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.

See Also

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


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).

Page Authors

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Added some notes.
Various changes.