Question

Carry out one step of the Euler method and of the improved Euler method using the step size \(h=0.1 .\) Suppose that a local truncation error no greater than 0.0025 is required. Estimate the step size that is needed for the Euler method to satisfy this requirement at the first step. $$ y^{\prime}=\left(y^{2}+2 t y\right) /\left(3+t^{2}\right), \quad y(0)=0.5 $$

Short answer

Using the Euler method, the first step approximation for the given differential equation is approximately y₁ ≈ 0.5083. For the Improved Euler method, the first step approximation is y₁ ≈ 0.5084. To achieve a local truncation error no greater than 0.0025, the step size required for the Euler method is approximately h ≈ 0.023.

Step-by-step solution

(1) Euler Method - First Step

Start with the differential equation \(y'(t) = \frac{y^2 + 2ty}{3+t^2}\), and apply the Euler method: $$y_{n+1} = y_n + hf(t_n, y_n),$$ where h = 0.1, \(f(t_n, y_n)\) is the value of the differential equation at the point \((t_n, y_n)\), and \((t_0, y_0) = (0, 0.5)\). First, calculate the value of the differential equation at the initial point, \(f(t_0, y_0) = f(0, 0.5) = \frac{(0.5)^2 + 2(0)(0.5)}{3 + (0)^2} = \frac{0.25}{3}\). Next, apply the Euler method with h = 0.1: $$y_1 = y_0 + 0.1\cdot f(t_0, y_0) = 0.5 + 0.1\cdot \frac{0.25}{3} \approx 0.5083.$$

(2) Improved Euler Method - First Step

Now, apply the Improved Euler method (also known as Heun's method). The formula for the improved Euler method is: $$y_{n+1} = y_n + \frac{h}{2}\left( f(t_n, y_n) + f(t_{n+1}, y_n+hf(t_n, y_n)) \right).$$ Use the previous result, \(f(t_0, y_0) = \frac{0.25}{3}\), and calculate \(f(t_1, y_1) = f(0.1, 0.5083) \approx \frac{(0.5083)^2 + 2(0.1)(0.5083)}{3 + (0.1)^2}\). Finally, apply the Improved Euler Method: $$y_1 \approx 0.5 + \frac{0.1}{2}\left( \frac{0.25}{3} \right) \approx 0.5084$$.

(3) Determine step size for required local truncation error

A local truncation error no greater than 0.0025 is required. The local truncation error for the Euler method can be estimated as: $$\text{Local Truncation Error} \approx \frac{hL}{2}|y(t_1) - y_0|,$$ where L is the Lipschitz constant. First, we need to find L. To do so, compute the partial derivative of the differential equation with respect to y: $$\frac{\partial f}{\partial y} = \frac{2y (3 + t^2) - 2yt (y^2 + 2ty)}{(3+t^2)^2}$$. Notice that the maximum value of the denominator occurs when t = 0, that is: $$(3+t^2)^2 \geq 3^2 = 9$$. Now, we estimate the maximum value of the numerator. Since \(y_0 = 0.5\), \(t=0\), we can find it as follows: $$\max_{0\leq y \leq y_1} |2y (3 + t^2) - 2yt (y^2 + 2ty)| \leq |2y_1 (3 + 0) - 2y_1^2|$$. So, we can estimate L as: $$L \leq \frac{|2y_1 (3) - 2y_1^2|}{9}$$. Now, plug L into the formula for the local truncation error and solve for h: $$0.0025 \geq \frac{hL}{2}|y_1 - y_0| \Rightarrow h \leq \frac{0.005}{L|y_1-y_0|}$$. Use the previous result and plug in the estimated L to find the appropriate step size h: $$h \leq \frac{0.005}{\frac{|2y_1 (3) - 2y_1^2|}{9}|y_1-y_0|} \approx 0.023$$. As such, the step size required to satisfy the local truncation error requirement is approximately h ≈ 0.023.

Key concepts

Euler Method

The Euler Method is one of the simplest numerical approaches for solving ordinary differential equations (ODEs). It provides a straightforward way to approximate solutions, particularly when analytical solutions are hard to find.
Let's break it down:

• Start with a known initial value, often denoted as \(y_0\), at a point \(t_0\). In our case, this point is (0, 0.5).
• The differential equation tells us the slope at this point. Using \(y' = (y^2 + 2ty) / (3 + t^2)\), we calculate the slope at \(t_0\).

The Euler method then uses this slope times the step size \(h\) to estimate the value at the next point. Mathematically, it's represented as: \[y_{n+1} = y_n + h \, f(t_n, y_n)\]In the exercise, with \(h = 0.1\), \(y_0 = 0.5\), and calculated slope \(f(t_0, y_0) \approx 0.0833\), the first-step estimate becomes:\[y_1 \approx 0.5 + 0.1 \cdot 0.0833 = 0.5083\]The Euler Method is easy to use, but note that its simplicity also limits its accuracy. Hence, other methods like the Improved Euler Method offer greater precision.

Improved Euler Method

The Improved Euler Method, also known as Heun's Method, is an enhancement of the classical Euler Method. It provides better accuracy by using an average of slopes rather than just one.
Here’s how it works:

• Like the Euler method, it starts with an initial condition, e.g., \(y_0\) at \(t_0\).
• First, calculate the slope at this initial point using the original function. This is similar to the first step in the Euler Method.
• Next, make an initial estimate of the new point \(y_1'\) using the first slope.
• Calculate the slope at this new point.
• The final \(y_1\) estimate averages these two slopes.

The formula looks like this:\[y_{n+1} = y_n + \frac{h}{2} \left(f(t_n, y_n) + f(t_{n+1}, y_n + h f(t_n, y_n)) \right)\]Using the provided values, the Improved Euler method gives us:\[y_1 \approx 0.5 + \frac{0.1}{2} \left(0.0833 + f(0.1, 0.5083)\right)\]Ultimately, this results in a slightly more accurate estimate: \(y_1 \approx 0.5084\). By averaging the slopes, the Improved Euler Method significantly reduces error over one-step approximations.

Local Truncation Error

Local Truncation Error is crucial in determining the accuracy of numerical methods. It refers to the error made in one step of a method, due to approximations replacing the true underlying continuous function.For the Euler Method, the local truncation error can be approximated by:\[\text{Local Truncation Error} \approx \frac{hL}{2} |y(t_1) - y_0|\]where \(L\) is the Lipschitz constant, related to how steeply the function can change and often derived from the partial derivatives of the function.Calculating \(L\) involves examining the function's derivative relative to \(y\):\[\frac{\partial f}{\partial y} = \frac{2y (3 + t^2) - 2yt (y^2 + 2ty)}{(3+t^2)^2}\]This helps identify bounds on how rapidly results can change, guiding us in choosing an appropriate step size. For our requirement of error no greater than 0.0025, a step size of approximately \(h = 0.023\) was determined. This smaller step size ensures precision and accuracy, fulfilling the error criteria by taking into account potential changes in the function over a short interval. Choosing the correct step size is crucial in managing and minimizing local truncation error.