MAPLE IMPLEMENTATION

Maple contains a large range of numerical procedures for integrating non-autonomous dynamical systems of the type

All of these procedures are invoked using the Maple dsolve command with the numeric option. The syntax for the command is

dsolve(deqns, vars, type=numeric, options)

where deqns defines the system of differential equations and initial values, vars defines the dependent and independent variables, type=numeric tells Maple to use a numerical algorithm and options allows a choice of method, stepsize and other options associated with the method. The default method is a Fehlberg fourth-fifth order Runge-Kutta method.

- Worked Example 1 - Van der Pol Equation

As an example consider the solution of the Van der Pol equation written as the first order system

Maple Solution

Classical Methods

Maple contains a number of one-step methods for the numerical solution of initial value problems. These are referred to as classical methods and are invoked by including the option method=classical[type] in the call to dsolve. Here type can be one of

foreuler | the forward Euler method;
heunform | the Heun formula (also known as the trapezoidal rule, or the improved Euler method);
impoly | the improved polygon method (also known as the modified Euler method);
rk2 | the second-order classical Runge-Kutta method;
rk3 | the third-order classical Runge-Kutta method;
rk4 | the fourth-order classical Runge-Kutta method;
adambash | the Adams-Bashford method (a "predictor" method);
abmoulton | the Adams-Bashford-Moulton method (a "predictor-corrector" method).

If no type is specified the forward Euler method is used.

Worked Example 2 - The Forward Euler Method

Consider the IVP

Maple Solution

Worked Example 3 - The Classical Second Order Runge-Kutta Method

Consider the solution of the IVP

Maple Solution