Algorithmic Representation

Numerical solution of DE’s is accomplished by feeding in first order differential form of the series of equations to be solved to an algorithm like Runge Kutta or Rosenbrock. First order ODE’s can be easily represented in this manner. With higher order ODE’s, the higher orders are first converted into 1^st order ones before an algorithm can be implemented.

Take for example the 2^nd order mass-spring-damper system model:

This equation can be numerically solved using the conversion

      

 

To be more specific, let us assume that the variables to be solved are var1 and var2. The algorithmic form would then become:

 

For a series of higher order ODE’s, one has to convert the successive higher orders into first order equivalents and then use the ODE solvers.