Differential Equations

Euler developed a method for finding the approximate solution to initial value problems. The differential equations to be solved have the form of:

dy/dx = f ( x, y ) x range is [a, b], initial value y(a) is known.

The approximation of dy/dx is, therefore the next data point, y_i+1 can be calculated by present data point, x_i and y_i. The x range of [a, b] can be divided equally into N steps, the length of the step, ; there are N+1 data points; the x data points are . The y data points are with y₁=y(a).

For example: to solve the differential equation, dy/dx = xy, in the x range of [0, 1], with x increment, dx = 0.1 and initial value, y(0)=1,

we can first get x data points, then use the Euler method to get y values.

DO i = 1, N+1 ! x data points

x(i) = a + (i-1)*dx

END DO

y(1) = 1 ! initial y value

DO i = 1, N ! y data points

y(i+1) = y(i) + dx*x(i)*y(i) ! f(x,y)=xy

END DO

Exercise

The exact solution of the differential equation in the example is. Find the error of the Euler’s method for N=10,100 and 1000 at x = 1.