A functional differential equation is a
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
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