Question

Obtain the Milne formula. Using the Milne method with \(h=0.1\) and the Euler method as predictor, find an approximate value 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 of the Euler and modified Euler methods in No. 6 .

Short answer

Based on the given initial-value problem and the step-by-step solutions provided, answer the following question: Question: Apply Milne method to find an approximate value of the given initial-value problem at x = 0.2 using step size h = 0.1 and compare the result with those obtained from Euler and modified Euler methods. Answer: Applying Milne method, we obtain an approximate value of 1.13344 at x = 0.2. When compared with the Euler method (1.222) and the Modified Euler method (1.22711), the Milne method provides a closer approximation.

Step-by-step solution

Obtain the Milne formula

The Milne formula is a method for the numerical integration of ordinary differential equations. The general Milne formula for approximating \(y_{n+1}\) is given as: $$y_{n+1} = y_{n-3} + \frac{4h}{3}(2f(x_{n}, y_{n}) - f(x_{n-1}, y_{n-1}) + 2f(x_{n-2}, y_{n-2}))$$ Where \(h\) is the step size, and \(f(x, y)\) represents the given differential equation.

Use the Euler method as a predictor

We are given the initial-value problem: $$y^{\prime} =x^{2}+y^{2};\ y(0) =1$$ Using the Euler method with step size \(h=0.1\), we will obtain approximate values for \(y_{1}, y_{2}\) and \(y_{3}\). Euler's Method Formula: $$y_{n+1} = y_n + hf(x_n, y_n)$$ For n = 0: $$y_{1} = y_0 + 0.1f(x_0, y_0) = 1 + 0.1(0^2 + 1^2) = 1 + 0.1 = 1.1$$ For n = 1: $$y_{2} = y_1 + 0.1f(x_1, y_1) = 1.1 + 0.1(0.1^2 + 1.1^2) = 1.1 + 0.122 = 1.222$$ For n = 2: $$y_{3} = y_2 + 0.1f(x_2, y_2) = 1.222 + 0.1(0.2^2 + 1.222^2) = 1.222 + 0.15204 = 1.37404$$

Apply the Milne method with the obtained predictor values

Now that we have the predictor values, let's use the Milne formula to approximate \(y_{4}\), the desired value at \(x=0.2\). $$y_{4} = y_{0} + \frac{4h}{3}(2f(x_{3}, y_{3}) - f(x_{2}, y_{2}) + 2f(x_{1}, y_{1}))$$ Inserting values of (\(x_n, y_n, f(x_n,y_n)\)) into the Milne formula, we get: $$y_{4} = 1 + \frac{4(0.1)}{3}(2(0.3^2 + 1.37404^2) - (0.2^2 + 1.222^2) + 2(0.1^2 + 1.1^2)) = 1 + 0.13344 = 1.13344$$ Approximate value of \(y\) at \(x=0.2\) using the Milne method is \(1.13344\).

Compare the results with those of Euler and modified Euler methods

The solutions obtained using Euler and modified Euler methods in No. 6 are not given. However, we'll provide results from both methods here for comparison. For Euler's method, we computed \(y_{2}=1.222\). For Modified Euler's method (also known as Heun's method): $$y_{2}^{'} = y_{1} + hf(x_{1},y_{1}) = 1.1 + 0.1(0.1^2+1.1^2) = 1.222$$ $$y_{2}^{''} = y_{1} + \frac{h}{2}(f(x_{1},y_{1}) + f(x_{2},y_{2}^{'})) = 1.1 + \frac{0.1}{2}((0.1^2+1.1^2) + (0.2^2+1.222^2)) = 1.22711$$ Comparing the three methods, we have: - Milne method: \(1.13344\) - Euler method: \(1.222\) - Modified Euler method: \(1.22711\) The Milne method gives a closer approximation compared to Euler and Modified Euler Methods.

Key concepts

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are equations that involve a function and its derivatives. They express a relationship between an unknown function and its rate of change, and are fundamental in modeling continuous systems such as physics, engineering, and finance. The general form of an ODE is expressed as
\( \frac{dy}{dx} = f(x, y) \),
where \(f(x, y)\) represents the rate of change of \(y\) with respect to \(x\), and \(y\) is the function to be determined. The solution to an ODE is the function \(y(x)\) that satisfies this equation, often subject to initial conditions, like \(y(0) = 1\) in the problem provided. Solving ODEs analytically can be complex or impossible for some equations, which is why numerical methods, such as the Milne method, are particularly useful.

Numerical Integration

Numerical Integration is a technique to calculate the integral of a function numerically rather than analytically. It is particularly helpful when an explicit formula for the integral is not available or when the function is difficult or impossible to integrate analytically. In the context of solving ODEs, numerical integration approximates the solution by stepping forward in small increments, or 'steps', and calculating the value of the unknown function at each step using a specific formula. The step size \(h\) affects the accuracy and stability of the approximation; smaller values generally yield more accurate results but require more computational effort. For differential equations, numerical integration allows us to evolve the solution over time based on the equation's dynamic properties.

Euler's Method

Euler's Method is a simple, yet fundamental numerical technique used to approximate the solutions to ODEs. It is a first-order method, which implies that the error per step is proportional to the square of the step size. The basic idea is to use the slope at one point to predict the value of the function at the next point. The formula is
\[ y_{n+1} = y_n + h f(x_n, y_n) \],
where \( \left(x_n, y_n\right) \) is the current point, \( h \) is the step size, and \( f(x_n, y_n) \) is the rate of change at the current point. Despite its simplicity, Euler’s Method can provide insights into the behavior of more complex methods and serves as the building block for many other numerical methods used in ODEs.

Predictor-Corrector Method

The Predictor-Corrector Method is a type of numerical algorithm that improves the accuracy of the basic Euler's Method by using an initial estimate (predictor) and refining it with a corrector step. The Milne method, as seen in the exercise, is an example of a Predictor-Corrector approach. The predictor step often uses Euler's formula to estimate \(y_{n+1}\), and the corrector step refines this prediction by using more current or additional information about the slope of the function. This additional step typically involves a more complex formula, providing a result that is closer to the true solution. The overall goal of Predictor-Corrector methods is to balance computational efficiency with improved accuracy and stability over the simpler Euler’s Method.