Functional differential equation

The simplest, fundamental functional differential equation is the linear first-order delay differential equation[4] which is given by

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

where ${\displaystyle \alpha _{1},\alpha _{2},\tau }$ are constants, ${\displaystyle f(t)}$ is some continuous function, and ${\displaystyle x}$ is a scalar. Below is a table with a comparison of several ordinary and functional differential equations.

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

Differential difference equations are functional differential equations in which the argument values are discrete.[1] The general form for functional differential equations of finitely many discrete deviating arguments is

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

where ${\displaystyle x(t)\in \mathbb {R} ^{m},\,n_{1},n_{2},\ldots ,n_{i}\geq 0,}$ and ${\displaystyle \tau _{1}(t),\tau _{2}(t),\ldots ,\tau _{i}(t)\geq 0}$

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]

Classifications of differential difference equations[6]
Retarded	${\displaystyle x'(t)=f{\biggl (}t,x(t),x(t-\tau ){\biggr )}}$
Neutral	${\displaystyle x'(t)=f{\biggl (}t,x(t),x(t-\tau ),x'(t-\tau ){\biggr )}}$
Advanced	${\displaystyle x'(t-\tau )=f{\biggl (}t,x(t),x(t-\tau ){\biggr )}}$

Functional differential equations of retarded type occur when ${\displaystyle \max\{n_{1},n_{2},\ldots ,n_{k}\ \}<n}$ for the equation given above. In other words, this class of functional differential equations depends on the past and present values of the function with delays.

A simple example of an retarded functional differential equation is

${\displaystyle x'(t)=-x(t-\tau )}$

whereas a more general form for discrete deviating arguments can be written as

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

Functional differential equations of neutral type, or neutral differential equations occur when

${\displaystyle \max\{n_{1},n_{2},\ldots ,n_{k}\}=n.}$

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.

Integro-differential equations of Volterra type are functional differential equations with continuous argument values.[1] Integro-differential equations involve both the integrals and derivatives of some function with respect to its argument.

The continuous integro-differential equation for retarded functional differential equations, ${\displaystyle x'(t)=f{\biggl (}t,x(t-\tau _{1}(t)),x(t-\tau _{2}(t)),\ldots ,x(t-\tau _{k}(t)){\biggr )}}$, can be written as

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

A logistic equation for population growth is given by

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

where ρ is the reproduction rate and k is the carrying capacity. ${\displaystyle x(t)}$ represents the population size at time t, and ${\displaystyle \rho {\Biggl (}1-{\frac {x(t)}{k}}{\Biggr )}}$ is the density-dependent reproduction rate.[7]

If we were to now apply this to an earlier time ${\displaystyle t-\tau }$, we get

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

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 ${\displaystyle r}$ 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 ${\displaystyle V}$ be the amount in liters of salt water in the container and ${\displaystyle x(t)}$ be the uniform concentration in grams per liter of salt water at time ${\displaystyle t}$. Then, we have the differential equation[8]

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

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 ${\displaystyle x(t)}$ be the average concentration at time ${\displaystyle t}$, rather than uniform. Then, let's assume the solution leaving the container at time ${\displaystyle t}$ is equal to ${\displaystyle x(t-\tau ),\tau >0}$, the average concentration at some earlier time. Then, the equation is a delay-differential equation of the form[8]

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

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:

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

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

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

where ${\displaystyle p(t)}$ denotes the prey population density at time t, ${\displaystyle P_{1}(t)}$ and ${\displaystyle P_{2}(t)}$ denote the density of the predator population at time ${\displaystyle t,r_{i},a_{ij}\in C(\mathbb {R} ,[0,\infty ))}$ and ${\displaystyle \tau _{ij}\in C(\mathbb {R} ,\mathbb {R} )}$

Note that this model uses linear partial differential equations.

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]