Question

In each exercise, (a) Write the Euler's method iteration \(y_{k+1}=y_{k}+h f\left(t_{k}, y_{k}\right)\) for the given problem. Also, identify the values \(t_{0}\) and \(y_{0}\). (b) Using step size \(h=0.1\), compute the approximations \(y_{1}, y_{2}\), and \(y_{3}\). (c) Solve the given problem analytically. (d) Using the results from (b) and (c), tabulate the errors \(e_{k}=y\left(t_{k}\right)-y_{k}\) for \(k=1,2,3\). \(y^{\prime}=2 t-1, \quad y(1)=0\)

Short answer

A) 0.01 B) 0.04 C) 0.09 D) 0.25

Step-by-step solution

Write Euler's Method Iteration and Identify \(t_0\) and \(y_0\)

Given the problem, we have the differential equation \(y^{\prime} = 2t - 1\) and the initial condition \(y(1) = 0\). Therefore, we have: $$t_0 = 1 \quad \text{and} \quad y_0 = 0$$ The Euler's method iteration for this problem is: $$y_{k+1} = y_k + h (2t_k - 1)$$ where \(h = 0.1\) is the step size.

Compute the Approximations \(y_1\), \(y_2\), and \(y_3\)

Using the Euler's method iteration, we'll compute the approximations: 1. \(y_1 = y_0 + h(2t_0 - 1) = 0 + 0.1(2(1) - 1) = 0.1\) 2. \(y_2 = y_1 + h(2t_1 - 1) = 0.1 + 0.1(2(1.1) - 1) = 0.1 + 0.1(0.2) = 0.12\) 3. \(y_3 = y_2 + h(2t_2 - 1) = 0.12 + 0.1(2(1.2) - 1) = 0.12 + 0.1(0.4) = 0.16\) So, the approximations are \(y_1 = 0.1\), \(y_2 = 0.12\), and \(y_3 = 0.16\).

Solve the Problem Analytically

To solve the given problem analytically, we need to find the solution of the differential equation \(y^{\prime} = 2t - 1\) and use the initial condition \(y(1) = 0\). The analytical solution will be of the form: $$y(t) = \int (2t - 1) dt = t^2 - t + C$$ Applying the initial condition \(y(1) = 0\): $$0 = 1^2 - 1 + C \Rightarrow C = 0$$ So, the analytical solution is: $$y(t) = t^2 - t$$

Tabulate the Errors \(e_k = y(t_k) - y_k\)

Now, we'll compute the errors using the analytical solution \(y(t) = t^2 - t\): 1. \(e_1 = y(1.1) - y_1 = (1.1^2 - 1.1) - 0.1 = 0.01\) 2. \(e_2 = y(1.2) - y_2 = (1.2^2 - 1.2) - 0.12 = 0.04\) 3. \(e_3 = y(1.3) - y_3 = (1.3^2 - 1.3) - 0.16 = 0.09\) The errors are tabulated as follows: | \(k\) | \(t_k\) | \(y_k\) | \(y(t_k)\) | \(e_k\) | |-----|-------|-------|----------|-------| | 1 | 1.1 | 0.1 | 0.11 | 0.01 | | 2 | 1.2 | 0.12 | 0.16 | 0.04 | | 3 | 1.3 | 0.16 | 0.25 | 0.09 |

Key concepts

Numerical Methods

Numerical methods are techniques that enable us to find approximate solutions for complex mathematical problems that cannot be easily solved analytically. One popular numerical method is Euler's Method, which is particularly useful for solving differential equations. In Euler's Method, we approximate the solution by taking small steps from a known initial value, using a slope calculated from the differential equation.

To use Euler's Method, we follow these steps:

• Start with an initial condition, typically denoted as \( t_0 \) and \( y_0 \).
• Select a step size \( h \), which determines how far we move along the curve at each step.
• Apply the iterative formula \( y_{k+1} = y_k + hf(t_k, y_k) \) to compute subsequent values.

This method transforms a continuous problem into a series of discrete calculations that provide approximate solutions. The smaller the step size \( h \), the more accurate the approximation, but it requires more computations.

Differential Equations

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are used to model various phenomena in engineering, physics, biology, economics, and many other fields. The given exercise involves a first-order differential equation: \( y' = 2t - 1 \). This equation implies that the rate of change of \( y \) with respect to \( t \) is governed by the expression \( 2t - 1 \).

To solve a differential equation analytically means finding an explicit formula for \( y(t) \), considering all conditions provided.

• The general process involves integrating the differential equation.
• After integration, apply the initial conditions to determine any constants of integration.

For our exercise, integration of \( 2t - 1 \) yields \( y = t^2 - t + C \). Using the initial condition \( y(1) = 0 \), we find \( C = 0 \), resulting in the solution: \( y(t) = t^2 - t \).

Error Analysis

Error analysis is a crucial part of numerical methods, helping to assess the accuracy of the approximations. This involves calculating the difference between the numerical solution and the exact or analytical solution. The error \( e_k \) at each step \( k \) is calculated as: \( e_k = y(t_k) - y_k \), where \( y(t_k) \) is the value obtained from the analytical solution.

In our exercise, the errors calculated at \( t_1 = 1.1 \), \( t_2 = 1.2 \), and \( t_3 = 1.3 \) are respectively 0.01, 0.04, and 0.09. This error analysis provides insight into the effectiveness of Euler's Method for a given problem and step size.

• Errors typically increase with the step size used in numerical methods.
• Comparing errors at different steps helps in deciding whether a smaller step size might provide a more accurate solution.

Understanding the nature and size of errors allows for better refinements in the computation process and choice of methods.