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In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives 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 : \fracu(x) + \int_^x f(t,u(t))\,dt = g(x,u(x)), \qquad u(x_0) = u_0, \qquad x_0 \ge 0. As is typical with differential equations, 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, : u'(x) + 2u(x) + 5\int_^u(t)\,dt = \theta(x) \qquad \text \qquad u(0)=0, where : \theta(x) = \left\{ \begin{array}{ll} 1, \qquad x \geq 0\\ 0, \qquad x < 0 \end{array} \right. is the Heaviside step function. The Laplace transform is defined by, : U(s) = \mathcal{L} \left\{u(x)\right\}=\int_0^{\infty} e^{-sx} u(x) \,dx. 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, : s U(s) - u(0) + 2U(s) + \frac{5}{s}U(s) = \frac{1}{s}. Thus, : U(s) = \frac{1}{s^2 + 2s + 5} . Inverting the Laplace transform using contour integral methods then gives : u(x) = \frac{1}{2} e^{-x} \sin(2x) \theta(x) . 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 perfe ...
and use a table of Laplace transforms ("exponentially decaying sine wave") or recall from memory to proceed: : U(s) = \frac{1}{s^2 + 2s + 5} = \frac{1}{2} \frac{2}{(s+1)^2+4} \Rightarrow u(x) = \mathcal L^{-1}\left\{ U(s) \right\} = \frac{1}{2} e^{-x} \sin(2x) \theta(x) .


Applications

Integro-differential equations model many situations from science and engineering, such as in circuit analysis. By Kirchhoff's second law, the net voltage drop across a closed loop equals the voltage impressed E(t) . (It is essentially an application of energy conservation.) An RLC circuit therefore obeys L \frac{d}{dt}I(t) + RI(t) + \frac{1}{C} \int_{0}^{t} I(\tau) d\tau = E(t), where I(t) is the current as a function of time, R is the resistance, L the inductance, and C 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, the mathematical modeling of
epidemic An epidemic (from 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 of infectious d ...
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 *
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 functio ...


References


Further reading

* Vangipuram Lakshmikantham, M. Rama Mohana Rao, �
Theory of Integro-Differential Equations
��, CRC Press, 1995


External links


Interactive Mathematics


of 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