Euler Method

• In an initial-value problem, the numerical solution begins at the initial point and marches from there to increasing values for the independent variable.
• The Euler method. Describes a method that is easy to use but is not very precise unless the step size, the intervals for the projection of the solution, is very small.
• Consider the following first-order differential equation:

  $$\frac{dy}{dx}=y'(x)=f(x,y)~~~\&~~~y(x_0)=y_0$$ (5.1)

• Here x is the variable, y(x) and f(x,y) are real functions, and the initial condition $y_0$ is a real number.
• From the solution of this equation, we get $y_1, y_2, \ldots, y_n$ values for the function at the points $x_1,x_2, \ldots, x_n$ with equal step lengths $h$.
• Equations of higher order are solved by converting them to a system of linear equations.

• The expression given by Equation 5.1 is written as the forward-difference approximation at a point $x_i$ by Euler's method.
  $$\frac{y_{i+1}-y_i}{h}+O(h)=f(x_i,y_i)$$

• If we solve this expression for $y_{i+1}$, we get the Euler method formula:
  $$\boxed{y_{i+1}=y_i+hf(x_i,y_i)+O(h^2)}$$

• This expression shows that the error in one step of Euler method is $O(h^2)$. But, this error is just the local error. Over many steps, the global error becomes $O(h)$ (as $NO(h^2) \approx O(h)$ for N steps).
• The method is easy to program when we know the formula for $y'(x)(\equiv f(x_i,y_i))$ and a starting value, $y_0 = y(x_0)$.

• Let's see the application of this method on an example. Given differential equation,

  $$\frac{dy}{dx}=x+y$$

• The analytical solution of this equation is given as $y(x)=2e^x-x-1$. Initial condition: $y(x=0)=1$

Table 5.1: Solution of the differential equation $dy/dx=x+y$ in the interval [0, 1] by Euler method.

(Example py-file: myeuler.py) As can be seen from the table, the margin of error is large in the Euler method.

Figure 5.1: Solution of the differential equation $dy/dx=x+y$ in the interval [0, 1] by Euler method.