mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a summation equation or discrete integral equation is an equation in which an unknown function appears under a

summation In mathematics, summation is the addition of a sequence of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, pol ...

sign. The theories of summation equations and integral equations can be unified as ''integral equations on time scales''Volterra integral equations on time scales: Basic qualitative and quantitative results with applications to initial value problems on unbounded domains
Tomasia Kulik, Christopher C. Tisdell, September 3, 2007 using time scale calculus. A summation equation compares to a difference equation as an integral equation compares to a differential equation. The Volterra summation equation is:

x(t) = f(t) + \sum_^n k \bigl( t, s, x(s) \bigr)

where is the unknown function, are

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s, and are known functions.

References

Summation equations or discrete integral equations
Integral equations {{mathanalysis-stub