Higher Order Methods


Euler Method for the Higher Order Differential Equations

The method described above is often sufficient to approximate first order differential equations, it may not be obvious how to apply it to the approximation of differential equations of higher order.  Trick here is to break down the higher order differential equation into several first order differential equations.  Technique is also called state variable analysis.

Consider the differential equation:

 

Assume initial conditions are all zero.

Note: this differential equation represents the rectifier circuit from section 7-3 of Circuits Devices and Systems, by Smith  (your E11 text).  The circuit parameter are R=400W, C=25mF, L=4H, w=377 rad/sec(60 Hz), and Vm=1.   Taking the absolute value of the sine wave is equivalent to rectification, y(t) is equal to the output voltage.

This problem would be extremely hard to solve analytically, poses no particular problem for Euler's method.  First thing we must do is to recast the problem in terms of first order equations. So we introduce two new variables, x1(t) and x2(t).  Let x1(t)=y(t). Then we can write two coupled first order equations.

 

These equations also expressed as first order matrix equation.

 

We can now use the Euler's method to solve both first order equations simultaneously by following the following procedure (which is a simple modification of the procedure for first order equations).

1) Starting at time to, choose a value for h, find initial conditions for all state variables x1(to), x2(to), ...

2) From the values of  xi(to) calculate derivatives for each xi(t) at t=to. Call these k1i

 

3) From this value we can find an approximate value for each xi*(to+h).

 

4) Let to=to+h, and for each xi, let xi (to) =xi*(to+h).

5) Repeat the steps 2 through 4 until the solution is finished.

 

Numerical Example (2nd order differential equation - Euler)

Here is a link to the Matlab code for this problem, link to the C Code.  Shown below are the results. The solution is quite a complex function.

 

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