๐ Deriving Euler's Method from Taylor Series
Euler's method is a numerical technique used to approximate the solution of an ordinary differential equation (ODE). It's a first-order method, meaning it uses information from only the previous time step to estimate the solution at the next time step. The derivation stems directly from the Taylor series expansion.
๐ History and Background
Leonhard Euler, a Swiss mathematician, developed this method in the 18th century. It's one of the most fundamental and intuitive numerical methods for solving ODEs. While not as accurate as higher-order methods, its simplicity makes it an excellent starting point for understanding numerical solutions.
๐ Key Principles
- ๐ Taylor Series Expansion: The core idea is to approximate a function $y(t)$ around a point $t_0$ using its Taylor series:
$y(t_0 + h) = y(t_0) + hy'(t_0) + \frac{h^2}{2!}y''(t_0) + \frac{h^3}{3!}y'''(t_0) + ...$, where $h$ is a small step size.
- โ๏ธ Truncation: Euler's method truncates the Taylor series after the first-order term. This means we only consider the terms $y(t_0)$ and $hy'(t_0)$.
- โ๏ธ Approximation: We approximate $y(t_0 + h)$ as $y(t_0 + h) \approx y(t_0) + hy'(t_0)$. This is the heart of Euler's method.
- ๐ฑ Differential Equation: We're solving an ODE of the form $y'(t) = f(t, y(t))$. This means the derivative of $y$ is a function of both $t$ and $y$.
- ๐ Iterative Process: We start with an initial condition $y(t_0) = y_0$ and iteratively apply the formula $y_{i+1} = y_i + h f(t_i, y_i)$ to approximate the solution at subsequent time steps.
๐ช Step-by-Step Derivation
- Start with the Taylor Series: Begin with the Taylor series expansion of $y(t + h)$ around $t$: $y(t + h) = y(t) + hy'(t) + \frac{h^2}{2!}y''(t) + ...$
- Truncate the Series: Truncate the series after the first-order term: $y(t + h) \approx y(t) + hy'(t)$
- Substitute the ODE: Substitute $y'(t) = f(t, y(t))$ into the truncated series: $y(t + h) \approx y(t) + hf(t, y(t))$
- Iterative Formula: Let $t_{i+1} = t_i + h$ and $y_{i+1} \approx y(t_{i+1})$. We get the iterative formula: $y_{i+1} = y_i + h f(t_i, y_i)$. This is Euler's method.
๐ Real-world Examples
- ๐ฆ Population Growth: Modeling population growth where the rate of change of population is proportional to the current population.
- ๐ก๏ธ Cooling of an Object: Simulating the cooling of an object using Newton's law of cooling.
- ๐ Trajectory Prediction: Approximating the trajectory of a projectile under the influence of gravity and air resistance.
๐ก Conclusion
Euler's method provides a simple and intuitive way to approximate solutions to ODEs. While its accuracy is limited by its first-order nature, it serves as a foundational concept for understanding more advanced numerical methods. By understanding its derivation from the Taylor series, you gain a deeper appreciation for its underlying principles and limitations.
