Functional differential equation

A functional differential equation (FDE) is a differential equation with deviating argument. That is, an FDE is an equation that contains some function and some of its derivatives to different argument values.

FDEs are used in mathematical models that assume the specified behavior or phenomenon is dependent on the present as well as the past events. In other words, the past events influence the future results. For this reason, FDEs are used to in many applications rather than ordinary differential equations (ODE), which implicitly assume future behavior is independent of the past.

Unlike ODEs, which are equations containing a function of one variable and its derivative for the same input, functional differential equations are equations containing a function and some of its derivatives for different input values.

The simplest type of FDE, called the retarded functional differential equation (RDE) or retarded differential difference equation, is of the form

FDE is the general form for more specific types of differential equations that are used in numerous applications. There are delay differential equations, integro-differential equations, and so on.

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.

Examples of other models that have used FDEs, namely RFDEs, are given below:

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