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 \(y(T)\). $$\begin{array}{l}y^{\prime}(t)=t-y, y(0)=4 ; \Delta t=0.2, T=4; \\\y(t)=5 e^{-t}+t-1\end{array}$$

Short answer

Answer: The error in the approximation of Euler's method decreases when the time step is reduced from 0.2 to 0.1. This is because a smaller time step leads to more accurate approximations in solving the differential equation.

Step-by-step solution

Apply Euler's method with a time step of 0.2

To apply Euler's method, we will use the differential equation \(\frac{dy}{dt} = t - y\). The initial value is \(y(0) = 4\). We need to approximate the value of \(y(T)\), where \(T = 4\), and \(\Delta t = 0.2\). We have: 1. \(y_0 = 4\) 2. \(y_1 = y_0 + \Delta t(t_0 - y_0) = 4 + 0.2(0 - 4) = 3.2\) 3. \(y_2 = y_1 + \Delta t(t_1 - y_1) = 3.2 + 0.2(0.2 - 3.2) = 2.64\) And we continue iterating until we reach the end of the interval. After we finish the iterations, we can find the approximation \(y(T)\).

Calculate the exact solution and error

Use the exact solution \(y(t) = 5e^{-t} + t - 1\). We can find the exact value of \(y(T)\) by substituting \(T=4\) into the function: \(y(4) = 5e^{-4} + 4 - 1\). Now, let's calculate the error between the Euler's method approximation and the exact solution: \(error = |y(4) - y(T)|\).

Apply Euler's method with a time step of 0.1

Now, repeat the process of Step 1, but with a time step of \(\Delta t = 0.1\). Calculate the new approximation of \(y(T)\) using Euler's method with this smaller time step.

Calculate the new exact solution and error

Again, use the exact solution \(y(t) = 5e^{-t} + t - 1\) to find the exact value of \(y(T)\) for this time step. Calculate the new error between the Euler's method approximation with this new time step and the exact solution: \(new\_error = |y(4) - y(T)|\).

Compare the errors

With both errors calculated (for time steps of 0.2 and 0.1), we can now compare them to see how the error changes with differing time steps. A smaller error implies a better approximation.

Key concepts

Numerical Analysis

Numerical Analysis is a field of mathematics that uses numerical computations to solve mathematical problems and analyze the results. When dealing with complex differential equations, analytical solutions can be difficult to find.
Thus, numerical methods like Euler's Method become useful tools.

Euler's Method is a straightforward approach used to approximate solutions to differential equations.

• It assumes that the solution can be represented as a series of tangential line segments.
• The method begins with an initial value problem and iteratively estimates values using a small step size.
• By using discrete steps, Euler's Method provides a sequence of approximate values that represent the solution curve of the differential equation over an interval.

While practical for its simplicity, Euler's Method is not always the most accurate numerical technique. It tends to work better with smaller step sizes but at the cost of increased computational effort.
This is why numerical analysts often explore more advanced methods for particular kinds of differential equations.

Differential Equations

Differential equations are equations that describe how a quantity changes relative to the change of another. They are fundamental in expressing a variety of physical phenomena, from the growth of populations to the motion of planets.

In the context of the exercise, we are given the ordinary differential equation (ODE):
\( \frac{dy}{dt} = t - y \), with an initial condition of \( y(0) = 4 \).

This equation tells us how the derivative of \( y \), commonly related to its rate of change, depends on the current value of \( t \) and \( y \) itself.

• The initial condition specifies the starting point, ensuring a uniquely determined solution.
• The objective is to find \( y(T) \), the value of \( y \) at some point \( T \), in this case, 4.
• The exact solution to this ODE is given by \( y(t) = 5e^{-t} + t - 1 \), derived from integrating and solving the equation analytically.

Solving differential equations provides insights into the continuity and change observed in various systems across science and engineering.
The exact solutions are often expected to align closely with numerical ones where possible.

Error Analysis

Error Analysis in numerical computations involves assessing the approximation error relative to the exact solution. When using methods such as Euler's Method, understanding and managing error is essential for validating outcomes.

The exercise demonstrates how the error between an approximate and exact solution can be calculated.

• The error is defined as the absolute difference between the exact value, \( y(T) \), and the approximated value obtained through Euler's Method.
• In this context, the error helps us understand the accuracy of our numerical approximation.
• When repeating the Euler's calculation with a smaller time step, the error generally decreases, illustrating a trade-off between computational cost and precision.

Error analysis not only guides the practice of choosing suitable step sizes but also informs us on how close our numerical solution is to the true behavior of the differential equation.
Refining this analysis could involve comparing results with various methods or employing higher-order techniques to achieve better accuracy.