## Ordinary Differential Equations

Ordinary differential equations, or ODE's, used in numerical analysis, express a function via its derivative, and are often used to model physical systems mathematically.

Bernoulli
Differentiation and steepest-descent
DsTool
A Tk program for exploring dynamical systems.
minsky
A program for simulating of models (particularly from economics) defined in terms of couple ordinary differential equations.
Runge-Kutta
A numeric method for solving ODE's.
Runge-Kutta-Fehlberg
Another numeric method, derived from Runge-Kutta, for solving ODE's.
math
AM A number of the commonly used numerical methods can be found in Tcllib's math module.
tclode
A Tcl extension that uses ODEPACK to solve differential equations.

## Description

Radioactive decay, for example, proposes that the rate of decay is purely dependent on the number of atoms that have not yet decayed, so Rate of Change of Number = -constant*number.

or d/dt(Number) = -constant*number

The solution to this trivial (can such an important equation be trivial?) equation is

N=N0 * exp (-constant * time).

where N0 is the number of atoms at time = 0. (Just differentiate the function).

If constant is negative then the number of atoms would grow. This situation can occur for living organisms - the number of new babies is proportional to the number of adults.

Partial Differential Equations are much tougher to solve, particularly in 3 dimensions. Many solver programs exist, but few are suitable for Tcl (a fully compiled language is indicated).

anonymous
Original author.
am