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{{Short description|Equation involving both integrals and derivatives of a function}}
In [[mathematics]], an '''integro-differential equation''' is an [[equation]] that involves both [[integral]]s and [[derivative]]s of a [[function (mathematics)|function]]. ▼
{{Differential equations}}
▲In [[mathematics]], an '''integro-differential equation''' is an [[equation]] that involves both [[integral]]s and [[derivative]]s of a [[function (mathematics)|function]].
==General first order linear equations==
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===Example===
Consider the following
: <math>
u'(x) + 2u(x) + 5\int_{0}^{x}u(t)\,dt = \theta(x)
\left\{ \begin{array}{ll}▼
</math>
where
: <math>
▲ \theta(x) = \left\{ \begin{array}{ll}
1, \qquad x \geq 0\\
0, \qquad x < 0 \end{array}
\right.
▲\right. \qquad \text{with} \qquad u(0)=0.
</math>
is the [[Heaviside step function]]. The [[Laplace transform]] is defined by,
:<math> U(s) = \mathcal{L} \left\{u(x)\right\}=\int_0^{\infty} e^{-sx} u(x) \,dx. </math>
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:<math> U(s) = \frac{1}{s^2 + 2s + 5} </math>.
Inverting the Laplace transform using [[
:<math> u(x) = \frac{1}{2} e^{-x} \sin(2x) \theta(x) </math>.
Alternatively, one can [[complete the square]]
:<math> 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) </math>.
== Applications ==▼
▲== Applications ==
Integro-differential equations model many situations from [[science]] and [[engineering]], such as in circuit analysis. By [[Kirchhoff's circuit laws|Kirchoff's second law]], the net voltage drop across a closed loop equals the voltage impressed <math> E(t) </math>. (It is essentially an application of energy conservation.) An RLC circuit therefore obeys▼
▲Integro-differential equations model many situations from [[science]] and [[engineering]], such as in circuit analysis. By [[Kirchhoff's circuit laws|
<math display="block"> L \frac{
where <math>I(t)</math> is the current as a function of time, <math>R</math> is the resistance, <math>L</math> the inductance, and <math>C</math> the capacitance.<ref>Zill, Dennis G., and Warren S. Wright. “Section 7.4: Operational Properties II.” ''[https://books.google.com/books?id=0UX8e0xdOr0C&q=Integrodifferential Differential Equations with Boundary-Value Problems]'', 8th ed., Brooks/Cole Cengage Learning, 2013, p. 305. {{ISBN|978-1-111-82706-9}}. Chapter 7 concerns the Laplace transform.</ref>▼
▲where <math>I(t)</math> is the current as a function of time, <math>R</math> is the resistance, <math>L</math> the inductance, and <math>C</math> the capacitance.<ref>Zill, Dennis G., and Warren S. Wright. “Section 7.4: Operational Properties II.” ''Differential Equations with Boundary-Value Problems'', 8th ed., Brooks/Cole Cengage Learning, 2013, p. 305. Chapter 7 concerns the Laplace transform.</ref>
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]].
The [[Whitham equation]] is used to model nonlinear dispersive waves in fluid dynamics.<ref>{{Cite book |last=Whitham |first=G.B. |title=Linear and Nonlinear Waves |publisher=Wiley |year=1974 |location=New York |isbn=0-471-94090-9 }}</ref>
=== Epidemiology ===
Integro-differential equations have found applications in [[epidemiology]], the mathematical modeling of [[epidemic]]s, particularly when the models contain [[Population pyramid|age-structure]]<ref>{{Cite book|date=2008|editor-last=Brauer|editor-first=Fred|editor2-last=van den Driessche|editor2-first=Pauline|editor2-link=Pauline van den Driessche|editor3-last=Wu|editor3-first=Jianhong|title=Mathematical Epidemiology|series=Lecture Notes in Mathematics|volume=1945|pages=205–227|doi=10.1007/978-3-540-78911-6|isbn=978-3-540-78910-9|issn=0075-8434}}</ref> or describe spatial epidemics.<ref>{{Cite web|url=http://people.oregonstate.edu/~medlockj/other/IDE.pdf|title=Integro-differential-Equation Models for Infectious Disease|last=Medlock|first=Jan|date=March 16, 2005|website=Yale University|archive-url=https://web.archive.org/web/20200321190642/http://people.oregonstate.edu/~medlockj/other/IDE.pdf|archive-date=2020-03-21}}</ref> The [[Kermack–McKendrick theory|Kermack-McKendrick theory]] of infectious disease transmission is one particular example where age-structure in the population is incorporated into the modeling framework.
== See also ==
* [[Delay differential equation]]
*[[Differential equation]]
* [[Integral equation]]
* [[Integrodifference equation]]
== References ==
{{
==Further reading==
* Vangipuram Lakshmikantham, M. Rama Mohana Rao,
==External links==
* [http://www.intmath.com/Laplace-transformation/9_Integro-differential-eqns-simultaneous-DE.php Interactive Mathematics]
* [http://www.chebfun.org/examples/integro/WikiIntegroDiff.html Numerical solution] of the example using [[Chebfun]]
{{Differential equations topics}}
{{Authority control}}
[[Category:Differential equations]]
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