Question

Formulate a version of the improved Euler method (Section 3.2) for the numerical solution of the initial value problem $$ \begin{array}{ll} y_{1}^{\prime}= & g_{1}\left(t, y_{1}, y_{2}\right), \quad y_{1}\left(t_{0}\right)=y_{10}, \\ y_{2}^{\prime}=g_{2}\left(t, y_{1}, y_{2}\right), & y_{2}\left(t_{0}\right)=y_{20}, \end{array} $$ on an interval \(\left[t_{0}, b\right]\).

Short answer

Based on the given system of ODEs, we have applied the improved Euler method and derived the following iterative scheme for both \(y_1\) and \(y_2\): For \(y_1\): 1. Predictor step: \(\tilde{y}_{1, i+1} = y_{1, i} + h g_{1}(t_i, y_{1, i}, y_{2, i})\) 2. Corrector step: \(y_{1, i+1} = y_{1, i} + \frac{h}{2}\left(g_{1}(t_i, y_{1, i}, y_{2, i}) + g_{1}(t_{i+1}, \tilde{y}_{1, i+1}, y_{2, i})\right)\) For \(y_2\): 1. Predictor step: \(\tilde{y}_{2, i+1} = y_{2, i} + h g_{2}(t_i, y_{1, i}, y_{2, i})\) 2. Corrector step: \(y_{2, i+1} = y_{2, i} + \frac{h}{2}\left(g_{2}(t_i, y_{1, i}, y_{2, i}) + g_{2}(t_{i+1}, y_{1, i}, \tilde{y}_{2, i+1})\right)\) Using these equations and the given initial values \(y_{1}(t_0) = y_{10}\) and \(y_{2}(t_0) = y_{20}\), we can numerically solve the system of ODEs on the interval \([t_0, b]\) using the improved Euler method.

Step-by-step solution

Write down the given system of ODEs and initial values

The given system of ODEs and initial value problem is: $$ \begin{array}{ll} y_{1}^{\prime}= & g_{1}\left(t, y_{1}, y_{2}\right), \quad y_{1}\left(t_{0}\right)=y_{10}, \\\ y_{2}^{\prime}=g_{2}\left(t, y_{1}, y_{2}\right), & y_{2}\left(t_{0}\right)=y_{20}, \end{array} $$ Our goal is to apply the improved Euler method for both ODEs to find a numerical solution for \(y_1\) and \(y_2\) on the interval \([t_0, b]\).

Recall the algorithm of the improved Euler method

The improved Euler method is a two-step method for solving ODEs: 1. Predictor step: Calculate an Euler step, giving an intermediate approximation. $$\tilde{y}_{i+1} = y_i + hf(t_i, y_i)$$ 2. Corrector step: Use the intermediate approximation to refine the approximation at the next step. $$y_{i+1} = y_i + \frac{h}{2}\left(f(t_i, y_i) + f(t_{i+1}, \tilde{y}_{i+1})\right)$$ Here, \(f\) represents the function on the right-hand side of the ODE, and \(h\) is the step size.

Apply the improved Euler method for the first ODE

For the first ODE, we have: $$y_{1}^{\prime} = g_{1}\left(t, y_{1}, y_{2}\right)$$ 1. Predictor step for \(y_1\): $$\tilde{y}_{1, i+1} = y_{1, i} + h g_{1}(t_i, y_{1, i}, y_{2, i})$$ 2. Corrector step for \(y_1\): $$y_{1, i+1} = y_{1, i} + \frac{h}{2}\left(g_{1}(t_i, y_{1, i}, y_{2, i}) + g_{1}(t_{i+1}, \tilde{y}_{1, i+1}, y_{2, i})\right)$$

Apply the improved Euler method for the second ODE

For the second ODE, we have: $$y_{2}^{\prime} = g_{2}\left(t, y_{1}, y_{2}\right)$$ 1. Predictor step for \(y_2\): $$\tilde{y}_{2, i+1} = y_{2, i} + h g_{2}(t_i, y_{1, i}, y_{2, i})$$ 2. Corrector step for \(y_2\): $$y_{2, i+1} = y_{2, i} + \frac{h}{2}\left(g_{2}(t_i, y_{1, i}, y_{2, i}) + g_{2}(t_{i+1}, y_{1, i}, \tilde{y}_{2, i+1})\right)$$

Combine the results

We now have the improved Euler method applied to the given system of ODEs: For \(y_1\): 1. Predictor step: \(\tilde{y}_{1, i+1} = y_{1, i} + h g_{1}(t_i, y_{1, i}, y_{2, i})\) 2. Corrector step: \(y_{1, i+1} = y_{1, i} + \frac{h}{2}\left(g_{1}(t_i, y_{1, i}, y_{2, i}) + g_{1}(t_{i+1}, \tilde{y}_{1, i+1}, y_{2, i})\right)\) For \(y_2\): 1. Predictor step: \(\tilde{y}_{2, i+1} = y_{2, i} + h g_{2}(t_i, y_{1, i}, y_{2, i})\) 2. Corrector step: \(y_{2, i+1} = y_{2, i} + \frac{h}{2}\left(g_{2}(t_i, y_{1, i}, y_{2, i}) + g_{2}(t_{i+1}, y_{1, i}, \tilde{y}_{2, i+1})\right)\) With these equations and given initial values \(y_{1}(t_0) = y_{10}\) and \(y_{2}(t_0) = y_{20}\), we can numerically solve the system of ODEs on the interval \(\left[t_{0}, b\right]\) using the improved Euler method.

Key concepts

Numerical Solutions of ODEs

Solving ordinary differential equations (ODEs) analytically can be challenging, especially when dealing with complex or nonlinear equations. This is where numerical solutions come into play. Numerical methods provide approximate solutions to ODEs, allowing us to use computers to calculate and interpret these solutions.
The numerical solution involves approximating the behavior of the differential equation over an interval by discretizing time. One of the most common numerical methods is the Euler method, which lays the groundwork for more advanced techniques.
The Improved Euler Method, also known as Heun's method, enhances the basic Euler approach by using a predictor-corrector mechanism. This typically results in more accurate approximations. The main idea is to take an average of the derivatives at the current and the initially predicted points, giving a better approximation of the solution's behavior.

Initial Value Problems

In the context of ODEs, an initial value problem (IVP) is a type of problem where the solution to the differential equation is to be determined over an interval, starting from an initial condition. These problems consist of a differential equation and a specified value, or set of values, at the starting point of the interval.
Initial conditions are crucial as they determine the behavior and uniqueness of the solution. For instance, the values of variables at time zero influence how the solution evolves. In our example, the initial values for \(y_1\) and \(y_2\) are given at time \(t_0\) as \(y_{10}\) and \(y_{20}\).
Efficiently solving initial value problems enables the modeling of many real-world phenomena, like the motion of planets, population growth, or chemical reactions. The Improved Euler Method effectively handles initial value problems by progressively building a solution using the known starting point.

Predictor-Corrector Methods

Predictor-Corrector methods are a class of numerical techniques used to approximate solutions of ODEs. These methods work by initially predicting a solution (predictor) and then refining this prediction (corrector) to improve accuracy.
The Improved Euler Method employs this technique effectively. In the predictor step, a simple approximation is calculated using Euler's method. This step gives a rough estimate, sometimes referred to as the forward or initial Euler step.
Following this, the corrector step adjusts this initial estimate by using a more refined formula that considers the slope at both the old and new points. This kind of averaging generally leads to a more accurate solution.
These two steps work together: the predictor gives a preliminary guess, and the corrector provides a way to adjust this guess based on the actual trend of the differential equation. This collaboration between prediction and correction is key to improved accuracy and stability in numerical solutions of ODEs.