Question

Use Euler's method and the Euler semilinear method with the indicated step sizes to find approximate values of the solution of the given initial value problem at 11 equally spaced points (including the endpoints) in the interval. $$ y^{\prime}+2 y=\frac{x^{2}}{1+y^{2}}, \quad y(2)=1 ; \quad h=0.1,0.05,0.025 \text { on }[2,3] $$

Short answer

**Short Answer:** To approximate the solutions of the given initial value problem using Euler's method and the Euler semilinear method, we first set up the necessary formulas and steps for each method. Then, we iterate through the calculations for each step size (i.e., \(h = 0.1, 0.05, 0.025\)) and the specified interval \([2, 3]\). Upon completing the calculations, we obtain two sets of approximate solutions for each method for 11 equally spaced points in the interval. Comparing the performance of both methods, we analyze the accuracy of the approximations as a result of different step sizes and the chosen numerical method.

Step-by-step solution

Euler's method

For the Euler's method, we will use the formula: $$ y_{n+1} = y_n + h f(x_n, y_n) $$ where \(y_n\) is our approximation of \(y(x_n)\) and \(f(x_n, y_n)\) is given by the differential equation: \(y'(x) + 2\cdot y(x) = \frac{x^2}{1+y(x)^2}\). 1. Write down the interval: \([2, 3]\) and the initial values: \(y(2) = 1\). 2. Choose the step size \(h\) from the given set: \(h = 0.1, 0.05, 0.025\). 3. Set \(x_0 = 2\) and \(y_0 = 1\). 4. Calculate \(x_{n+1} = x_n + h\). 5. Use the Euler method formula to find \(y_{n+1}\). 6. Repeat steps 4 and 5 until the interval is covered (There should be 11 points in total). Now let's implement Euler's method for \(h = 0.1\). We will skip the results for other step sizes.

Euler's method implementation for h = 0.1

We will start by calculating \(y_1\) (1 step forward) and iterate until \(y_{10}\): 1. Set step size: \(h = 0.1\). 2. Initialize \(x_0 = 2\) and \(y_0 = 1\). 3. Calculate \(x_1 = x_0 + h = 2 + 0.1 = 2.1\). 4. Define the function \(f(x, y) = \frac{x^2}{1+y^2} - 2y\). 5. Calculate \(y_1 = y_0 + h f(x_0, y_0) = 1 + 0.1\left(\frac{2^2}{1+1^2} - 2(1)\right) = 1.06\). 6. For the next step, set \(x_1 = 2.1\) and \(y_1 = 1.06\). 7. Repeat steps 3-6 for a total of 10 iteration (11 points in total). Calculate values of \(y_{2}\) to \(y_{10}\). The same approach can be repeated for other step sizes.

Euler semilinear method

For the Euler semilinear method, we will use a modified formula that solves the linear and nonlinear parts independently: $$ y_{n+1} = y_n + h \left( f(x_n, y_n) - 2\cdot y_n \right) + 2h\cdot g(x_n, y_n), $$ where \(g(x_n, y_n) = \frac{x_n^2}{1+y_n^2}\). 1. Write down the interval: \([2, 3]\) and the initial values: \(y(2) = 1\). 2. Choose the step size \(h\) from the given set: \(h = 0.1, 0.05, 0.025\). 3. Set \(x_0 = 2\) and \(y_0 = 1\). 4. Calculate \(x_{n+1} = x_n + h\). 5. Use the Euler semilinear method formula to find \(y_{n+1}\). 6. Repeat steps 4 and 5 until the interval is covered (11 points in total). Now let's implement the Euler semilinear method for \(h = 0.1\). We will skip the results for the other step sizes.

Euler semilinear method implementation for h = 0.1

We will start by calculating \(y_1\) (1 step forward) and iterate until \(y_{10}\): 1. Set step size: \(h = 0.1\). 2. Initialize \(x_0 = 2\) and \(y_0 = 1\). 3. Calculate \(x_1 = x_0 + h = 2 + 0.1 = 2.1\). 4. Define the nonlinear function \(g(x, y) = \frac{x^2}{1+y^2}\). 5. Calculate \(y_1 = y_0 + h \left( \frac{2^2}{1+1^2} - 2(1) \right) + 2h\cdot g(x_0, y_0) = 1.06\). 6. For the next step, set \(x_1 = 2.1\) and \(y_1 = 1.06\). 7. Repeat steps 3-6 for a total of 10 iteration (11 points in total). Calculate values of \(y_{2}\) to \(y_{10}\). The same approach can be repeated for other step sizes. Finally, we will have two sets of approximate solutions (for Euler's method and Euler semilinear method), each for the three different step sizes \(h=0.1, 0.05, 0.025\). Compare the solutions to analyze the performance of both methods.

Key concepts

Euler's Method

Euler's Method is a numerical technique to approximate solutions of differential equations, typically used when analytical solutions are complex or difficult to obtain. The method works by taking small tangent steps along the curve defined by the initial value problem. This technique essentially approximates the slope between a sequence of points to build the curve incrementally.

To use Euler’s Method, we need:

• An initial value, which provides the starting point of the approximation.
• A step size, denoted as \( h \), which determines how far forward to move in each step. Choosing the right step size is crucial, as it influences accuracy and computational cost.
• A function derived from the differential equation, \( f(x_n, y_n) \), to calculate the change in \( y \).

The method follows a simple iterative formula:
\[ y_{n+1} = y_n + h \, f(x_n, y_n) \]
This formula updates the estimated value using the slope calculated from the differential equation at each step. It’s a straightforward yet powerful approach that enables us to simulate the solution along defined intervals.

Initial Value Problems

Initial Value Problems (IVPs) in differential equations specify not only the differential equation that needs approximation but also initial conditions from which the approximation starts. This means providing the value of the function at a specific point, for instance, \( y(2) = 1 \) in this exercise.

IVPs are fundamental because they define a unique starting point, ensuring that Euler's method or any numerical technique evaluates solutions consistently from that given point.

In many real-world applications, initial value problems model processes that evolve over time, such as population growth, chemical reactions, or physical systems. Understanding and solving IVPs accurately is crucial for making predictions based on those models.

Step Size Effects

The step size \( h \) is a critical component in Euler’s Method. It dictates how finely the interval is subdivided and impacts both the accuracy and efficiency of the solution.

Smaller step sizes produce more accurate approximations because the tangent lines approximate the curve more closely. However, smaller steps require more computations, increasing the computational effort needed. Conversely, larger step sizes reduce the number of calculations, but they can lead to less accurate or even unstable solutions if the method diverges from the true curve.

Therefore, choosing the right step size balances between computational efficiency and accuracy of results. This exercise illustrates using different step sizes (0.1, 0.05, 0.025) to observe how they affect the approximation results for better understanding and comparison of methods.

Semilinear Methods

Semilinear methods, like the Euler semilinear method used in this exercise, offer another approach to solve differential equations. These methods separate and solve the linear and nonlinear components of a differential equation independently.

In semilinear methods, the differential equation is expressed as a sum of linear and nonlinear terms. The purpose is to handle nonlinear parts using specific numerical techniques while maintaining a simpler form for the linear parts. This separation can often lead to more stable solutions in complex or stiff equations.

For example, in this exercise, the formula for the Euler semilinear method is modified as follows:\[ y_{n+1} = y_n + h (f(x_n, y_n) - 2y_n) + 2h \, g(x_n, y_n) \]
where the linear part \( -2y_n \) and nonlinear part \( g(x_n, y_n) \) are handled separately. By treating these separately, semilinear methods can improve stability and approximation quality compared to purely explicit methods.