Differentialdifference

Differential-difference, or differential–difference, equations are equations that involve derivatives of a function with respect to a continuous variable and terms in which the function is evaluated at shifted or delayed arguments. They generalize ordinary differential equations by incorporating delays or advances in the dependent variable, often written in the form y'(t) = F(t, y(t), y(t - τ1), y(t - τ2), …), where τi ≥ 0 are fixed delays. More generally, they may include linear combinations of derivatives and delayed values, such as L[y](t) = sum ai(t) y^{(i)}(t) + sum bj(t) y(t - τj).

Commonly studied instances are retarded differential-difference equations, where the right-hand side depends only on past values, and neutral equations, where derivatives also depend on delayed derivatives. For retarded types, the initial data must specify the history function y(t) on an interval [−τ, 0], not just a single initial value.

A simple example is the linear retarded equation y'(t) = a y(t) + b y(t − τ). Its solutions require the history y(t) for t in [−τ, 0], and its qualitative behavior is influenced by the delay τ and the coefficients a, b. Linear constant-coefficient forms lead to characteristic equations whose roots determine stability.

Applications span biology, physics, engineering, and beyond, including population dynamics with gestation or maturation delays, control systems with actuation delays, epidemiological models with incubation periods, and neural networks with synaptic delays. Related concepts include ordinary and partial differential equations, difference equations, and the broader class of functional differential equations, of which differential-difference equations form a key subset. Numerical methods for these equations often employ the method of steps or collocation to handle history-dependent terms.