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 f ′ ( x ) = 2 f ( x ) + 1 {\displaystyle f'(x)=2f(x)+1}
In comparison, a functional differential equation would be f ′ ( x ) = 2 f ( x + 3 ) − [ f ( x − 1 ) ] 2 {\displaystyle f'(x)=2f(x+3)-[f(x-1)]^{2}}
The simplest type of functional differential equation called the retarded functional differential equation or retarded differential difference equation, is of the form
x ′ ( t ) = f ( t , x ( t ) , x ( t − r ) ) {\displaystyle x'(t)=f{\bigl (}t,x(t),x(t-r){\bigr )}}
Examples A simple functional differential equation is the linear first-order delay differential equation which is given by
x ′ ( t ) = α 1 x ( t ) + α 2 x ( t − τ ) + f ( t ) , t ≥ 0 {\displaystyle x'(t)=\alpha _{1}x(t)+\alpha _{2}x(t-\tau )+f(t),t\geq 0}
where α 1 , α 2 , τ {\displaystyle \alpha _{1},\alpha _{2},\tau } are constants, f ( t ) {\displaystyle f(t)} is some continuous function, and x {\displaystyle x} 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.
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
x ( n ) ( t ) = f ( t , x ( n 1 ) ( t − τ 1 ( t ) ) , x ( n 2 ) ( t − τ 2 ( t ) ) , … , x ( n k ) ( t − τ k ( t ) ) ) {\displaystyle x^{(n)}(t)=f{\Bigl (}t,x^{(n_{1})}{\bigl (}t-\tau _{1}(t){\bigr )},x^{(n_{2})}{\bigl (}t-\tau _{2}(t){\bigr )},\ldots ,x^{(n_{k})}{\bigl (}t-\tau _{k}(t){\bigr )}{\Bigr )}}
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