ODE Solver

The ode class is used to solve a system of ordinary differential equations.  The ODE is expected to be provided to the constuctor in vector field form.  Take the following ODE for example:

x''' - x'' + 2*x' - 3*x = sin(t)         (1)

This can be written in first order form, by assigning a new variable for all but the highest order derivative:

x' = y                                   (2)
y' = z  (i.e. x'' = z)

Note that the higest order derivative can then be replaced with a first order derivative (z’ = x’’’).  If we substitute (2) and z’ into (1) and solve for z’ (i.e. x’’’) we get:

z' = z - 2*y + 3*x + sin(t)              (3)

If we combine the equations from (2) and (3) we now have a system of equations in first order form.

x' = y
y' = z
z' = z - 2*y + 3*x + sin(t)

This can be represented as a vector or an array.

[x',y',z'] = [y, z, z-2*y+3*x+sin(t)]    (4)

To use the ODE class, a javascript function needs to be created that replicates a first order vector field such as equation (4) above.  The javascript function takes the independent derivative variable as a first value (if we assume x’ = dx/dt then t is our independent variable), and the array of dependent variables (in this case [x,y,z]) as the second term.  Here is an example function.

var V = function(t, pt) {
   var x = pt[0], y= pt[1], z = pt[2];
   return [y, z, z - 2*y + 3*x + sin(t)];
}

var ode_ = ODE(V);