Last modified: April 28, 2018
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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:
$$u' = f(t, u),$$ $$u(t_0) = u_0.$$
This IVP can be solved by Euler's method, where the method employs the following approximation:
$$u_{n+1} = u_n + h*f(t_n, u_n),$$
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:
$$ u(t+h)=u(t)+h u'(t) + O(h^2) $$
We may alternatively rewrite the above equation as follows:
$$ u(t+h)=u(t)+ h f(u(t),t)+ O(h^2).$$
Which is roughly equivalent to:
$$ u(t+h)=u(t)+ h f(u(t),t)$$
Algorithm Steps
- Start with initial conditions $t_0$ and $u_0$.
- Calculate $u_{n+1}$ using the formula: $u_{n+1} = u_n + h*f(t_n, u_n)$.
- Repeat the above step for a given number of steps or until the final value of $t$ is reached.
Example
$$ u'(t)=u(t),$$
$$ u(0)=1$$
$$u(0.1)=?$$
Let's choose the step value: $h = 0.05$
We start at $t=0$:
$$ u(0.05) \approx u(0)+0.05u'(0) $$
$$ u(0.05) \approx1+0.05u(0) $$
$$ u(0.05) \approx1+0.05 \cdot 1 $$
$$ u(0.05) \approx 1.05 $$
Now that we know $u(0.05)$, we can calculate the second step:
$$ u(0.1) \approx u(0.05)+0.05u'(0.05) $$
$$ u(0.1) \approx1.05+0.05u(0.05) $$
$$ u(0.1) \approx1.05+0.05 \cdot 1.05 $$
$$ u(0.1) \approx 1.1025 $$
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.