Last modified: September 30, 2023

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Euler's Method

Euler's Method is a numerical technique applied in the realm of initial value problems for ordinary differential equations (ODEs). The simplicity of this method makes it a popular choice in cases where the differential equation lacks a closed-form solution. The method might not always provide the most accurate result, but it offers a good trade-off between simplicity and accuracy.

Mathematical Formulation

Consider an initial value problem (IVP) represented as:

$$u^{\prime }=f(t,u),$$ $$u\left(t_0\right)=u_0.$$

This IVP can be solved by Euler's method, where the method employs the following approximation:

$$u_{n+1}=u_n+h\ast f(t_n,u_n),$$

where: - $u_{n+1}$ is the approximate value of $u$ at $t=t_n+h$, - $u_n$ is the approximate value of $u$ at $t=t_n$, - $h$ is the step size, - $f(t_n,u_n)$ is the derivative of $u$ at $t=t_n$.

Derivation

Let's start with the Taylor series:

$$u(t+h)=u\left(t\right)+h{u}^{\prime }\left(t\right)+O\left(h^2\right)$$

We may alternatively rewrite the above equation as follows:

$$u(t+h)=u\left(t\right)+hf\left(u\right(t),t)+O\left(h^2\right).$$

Which is roughly equivalent to:

$$u(t+h)=u\left(t\right)+hf\left(u\right(t),t)$$

Algorithm Steps

1. Start with initial conditions $t_0$ and $u_0$.
2. Calculate $u_{n+1}$ using the formula: $u_{n+1}=u_n+h\ast f(t_n,u_n)$.
3. Repeat the above step for a given number of steps or until the final value of $t$ is reached.

Example

$$u^{\prime }\left(t\right)=u\left(t\right),$$

$$u\left(0\right)=1$$

$$u\left(0.1\right)=?$$

Let's choose the step value: $h=0.05$

We start at $t=0$:

$$u\left(0.05\right)\approx u\left(0\right)+0.05u^{\prime }\left(0\right)$$

$$u\left(0.05\right)\approx 1+0.05u\left(0\right)$$

$$u\left(0.05\right)\approx 1+0.05\cdot 1$$

$$u\left(0.05\right)\approx 1.05$$

Now that we know $u\left(0.05\right)$, we can calculate the second step:

$$u\left(0.1\right)\approx u\left(0.05\right)+0.05u^{\prime }\left(0.05\right)$$

$$u\left(0.1\right)\approx 1.05+0.05u\left(0.05\right)$$

$$u\left(0.1\right)\approx 1.05+0.05\cdot 1.05$$

$$u\left(0.1\right)\approx 1.1025$$

Advantages

• Euler's method is easy to implement and serves as a foundational technique for introducing numerical solution methods for ODEs.
• It can provide reasonable approximations for well-behaved functions when the step size is sufficiently small.
• The simplicity of the method makes it computationally inexpensive for basic problems and suitable for initial experimentation.

Limitations

• Accuracy is limited, as the method can introduce significant errors if the step size is too large or if the function has rapid changes.
• The cumulative error from each step can grow significantly, making the method unsuitable for long-time integration.
• Stability is an issue, as Euler's method may fail to converge or produce reliable results for stiff or oscillatory ODEs.
• The method lacks the ability to adapt step sizes dynamically, which can lead to inefficiencies or inaccuracies in varying conditions.