That is, a functional differential equation is an equation that contains a function and some of its derivatives evaluated at different argument values.
[1] 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.
[2] 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.
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.
Below is a table with a comparison of several ordinary and 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.
[1] The general form for functional differential equations of finitely many discrete deviating arguments is where
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.
[5] Functional differential equations of retarded type occur when
A simple example of a retarded functional differential equation is whereas a more general form for discrete deviating arguments can be written as 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.
[1] Integro-differential equations involve both the integrals and derivatives of some function with respect to its argument.
, can be written as Functional differential equations have been used in models that determine future behavior of a certain phenomenon determined by the present and the past.
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.
where ρ is the reproduction rate and k is the carrying capacity.
Upon exposure to applications of ordinary differential equations, many come across the mixing model of some chemical solution.
Suppose there is a container holding liters of salt water.
Salt water is flowing in, and out of the container at the same rate
be the amount in liters of salt water in the container and
be the uniform concentration in grams per liter of salt water at time
The problem with this equation is that it makes the assumption that every drop of water that enters the contain is instantaneously mixed into the solution.
Then, let's assume the solution leaving the container at time
The Lotka–Volterra predator-prey model was originally developed to observe the population of sharks and fish in the Adriatic Sea; however, this model has been used in many other fields for different uses, such as describing chemical reactions.
Modelling predatory-prey population has always been widely researched, and as a result, there have been many different forms of the original equation.
One example, as shown by Xu, Wu (2013),[9] of the Lotka–Volterra model with time-delay is given below:
denotes the prey population density at time t,
denote the density of the predator population at time