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.^[1]

FDEs are used in mathematical models that assume the specified behavior or phenomenon is dependent on the present as well as the past events.^[2] 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.

Definition

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.

An example of an ODE would be: $f'(x)=2f(x)+1$
In comparison, an FDE would be: $f'(x)=2f(x+3)-[f(x-1)]^{2}$

The simplest type of FDE, called the retarded functional differential equation (RDE) or retarded differential difference equation, is of the form^[3]

x'(t)=f{\biggl (}t,x(t),x'(t-r){\biggr )}

Examples

The simplest, fundamental FDE is the linear first-order delay differential equation, also referred to as RDE,^[4] which is given by

x'(t)=\alpha _{1}(t)x(t)+\alpha _{2}x(t-\tau )+f(t),t\geq 0

where

\alpha _{1},\alpha _{2},\tau

are constants,

f(t)

is some continuous function, and

x

is a scalar.

Below is a table with a comparison of ODEs and FDEs

	Ordinary differential equation	Functional differential equation
Examples	$f'(x)=x^{2}-3$	$f'(x)=3x-f(x-4)$
	$f'(x)=f(x)-8$	$x'(t)=3x(2t)-{\biggl [}x(t-1){\biggr ]}^{2}$
	$F{\biggl (}t,x(t),x'(t),x''(t){\biggr )}=0$	$2x(3t+1)-5x(4t)=1$
	$f'(x)=4f(x)-3x$

Types of FDEs

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.

Differential difference equation

Differential difference equations (DDE) are FDEs in which the argument values are discrete.^[1]

The general form for FDEs of finitely many discrete deviating arguments is shown below:

$x^{(n)}(t)=f{\Biggl (}t,x^{(n_{1})}{\biggl (}t-\tau _{1}(t){\biggr )},x^{(n_{2})}{\biggl (}t-\tau _{2}(t){\biggr )},\ldots ,x^{(n_{k})}{\biggl (}t-\tau _{k}(t){\biggr )}{\Biggr )}$

where

x(t)\in \mathbb {R} ^{m},\,n_{1},n_{2},\ldots ,n_{i}\geq 0,

and

\tau _{1}(t),\tau _{2}(t),\ldots ,\tau _{i}(t)\geq 0

DDEs are also referred to as retarded FDEs, neutral FDEs, advanced FDEs, and mixed FDEs. This classification depends on whether the rate of change of the current state of the system depends on past values or future values or both.^[5]

Classifications of DDEs^[6]
Retarded DDE	$x'(t)=f{\biggl (}t,x(t),x(t-\tau ){\biggr )}$
Neutral DDE	$x'(t)=f{\biggl (}t,x(t),x(t-\tau ),x'(t-\tau ){\biggr )}$
Advanced DDE	$x'(t-\tau )=f{\biggl (}t,x(t),x(t-\tau ){\biggr )}$

Delay differential equation

Functional differential equations of retarded type are when

$\max\{n_{1},n_{2},\ldots ,n_{k}\ \}<n$ for the given equation (1).

In other words, this class of FDEs depends on the past and present values of the function with delays.

A simple example of an RFDE is

$x'(t)=-x(t-\tau )$

A more general form for discrete deviating arguments can be written as

$x'(t)=f{\Biggl (}t,x{\biggl (}t-\tau _{1}(t){\biggr )},x{\biggl (}t-\tau _{2}(t){\biggr )},\ldots ,x{\biggl (}t-\tau _{k}(t){\biggr )}{\Biggr )}$

Neutral differential difference equation

Functional differential equations of neutral type, or neutral differential equations (NDE) are when

$\max\{n_{1},n_{2},\ldots ,n_{k}\}<n$ for (1).

NDEs depend on past and present values of the function, similarly to RDEs, except it also depends on derivatives with delays. In other words, RDEs do not involve the given function's derivative with delays while NDEs do.

Integro-differential equation

Integro-differential equation (IDE) of Volterra type are FDEs in which the argument values are continuous.^[1] IDEs involve both the integrals and derivatives of some function.

The continuous integro-differential equation for RFDE

x'(t)=f{\biggl (}t,x(t-\tau _{1}(t)),x(t-\tau _{2}(t)),\ldots ,x(t-\tau _{k}(t)){\biggr )}

can be written as

$x'(t)=f{\Biggl (}t,\int _{t-\tau (t)}^{t}K(t,\theta ,x(\theta )){\Biggr )},\tau (t)\geq 0$

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.^[2] 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.^[1]^[4]

Population growth with time lag

A logistic equation for population growth is given by

${\mathrm {d} x \over \mathrm {d} t}=\rho \,x(t)\left(1-{\frac {x(t)}{k}}\right),$

where ρ is the reproduction rate and k is the carrying capacity.

x(t)

represents the population size at time t, and

\rho {\Biggl (}1-{\frac {x(t)}{k}}{\Biggr )}

is the density-dependent reproduction rate.^[7]

If we were to now apply this to an earlier time

t-\tau

, we get

${\mathrm {d} x \over \mathrm {d} t}=\rho \,x(t)\left(1-{\frac {x(t-\tau )}{k}}\right)$

Mixing model

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

r

of liters per second. In other words, the rate of water flowing in is equal to the rate of the salt water solution flowing out. Let

V

be the amount in liters of salt water in the container and

x(t)

be the uniform concentration in grams per liter of salt water at time

t

. Then, we have the differential equation^[8]

$x'(t)=-{\frac {r}{V}}x(t),{\frac {r}{V}}>0$

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. This can be eliminated by using a FDE instead of an ODE.

Let

x(t)

be the average concentration at time

t

, rather than uniform. Then, let's assume the solution leaving the container at time

t

is equal to

x(t-\tau ),\tau >0

, the average concentration at some earlier time. Then, the equation is a delay-differential equation of the form^[8]

$x'(t)=-{\frac {r}{V}}x(t-\tau )$

Volterra's predator-prey model

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:

$p'(t)=p(t){\Biggl [}r_{1}(t)-a_{11}(t)p{\biggl (}t-\tau _{11}(t){\biggr )}-a_{12}(t)P_{1}{\biggl (}t-\tau _{12}(t){\biggr )}-a_{13}(t)P_{2}{\biggl (}t-\tau _{13}(t){\biggr )}{\Biggr ]}$

$P_{1}'(t)=P_{1}(t){\Biggl [}-r_{2}(t)+a_{21}(t)p{\biggl (}t-\tau _{21}(t){\biggr )}-a_{22}(t)P_{1}{\biggl (}t-\tau _{22}(t){\biggr )}-a_{23}(t)P_{2}{\biggl (}t-\tau _{23}(t){\biggr )}{\Biggr ]}$

$P_{2}'(t)=P_{2}(t){\Biggl [}-r_{2}(t)+a_{31}(t)p{\biggl (}t-\tau _{31}(t){\biggr )}-a_{32}(t)P_{1}{\biggl (}t-\tau _{32}(t){\biggr )}-a_{33}(t)P_{2}{\biggl (}t-\tau _{33}(t){\biggr )}{\Biggr ]}$

where

p(t)

denotes the prey population density at time t,

P_{1}(t)

and

P_{2}(t)

denote the density of the predator population at time

t,r_{i},a_{ij}\in C(\mathbb {R} ,[0,\infty )

and

\tau _{ij}\in C(\mathbb {R} ,\mathbb {R} )

Note that this model uses linear partial differential equations.

Other models using FDEs

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

Controlled motion of a rigid body^[1]
Periodic motions^[8]
Flip-flop circuit as a NDE^[8]
Model of HIV epidemic
Math models of sugar quantity in blood^[1]
Evolution equations of single species^[1]
Spread of an infection between two species^[8]

References

1 2 3 4 5 6 7 Kolmanovskii, V.; Myshkis, A. (1992). Applied Theory of Functional Differential Equations. The Netherlands: Kluwer Academic Publishers. ISBN 0-7923-2013-1.
1 2 Hale, Jack K. (1971). Functional Differential Equation. United States: Springer-Verlag. ISBN 0-387-90023-3.
↑ Hale, Jack K.; Verduyn Lunel, Sjoerd M. (1993). Introduction to Functional Differential Equations. United States: Springer-Verlag. ISBN 0-387-94076-6.
1 2 Falbo, Clement E. "Some Elementary Methods for Solving Functional Differential Equations" (PDF). Somona State University.
↑ Guo, S.; Wu, J. (2013). Bifurcation Theory of Functional Differential Equations. New York: Springer. pp. 41–60. ISBN 978-1-4614-6991-9.
↑ Bellman, Richard; Cooke, Kenneth L. (1963). Differential-Difference Equations. New York, NY: Academic Press. pp. 42–49. ISBN 978-0124109735.
↑ Barnes, B.; Fulford, G. R. (2015). Mathematical Modelling with Case Studies. Taylor & Francis Group LLC. pp. 75–77. ISBN 978-1-4822-4772-5.
1 2 3 4 5 Schmitt, Klaus, ed. (1972). Delay and Functional Differential Equations and Their Applications. United States: Academic Press.
↑ Xu, Changjin; Wu, Yusen (2013). "Dynamics in a Lotka–Volterra Predator–Prey Model with Time-varying Delays". Abstract and Applied Analysis. 2013.

Functional differential equation

Definition

Examples

Types of FDEs

Differential difference equation

Delay differential equation

Neutral differential difference equation

Integro-differential equation

Application

Population growth with time lag

Mixing model

Volterra's predator-prey model

Other models using FDEs

See also

References

Further reading