Command-line ODE Solvers

rkfixed(init, x1, x2, npoints, D) Fourth-order Runge-Kutta fixed-step method is used.
Rkadapt(init, x1, x2, npoints, D) Fourth-order Runge-Kutta with adaptive step-size.
Bulstoer(init, x1, x2, npoints, D) Bulirsch-Stoer method is used, which is slightly more accurate than Runge-Kutta, but requires a smoothly varying system.

Each of these functions returns a matrix in which the first column contains the npoints between x1 and x2 at which the solution to the ODE is evaluated. Subsequent columns contain the corresponding values of the solution(s) to the nth-order ODE(s) and the n − 1 derivatives of the solution(s). Either a single ODE or a system of ODE equations are allowed.

Arguments:

• init is either a vector of n real initial values or a single scalar initial value, in the case of a single ODE.
• x1 and x2 are real, scalar endpoints of the interval over which the solution to the ODE(s) will be evaluated. Initial values in init are the values of the ODE function(s) evaluated at x1.
• npoints is the integer number of points beyond the initial point at which the solution is to be approximated. This determines the number of rows (1 + npoints) in the matrix of solutions.
• D(x,y) is an n-element vector-valued function of the independent variable, x, and a vector of functions, y, containing the equations for the first derivatives of all unknown functions in the system of ODEs. To create this vector, cast your equation(s) with a first derivative term by itself on the left-hand-side, with no multipliers, and no higher order derivatives in the equation. For example, a single ODE of the function y(x) which contains a second derivative must be written as a system of equations in y₀(x) and y₁(x), where the first derivative of y₀ is y₁. The single-function ODE

  

  is rewritten for the solver, using vector subscripts, as

  with implied left-hand-side

Notes:

• The vector of functions, D, is specified without its argument list to the ODE solvers, and the functions y_n(x) are specified in D without their argument. The second argument to D must be subscripted as a vector, even if there is only a single entry.
• Rkadapt uses non-uniform step sizes internally when it solves the differential equation, adding more steps in regions of greater variation of the solution, but returns the solution at the number of equally spaced points specified in npoints.
• You may wish to use ODE Solve Blocks to simplify the entry of your equations and initial conditions.
• To solve stiff or algebraic ODE problems, try Radau, Stiffb, or Stiffr.
• If you only require the solution at the last point on the integration interval, use the lowercase versions of these functions.

QuickSheet - First-Order ODE Initial Value Problem

QuickSheet - First-Order System of ODEs