In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an integro-differential equation is an
equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
that involves both
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s and
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
s of a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
.
General first order linear equations
The general first-order, linear (only with respect to the term involving derivative) integro-differential equation is of the form
:
As is typical with
differential equations
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
, obtaining a closed-form solution can often be difficult. In the relatively few cases where a solution can be found, it is often by some kind of integral transform, where the problem is first transformed into an algebraic setting. In such situations, the solution of the problem may be derived by applying the inverse transform to the solution of this algebraic equation.
Example
Consider the following second-order problem,
:
where
:
is the
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
. The
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
is defined by,
:
Upon taking term-by-term Laplace transforms, and utilising the rules for derivatives and integrals, the integro-differential equation is converted into the following algebraic equation,
:
Thus,
:
.
Inverting the Laplace transform using
contour integral methods then gives
:
.
Alternatively, one can
complete the square
:
In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form
:ax^2 + bx + c
to the form
:a(x-h)^2 + k
for some values of ''h'' and ''k''.
In other words, completing the square places a Square ...
and use a table of
Laplace transforms
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the comp ...
("exponentially decaying sine wave") or recall from memory to proceed:
:
.
Applications
Integro-differential equations model many situations from
science
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earliest archeological evidence for ...
and
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
, such as in circuit analysis. By
Kirchhoff's second law, the net voltage drop across a closed loop equals the voltage impressed
. (It is essentially an application of
energy conservation
Energy conservation is the effort to reduce wasteful energy consumption by using fewer energy services. This can be done by using energy more effectively (using less energy for continuous service) or changing one's behavior to use less service (f ...
.) An RLC circuit therefore obeys
where
is the current as a function of time,
is the resistance,
the inductance, and
the capacitance.
The activity of interacting ''
inhibitory
An inhibitory postsynaptic potential (IPSP) is a kind of synaptic potential that makes a postsynaptic neuron less likely to generate an action potential.Purves et al. Neuroscience. 4th ed. Sunderland (MA): Sinauer Associates, Incorporated; 2008. ...
'' and ''
excitatory
In neuroscience, an excitatory postsynaptic potential (EPSP) is a postsynaptic potential that makes the postsynaptic neuron more likely to fire an action potential. This temporary depolarization of postsynaptic membrane potential, caused by the ...
''
neurons
A neuron, neurone, or nerve cell is an electrically excitable cell that communicates with other cells via specialized connections called synapses. The neuron is the main component of nervous tissue in all animals except sponges and placozoa. N ...
can be described by a system of integro-differential equations, see for example the
Wilson-Cowan model.
Epidemiology
Integro-differential equations have found applications in
epidemiology
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and determinants of health and disease conditions in a defined population.
It is a cornerstone of public health, and shapes policy decisions and evidenc ...
, the mathematical modeling of
epidemic
An epidemic (from Ancient Greek, Greek ἐπί ''epi'' "upon or above" and δῆμος ''demos'' "people") is the rapid spread of disease to a large number of patients among a given population within an area in a short period of time.
Epidemics ...
s, particularly when the models contain
age-structure or describe spatial epidemics.
See also
*
Delay differential equation
In mathematics, delay differential equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times.
DDEs are also called time ...
*
Differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
*
Integral equation
In mathematics, integral equations are equations in which an unknown Function (mathematics), function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; ...
*
Integrodifference equation
In mathematics, an integrodifference equation is a recurrence relation on a function space, of the following form:
: n_(x) = \int_ k(x, y)\, f(n_t(y))\, dy,
where \\, is a sequence in the function space and \Omega\, is the domain of those funct ...
References
Further reading
* Vangipuram Lakshmikantham, M. Rama Mohana Rao,
Theory of Integro-Differential Equations, CRC Press, 1995
External links
Interactive Mathematicsof the example using
Chebfun
Chebfun is a free/open-source software system written in MATLAB for numerical computation with functions of a real variable. It is based on the idea of overloading MATLAB's commands for vectors and matrices to analogous commands for functions an ...
{{Authority control
Differential equations