16.6. Worked Example

Given: $\frac{dy}{dx} = x + y, \quad y(0) = 1, \quad h = 0.1$ Use 1-step Adams–Moulton (Trapezoidal Rule) to compute $y(0.1)$

In this worked example, we're given a first-order ordinary differential equation (ODE) $\frac{dy}{dx} = x + y$ with an initial condition $y(0) = 1$ and a step size of $h = 0.1$. The goal is to find the value of $y$ at x = 0.1 using the 1-step Adams–Moulton method, which is a numerical technique for approximating solutions to ODEs.

Think of this problem like tracking the growth of a plant that grows proportionally to its current height and the time passed. Essentially, we want to predict how tall the plant (y) will be after a short time (0.1 hours) given its initial height.

Step 1: Use RK2 or Euler for first point (say, use Euler): $y_{Euler} = y + h f(x_0, y_0) = 1 + 0.1(0 + 1) = 1.1$

In this first step, we need an initial estimation of the solution at $x = 0.1$. We can use Euler's method to do this. We calculate the slope at the initial point $(0, 1)$ which is given by the function $f(x, y) = x + y$. Substituting $x = 0$ and $y = 1$ into the function gives us 1. We then multiply this slope by our step size (0.1) and add it to our initial value of 1, leading to an estimated value of $y$ at $x=0.1$ of 1.1.

Imagine you are measuring how much a plant grows in the first few minutes. If it was initially at 1 cm and grows at a rate of 1 cm per minute, after 0.1 minutes, you’d expect it to be about 1.1 cm tall.

Step 2: Apply Trapezoidal Rule: $f_0 = f(0, 1) = 1, \quad f_1 = f(0.1, 1.1) = 0.1 + 1.1 = 1.2$ $y_{n+1} = y_n + \frac{h}{2} (f_0 + f_1) = 1 + \frac{0.1}{2} (1 + 1.2) = 1 + 0.11 = 1.11$

Next, we apply the Trapezoidal Rule to approximate $y(0.1)$. We first calculate the function values at the two points: $f_0 = f(0, 1)$ gives us 1, and then we evaluate the function at the newly estimated point $f_1 = f(0.1, 1.1)$, which computes to 1.2. We then average these two function values as we apply the trapezoidal formula, yielding an updated estimate of $y(0.1) = 1.11$.

Continuing with our plant example, after estimating the plant’s height at 0.1 minutes, we look at its growth rate at both the start (0 minutes) and the next point (0.1 minutes) to better understand how much it has grown in that interval and thus get a more accurate height.

So corrected $y(0.1) = 1.11$

After completing our calculations using the Adams-Moulton (Trapezoidal) method, we find the corrected value of the function at $x = 0.1$ to be $y(0.1) = 1.11$. This is our estimated solution based on the method we used, which integrates the slope information from the two points considered.

In terms of our plant growth analogy, after measuring how much the plant has grown in the first 0.1 minutes, we conclude that it is approximately 1.11 cm tall, taking into account its growth rate at the beginning and at the new height.

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