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| {{Differential equations}}
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| In [[mathematics]], an '''integro-differential equation''' is an [[equation]] that involves both [[integral]]s and [[derivative]]s of a [[function (mathematics)|function]].
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| ==General first order linear equations==
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| The general first-order, linear integro-differential equation is of the form
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| :<math> | |
| \frac{d}{dx}u(x) + \int_{x_0}^x f(t,u(t))\,dt = g(x,u(x)), \qquad u(x_0) = u_0, \qquad x_0 \ge 0.
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| </math>
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| 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.
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| ===Example===
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| Consider the following first-order problem,
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| : <math>
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| u'(x) + 2u(x) + 5\int_{0}^{x}u(t)\,dt =
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| \left\{ \begin{array}{ll}
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| 1, \qquad x \geq 0\\
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| 0, \qquad x < 0 \end{array}
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| \right. \qquad \text{with} \qquad u(0)=0.
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| </math>
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| The [[Laplace transform]] is defined by,
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| :<math> U(s) = \mathcal{L} \left\{u(x)\right\}=\int_0^{\infty} e^{-sx} u(x) \,dx. </math>
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| 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,
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| :<math> s U(s) - u(0) + 2U(s) + \frac{5}{s}U(s) = \frac{1}{s}. </math>
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| Thus,
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| :<math> U(s) = \frac{1}{s^2 + 2s + 5} </math>.
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| Inverting the Laplace transform using [[Methods_of_contour_integration|contour integral methods]] then gives
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| :<math> u(x) = \frac{1}{2} e^{-x} \sin(2x) </math>.
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| == Applications ==
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| Integro-differential equations model many situations from [[science]] and [[engineering]]. A particularly rich source is [[Network analysis (electrical circuits)|electrical circuit analysis]].{{citation needed|date=February 2013}}
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| The activity of interacting ''[[Inhibitory postsynaptic potential|inhibitory]]'' and ''[[Excitatory postsynaptic potential|excitatory]]'' [[neurons]] can be described by a system of integro-differential equations, see for example the [[Wilson-Cowan model]].
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| == See also ==
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| * [[Laplace transform]]
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| * [[Integrodifference equation]]
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| ==External links==
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| * [http://www.intmath.com/Laplace-transformation/9_Integro-differential-eqns-simultaneous-DE.php Interactive Mathematics]
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| * [http://www.maths.ox.ac.uk/chebfun/examples/integro/html/WikiIntegroDiff.shtml Numerical solution] of the example using [[Chebfun]]
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| == References ==
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| * Vangipuram Lakshmikantham, M. Rama Mohana Rao, “Theory of Integro-Differential Equations”, CRC Press, 1995
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| [[Category:Differential equations]]
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