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 \(y(T)\). $$y^{\prime}(t)=-2 y, y(0)=1 ; \Delta t=0.2, T=2 ; y(t)=e^{-2 t}$$

Short answer

Compare the errors in the approximations of \(y(2)\) with time steps of \(\Delta t = 0.2\) and \(\Delta t = 0.1\).

Step-by-step solution

Characteristics of the problem and the exact solution

To begin, we are given the following IVP: \(y^{\prime}(t) = -2y\), \(y(0) = 1\), which has an exact solution \(y(t) = e^{-2t}\). Our goal is to approximate the value of \(y(T)\) for \(T = 2\) using Euler's method.

Apply Euler's method with a given time step \(\Delta t\)

We will apply Euler's method with a time step of \(\Delta t = 0.2\) on the interval [0, T]. In Euler's method, we use the following update rule to approximate \(y(t)\): \(y_{n+1} = y_n + \Delta t\cdot f(t_n, y_n)\), where \(f(t, y) = -2y\) in our case. We need to perform these iterations until we reach \(t = T\). Therefore, we perform 10 iterations, since \((T-0)/\Delta t = (2-0)/0.2 = 10\).

Calculate the error in the approximation to \(y(T)\)

Next, we calculate the error in the approximation of \(y(T)\). To do this, we need to find the exact solution at \(t = T\), which is \(y(2) = e^{-2 \cdot 2} = e^{-4}\). Then, we subtract the exact solution from the approximation obtained in the previous step to calculate the error.

Repeat Steps 2-3 with half the time step

Now, we will perform the same steps but with the time step cut in half, \(\Delta t = 0.1\). We apply Euler's method and calculate the error in the approximation following the same procedure.

Compare the errors of both approximations

Finally, we compare the errors of our approximations from Step 3 and Step 4. This will show us how reducing the time step affects the accuracy of Euler's method in our case. The smaller the error, the better the approximation.

Key concepts

Numerical Methods

Numerical methods are strategies used for solving mathematical problems when an exact solution is difficult or impossible to find. These methods rely on numerical approximations rather than symbolic computations. For differential equations, where we seek a function instead of a mere value, numerical methods become particularly useful.

In the context of our exercise, Euler's Method stands out as a simple yet powerful numerical technique. It approximates solutions to differential equations by breaking the problem into smaller, manageable steps. This method eschews the analytical, exact solutions usually achieved by solving the differential equation symbolically.

Euler's Method is a straightforward approach that involves iterating over intervals using initial conditions. By applying a consistent update rule, this technique allows us to predict the future state of a function based on its current derivative.

Overall, while numerical methods like Euler's may not provide the exact answers, they effectively allow us to approximate solutions to problems that fields from engineering to physics face, making them indispensable tools in the realm of applied mathematics.

Initial Value Problem

An Initial Value Problem (IVP) is a type of differential equation along with some specified value, known as the initial condition. The solution to an IVP is the function that not only satisfies the differential equation but also meets the condition provided at the initial point.

In our exercise, we are given an IVP: \(y'(t) = -2y\) with an initial condition \(y(0) = 1\). This means when \(t = 0\), the value of \(y\) is 1. The goal is to find the solution function that fulfills both this differential equation and the initial condition.

IVPs are fundamental in mathematical modeling and simulation. They help predict the future behavior of dynamic systems by providing the necessary starting point from which calculations proceed.

For Euler’s Method, solving an IVP involves starting from the known initial value and iteratively projecting forward to approximate the unknown solution at future points. This makes IVPs crucially important as they serve as the starting blocks for numerical methods.

Approximation Error

Approximation error is the difference between the exact mathematical value and the estimated value obtained using a numerical method. In many cases, obtaining an exact solution is computationally intensive or impractical, making approximations necessary.

In our situation, Euler's Method provides an approximation to the differential equation \(y'(t) = -2y\) over a set interval by using discrete time steps. The approximation error is calculated by comparing the final calculated value with the exact solution at the endpoint \(t = 2\).

Errors in numerical methods can arise from multiple sources:

• Discretization error due to the finite step size in the numerical method
• Rounding errors from computer calculations

To minimize the approximation error, reducing the step size can be effective, though it may increase computational workload. Accurately estimating and controlling these errors is an essential part of using numerical methods effectively.

Step Size Reduction

Step size reduction is a method used to improve the accuracy of numerical solutions by decreasing the interval over which calculations are performed. In Euler’s Method, the step size, denoted as \(\Delta t\), is a critical factor affecting approximation accuracy.

For our exercise, initially, we used a step size of \(0.2\), which resulted in a certain approximation error when calculating \(y(2)\). By halving the step size to \(0.1\), we performed twice as many iterations but obtained a more accurate result with a reduced error.

A smaller step size generally leads to a closer estimate of the solution because it aligns finer intervals with the curve of the actual function being approximated. Nevertheless, this improvement comes at the cost of requiring more computations, which can increase processing time and resource use.

Therefore, finding a balance between step size and computational cost is important for the efficiency of numerical methods like Euler's. Smart choices regarding step size ensure accuracy without sacrificing computational efficiency.