A functional differential equation is a differential equation with deviating argument. That is, a functional differential equation is an equation that contains a function and some of its derivatives evaluated at different argument values.
Functional differential equations find use in mathematical models that assume a specified behavior or phenomenon depends on the present as well as the past state of a system. In other words, past events explicitly influence future results. For this reason, functional differential equations are more applicable than ordinary differential equations (ODE), in which future behavior only implicitly depends on the past.
Definition
Unlike ordinary differential equations, which contain a function of one variable and its derivatives evaluated with the same input, functional differential equations contain a function and its derivatives evaluated with different input values.
An example of an ordinary differential equation would be
In comparison, a functional differential equation would be
The simplest type of functional differential equation called the retarded functional differential equation or retarded differential difference equation, is of the form
Examples
The simplest, fundamental functional differential equation is the linear first-order delay differential equation which is given by
where are constants, is some continuous function, and is a scalar. Below is a table with a comparison of several ordinary and functional differential equations.
Types of functional differential equations
"Functional differential equation" is the general name for a number of more specific types of differential equations that are used in numerous applications. There are delay differential equations, integro-differential equations, and so on.
Differential difference equation
Differential difference equations are functional differential equations in which the argument values are discrete. The general form for functional differential equations of finitely many discrete deviating arguments is
where and
Differential difference equations are also referred to as retarded, neutral, advanced, and mixed functional differential equations. This classification depends on whether the rate of change of the current state of the system depends on past values, future values, or both.
Delay differential equation
Functional differential equations of retarded type occur when for the equation given above. In other words, this class of functional differential equations depends on the past and present values of the function with delays.
A simple example of a retarded functional differential equation is
whereas a more general form for discrete deviating arguments can be written as
Neutral differential equations
Functional differential equations of neutral type, or neutral differential equations occur when
Neutral differential equations depend on past and present values of the function, similarly to retarded differential equations, except it also depends on derivatives with delays. In other words, retarded differential equations do not involve the given function's derivative with delays while neutral differential equations do.
Integro-differential equation
Integro-differential equations of Volterra type are functional differential equations with continuous argument values. Integro-differential equations involve both the integrals and derivatives of some function with respect to its argument.
The continuous integro-differential equation for retarded functional differential equations, , can be written as
Application
Functional differential equations have been used in models that determine future behavior of a certain phenomenon determined by the present and the past. Future behavior of phenomena, described by the solutions of ODEs, assumes that behavior is independent of the past. However, there can be many situations that depend on past behavior.
FDEs are applicable for models in multiple fields, such as medicine, mechanics, biology, and economics. FDEs have been used in research for heat-transfer, signal processing, evolution of a species, traffic flow and study of epidemics.
Population growth with time lag
A logistic equation for population growth is given by
where ρ is the reproduction rate and k is the carrying capacity. represents the population size at time t, and is the density-dependent reproduction rate.