Question

By the Euler method obtain an approximate solution of the initial-value problem $$ \begin{aligned} y^{\prime} &=x-y, \\ y(0) &=1 \end{aligned} $$ at \(x=0.2\) using \(h=0.1\). Show that the exact solution is $$ \phi(x)=x-1+2 e^{-x} $$ By the Taylor expansion show that $$ e_{n}=\frac{1}{2} \phi^{\prime \prime}\left(\bar{x}_{n}\right) h^{2}, \quad x_{n}<\bar{x}_{n}

Short answer

Question: Use the Euler method to find an approximate solution of the given initial-value problem at \(x = 0.2\) with step size \(h = 0.1\). Verify that the exact solution is \(\phi(x) = x - 1 + 2e^{-x}\). Then, by Taylor expansion, find the error term and its formula. Lastly, compute the errors \(e_1\) and \(e_2\). Initial-value problem: \(y^{\prime} = x - y\), \(y(0) = 1\).

Step-by-step solution

Euler Method

To find an approximate solution at \(x=0.2\) using the Euler method with \(h=0.1\), we will follow the formula: \(y_{n+1} = y_n + hf(x_n, y_n)\), where \(f(x, y) = x - y\) and the initial value is \((x_0, y_0) = (0, 1)\). First, we find \(y_1 = y_0 + h f(x_0, y_0) = y_0 + h (x_0 - y_0)\), Then, we find \(y_2 = y_1 + h f(x_1, y_1) = y_1 + h (x_1 - y_1)\).

Verify Exact Solution

To verify that the exact solution is \(\phi(x)=x-1+2e^{-x}\), we will differentiate this function w.r.t. x and then check if the function satisfies the given initial-value problem. So, evaluate the derivative of the function as \(y^{\prime} =\phi^{\prime}(x)\). Now, check whether the function satisfies the given equation \(y^{\prime} = x - y\) and the initial condition \(y(0)=1\).

Taylor Expansion and Error Formula

To find the error term by Taylor expansion, we write the second-order Taylor expansion for \(\phi\): $$\phi(x_{n+1}) = \phi(x_n) + \phi^{\prime}(x_n) h + \frac{1}{2}\phi^{\prime\prime}(\bar{x}_n) h^2$$, where \(x_n < \bar{x}_n < x_{n+1}\). Now, subtract the Euler approximation \(y_{n+1} = y_n + hf(x_n, y_n)\) from the Taylor expansion and obtain the error term formula: \(e_n = \frac{1}{2} \phi^{\prime \prime}\left(\bar{x}_{n}\right) h^{2}\). Then, find the formula for the error term \(e_n = e^{-\bar{x}_n}h^2\) and show that \( |e_n| \leq h^2\).

Compute \(e_1\) and \(e_2\)

To compute \(e_1\) and \(e_2\), we will use the formula found in Step 3: \(e_n = e^{-\bar{x}_n}h^2\). For \(e_1\), let \(n=1\) and find \(e_1 = e^{-\bar{x}_1}h^2\). For \(e_2\), let \(n=2\) and find \(e_2 = e^{-\bar{x}_2}h^2\).

Key concepts

Understanding Initial-Value Problems

An initial-value problem is a type of differential equation accompanied by a specified value, called the initial condition, at a given point. In the context of our Euler method approximation exercise, the initial-value problem consists of an ordinary differential equation (ODE), \( y' = x - y \), and an initial condition \( y(0) = 1 \). This problem requires us to find a function \( y(x) \) that satisfies both the differential equation and the initial condition. Basic initial-value problems are fundamental in various fields such as physics, engineering, and finance for modeling phenomena that vary with time.

When working with initial-value problems, the main goal is to seek an exact solution. However, when an exact solution is difficult or impossible to find, as it often is with more complex equations, numerical methods such as the Euler method are invaluable for providing approximate solutions.

Taylor Expansion in Numerical Analysis

Taylor expansion is a mathematical method of approximating functions using the sum of terms calculated from the values of their derivatives at a single point. In the realm of numerical analysis, particularly for the Euler method exercise, Taylor expansion is crucial in understanding how the value of a function changes from one point to another. This expansion informs us about the nature of the function's curvature and allows us to make predictions about its behavior in new, nearby areas.

Our main exercise involves the use of Taylor expansion to estimate the local truncation error in the Euler's method as \( e_n = \frac{1}{2} \phi^{\prime \prime}(\bar{x}_n) h^{2} \). This error gives us a sense of how far off our Euler approximation might be from the true solution, which is essential when assessing the accuracy of numerical methods.

Ordinary Differential Equations and Their Significance

Ordinary differential equations (ODEs) are equations involving functions and their derivatives. They describe the rate of change of physical quantities, making them indispensable tools for modeling continuous systems. In our exercise, the ODE under consideration is a first-order equation given by \( y' = x - y \).

First-order ODEs, like the one in our problem, often represent dynamic processes where some quantity (in this case represented by the function \( y(x) \)) changes in response to another variable (here, \( x \)). Finding solutions to ODEs can help predict future states of systems and explain natural phenomena. But when analytical solutions are challenging to obtain, we must resort to numerical methods to tackle these problems.

The Role of Numerical Methods

Numerical methods are a collection of techniques for solving mathematical problems that are typically difficult or impossible to solve analytically. They allow us to approximate solutions with a desired level of accuracy. In the Euler method exercise, numerical methods enable us to approximate the solution of the ODE at specific points using iterative stepping procedures that are calculated from an initial starting condition.

To demonstrate, the Euler method starts at the known initial point \( (x_0, y_0) \) and takes steps of a fixed size \( h \), using the slope of the ODE to estimate the value of the function at the next step. Through doing this, we step through the problem's domain, building up an approximation of the function that is adequate for many practical purposes. Despite its simplicity, Euler's method is a powerful tool in cases where other numerical methods may not be necessary or where computational resources are limited.