AHL 5.16—Eulers method for 1st order DEs Notes

Euler's Method for First Order Differential Equations

• Euler's method is a numerical technique used to find approximate solutions to first-order ordinary differential equations (ODEs) with initial value problems.
• Named after the Swiss mathematician Leonhard Euler, this method provides a step-by-step approach to estimate the solution of an ODE when analytical methods are difficult or impossible to apply.

Basic Principle

• The fundamental idea behind Euler's method is to use the slope of the solution curve at a given point to estimate the solution's value at a nearby point.
• This process is then repeated to generate a series of approximate solution points.

For a differential equation of the form:

$\frac{dy}{dx} = f(x,y)$

with an initial condition $y(x_0) = y_0$, Euler's method approximates the solution using the following iterative formula:

$y_{n+1} = y_n + h \cdot f(x_n, y_n)$

where:

• $y_n$ is the current y-value
• $x_n$ is the current x-value
• $h$ is the step size
• $f(x_n, y_n)$ is the slope of the tangent line at the point $(x_n, y_n)$

Implementation Steps

1. Choose a step size $h$
2. Start with the initial condition $(x_0, y_0)$
3. Calculate the slope $f(x_0, y_0)$
4. Use the formula to find $y_1$
5. Repeat steps 3-4 for subsequent points

Using Spreadsheets for Euler's Method

Spreadsheet software like Microsoft Excel or Google Sheets can be powerful tools for implementing Euler's method, especially for larger numbers of iterations.

Setting Up a Spreadsheet

1. Create columns for $x$, $y$, and $f(x,y)$
2. Enter the initial values and step size
3. Use formulas to calculate subsequent values

Coupled Systems of Differential Equations

Euler's method can be extended to solve systems of coupled differential equations, which often arise in modeling complex phenomena like predator-prey relationships.

General Form

For a system of the form:

$$\begin{align*} \frac{dx}{dt} &= f_1(x,y,t) \ \frac{dy}{dt} &= f_2(x,y,t) \end{align*}$$

The Euler method equations become:

$$\begin{align*} x_{n+1} &= x_n + h \cdot f_1(x_n, y_n, t_n) \ y_{n+1} &= y_n + h \cdot f_2(x_n, y_n, t_n) \end{align*}$$

Predator-Prey Models

One common application of coupled systems is the Lotka-Volterra equations, which model predator-prey interactions:

$$\begin{align*} \frac{dx}{dt} &= ax - bxy \ \frac{dy}{dt} &= -cy + dxy \end{align*}$$

where:

• $x$ is the prey population
• $y$ is the predator population
• $a$, $b$, $c$, and $d$ are positive constants

Error in Euler’s Method: Local and Global Truncation Errors

1. Euler’s method, while simple and effective, introduces truncation errors due to its approximation of the derivative at discrete points.
2. These errors can be categorized into local truncation error and global truncation error.

• Local truncation error refers to the error made in a single step of Euler’s method.
  • It arises because the method assumes that the slope remains constant over the step size $h$, even though the actual solution curve may be curved.
  • This error is proportional to $h^2$, meaning that smaller step sizes reduce the per-step error significantly.
• Global truncation error accumulates over multiple steps, leading to a total deviation from the exact solution.
  • Since there are $\frac{1}{h}$​ steps in a given interval, the overall error is proportional to $h$, meaning that halving the step size approximately halves the total error.

To improve accuracy, one can reduce the step size or use higher-order numerical methods such as the Runge-Kutta method, which has a smaller global error. Understanding these errors helps in choosing an appropriate step size to balance accuracy and computational efficiency.