Difference Equation

A difference equation is the discrete analog of a Differential Equation. A difference equation involves a Function with Integer-valued arguments $f(n)$ in a form like

$$f(n) - f(n-1) = g(n),$$ (1)

where $g$ is some Function. The above equation is the discrete analog of the first-order Ordinary Differential Equation

$$f'(x) = g(x).$$ (2)

Examples of difference equations often arise in Dynamical Systems. Examples include the iteration involved in the Mandelbrot and Julia Set definitions,

$$f(n+1) = f(n)^2 + c,$$ (3)

with $c$ a constant, as well as the Logistic Equation

$$f(n+1) = r f(n)[1-f(n)],$$ (4)

with $r$ a constant.

See also Finite Difference, Recurrence Relation

References

Difference Equations

Batchelder, P. M. An Introduction to Linear Difference Equations. New York: Dover, 1967.

Bellman, R. E. and Cooke, K. L. Differential-Difference Equations. New York: Academic Press, 1963.

Beyer, W. H. ``Finite Differences.'' CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 429-460, 1988.

Brand, L. Differential and Difference Equations. New York: Wiley, 1966.

Goldberg, S. Introduction to Difference Equations, with Illustrative Examples from Economics, Psychology, and Sociology. New York: Dover, 1986.

Levy, H. and Lessman, F. Finite Difference Equations. New York: Dover, 1992.

Richtmyer, R. D. and Morton, K. W. Difference Methods for Initial-Value Problems, 2nd ed. New York: Interscience Publishers, 1967.