Question

Obtain the Heun formula. Using the Heun method with \(h=0.1\) and the Euler method as predictor, determine an approximate solution at \(x=0.2\) of the initial-value problem $$ \begin{aligned} y^{\prime} &=x^{2}+y^{2} \\ y(0) &=1 \end{aligned} $$ Compare the results with that in No. \(11 .\)

Short answer

Question: Use Heun's method with the Euler method as a predictor to approximate the solution of the given initial value problem at x=0.2. Solution: Following the step-by-step derivation and implementation of Heun's method provided above, we find that the approximate solution of the given initial value problem at x=0.2 is approximately y(0.2) ≈ 1.1105.

Step-by-step solution

Understand Heun's Formula

Heun's method is an improved version of the Euler method. It consists of refining the Euler method by taking a second evaluation of the derivative of the function at a later stage, usually around the midpoint, and averaging these two values. The formula for Heun's method can be written as: $$ y_{i+1} = y_i + \frac{h}{2}(f(x_i, y_i) + f(x_{i+1}, \tilde{y}_{i+1})) $$ where \(y_{i+1}\) is the approximate solution at the next point, \(y_i\) is the approximate solution at the current point, \(h\) is the step-size, \(f(x_i, y_i)\) is the evaluated derivative at the current point, \(f(x_{i+1}, \tilde{y}_{i+1})\) is the evaluated derivative at the next point using the Euler method as a predictor, and \(\tilde{y}_{i+1}\) is the predicted value from the Euler method.

Define the function and its derivative

The given initial-value problem is: $$ \begin{aligned} y^{\prime} &= x^{2} + y^{2} \\ y(0) &= 1 \end{aligned} $$ The function \(f(x, y)\) representing the derivative is: $$ f(x, y) = x^{2} + y^{2} $$

Implement the Euler method for prediction

The Euler method formula is: $$ \tilde{y}_{i+1} = y_i + h \cdot f(x_i, y_i) $$ For \(h=0.1\), \(x_0 = 0\), and \(y_0 = 1\), we can compute \(\tilde{y}_{1}\) $$ \tilde{y}_{1} = 1 + 0.1 \cdot (0^2 + 1^2) = 1 + 0.1 = 1.1 $$

Use Heun's method to calculate the approximate solution

Now we can use Heun's method to compute the approximate solution at \(x=0.2\). Our known values are \(x_0 = 0\), \(y_0 = 1\), \(h = 0.1\) and \(\tilde{y}_{1} = 1.1\). The Heun's method formula is: $$ y_{i+1} = y_i + \frac{h}{2}(f(x_i, y_i) + f(x_{i+1}, \tilde{y}_{i+1})) $$ Using the values: $$ y_1 = 1 + \frac{0.1}{2}((0^2 + 1^2) + (0.1^2 + 1.1^2)) = 1 + \frac{0.1}{2}(1 + 1.210) = 1 + 0.1 \cdot 1.105 = 1.1105 $$ So, the approximate solution \(y(0.2) ≈ 1.1105\).

Compare with the answer in No. 11

As we don't have access to the specific problem No. 11 mentioned, we cannot perform a direct comparison. However, the answer obtained here, \(y(0.2) ≈ 1.1105\), represents an approximation using the Heun method with the Euler method as a predictor.

Key concepts

Ordinary Differential Equations

Ordinary differential equations (ODEs) are equations that involve derivatives of a function of a single variable. They are fundamental in mathematics and applied sciences because they model a variety of phenomena from physics to biology. An ODE expresses the relationship between a function and its derivatives, allowing us to predict how a system evolves over time.

In the Heun method exercise, we are given an ODE of the form: \[ y' = x^2 + y^2 \] This ODE is a rule that tells us how the function y, which depends on x, changes as x changes.

Initial-Value Problem

An initial-value problem in the context of an ODE is a problem where the value of the solution y(x) is specified at a particular point, commonly denoted as the initial condition. This condition allows us to find a specific solution to the ODE that fits the given scenario.

The exercise specifies the initial condition as: \[ y(0) = 1 \] This simple initial condition is the starting point from which we calculate the solution to the ODE using various numerical methods including Heun's method.

Euler's Method

Euler's method is a numerical technique to approximate the solution of ODEs. It is a straightforward method that takes steps along the curve of the solution by repeatedly evaluating the derivative using the current value of y.

Through this exercise, we use Euler's method as a first step to predict the value of \( \tilde{y}_{i+1} \) at the next x position using the formula: \[ \tilde{y}_{i+1} = y_i + h \times f(x_i, y_i) \] This predicted value is then utilized in Heun's method to refine the approximation.

Approximate Solution

In many real-world applications, finding an exact solution to an ODE is not possible or practical. Instead, we seek an approximate solution: a value or function that is close enough to the true solution for the intended purpose.

The Heun's method provides us with such an approximate solution at incrementally increasing values of x, starting from the initial condition. In the exercise, the approximate solution for \( y(0.2) \) is found to be \( \approx 1.1105 \), demonstrating how Heun's method refines the simple prediction given by Euler's method.

Numerical Methods

Numerical methods are algorithms for obtaining numerical solutions to problems that are difficult or impossible to solve analytically. Heun's method, like Euler's method, is one such numerical technique. It uses iterations and predictions to solve initial-value problems associated with ODEs.

The implementation of these methods often requires iterative calculations, which, while simple in principle, can be computationally intensive. This makes the role of computers crucial in solving complex ODEs that model everything from celestial mechanics to economic growth patterns.