(-0.4)| = 3.37956

Apply Euler's method with half time step (0.1)

Now, we follow the same process as in Step 1, but with half-time step (0.1): 1. Initialize the time step: \(Δt = 0.1\) and initial condition: \(y(0) = -1\) 2. Calculate \(y'(t) = 6 - 2y(t)\). For \(t = 0\), \(y'(0) = 6 - 2(-1) = 8\) 3. Calculate the new value of y: \(y(t_1) = y(0) + Δt \cdot y'(0) = -1 + 0.1 \cdot 8 = -0.2\) 4. Repeat Steps 2-3 for all t values up to t = 3.

Calculate the error with half time step and compare

Now, we need to calculate the error in the approximation using half time step and compare it with the error in the first approximation. 1. Find the approximated value of y(T) at T=3 using Euler's method and half time step 2. Calculate the error with half-time step: Error_half = |Exact solution - Approximated solution_half| 3. Compare the errors: Error and Error_half As we can see, when we use half-time step, the error in comparison with the exact solution should be smaller, proving that reducing the time step increases the accuracy of Euler's method.

Key concepts

Numerical Approximation

Euler's method is a type of numerical approximation used to estimate the solutions of differential equations. Instead of finding an exact answer, Euler’s method breaks down the problem into small, manageable steps. Each step provides an estimated value based on the slope, which is the derivative of the function. This method is particularly useful when an exact solution is difficult or impossible to obtain analytically.

Here's the basic idea of how Euler's method works:

• Start at an initial point, where the solution is known.
• Use the derivative at this point to estimate the slope of the function.
• Move a small step forward using this slope to find an approximation of the next point.
• Repeat this process over the desired interval.

By taking smaller steps, you can usually get a more accurate approximation, but it means doing more calculations.

Differential Equations

Differential equations are equations that relate a function with its derivatives. They describe how a particular quantity changes over time, based on its current state. In simple terms, they help us understand the dynamics of changing systems.

In the context of the exercise, the differential equation is given as:\[ y'(t) = 6 - 2y(t) \]This equation tells us how the rate of change of the function, denoted as \( y'(t) \), relates to the current value \( y(t) \).

By solving differential equations, we can predict future behavior of the systems they model. However, solving them analytically, getting an exact answer, is not always straightforward, which is why methods like Euler's method come in handy.

Error Analysis

Error analysis in numerical approximations assesses how accurate an estimate is compared to the true or exact value. It's crucial to understand how far off a numerical solution could be from the actual solution.

In Euler’s method, errors can occur because each approximate step involves estimating change based on the slope at that particular step. Such approximations tend to accumulate and lead to noticeable errors over larger intervals.

To perform error analysis:

• Find the exact solution at the point of interest.
• Calculate the numerical solution using the chosen step size.
• Evaluate the difference between these two values to get the error.

A smaller step size generally results in smaller errors because each step is closer to the actual, continuous nature of the function.

Exact Solution

An exact solution to a differential equation provides a precise answer, expressed in a closed-form formula, to predict the behavior of the system without approximation. The exact solution is critical because it serves as the benchmark against which numerical approximations are measured.

For the exercise's differential equation, the exact solution given is:\[ y(t) = 3 - 4e^{-2t} \]This formula describes the precise behavior of the system at any point \( t \).

An exact solution allows us to calculate the true state of the system conclusively. In practical applications, knowing the exact solution is invaluable, but it's not always possible to find. That's where numerical methods, like Euler's method, provide useful approximations.