# Integro-differential equation

Differential equations :
Scope :
Fields
Natural sciencesEngineering
Astronomy
Physics
Chemistry
Biology
Geology
Applied mathematics
Continuum mechanics
Chaos theory
Dynamical systems
Social sciences
Economics
Population dynamics
List of named differential equations :
Classification :
Types
Ordinary
Partial
Differential-algebraic
Integro-differential
Fractional
Linear
Non-linear
By variable type
Dependent and independent variables
Autonomous
Coupled / Decoupled
Exact
Homogeneous / Nonhomogeneous
Features
Order
Operator
Notation :
Relation to processes
Difference (discrete analogue)
Stochastic
Stochastic partial
Delay :
Solution :
Existence and uniqueness
Picard–Lindelöf theorem
Peano existence theorem
Carathéodory's existence theorem
Cauchy–Kowalevski theorem :
General topics
Initial conditions
Boundary values
Dirichlet
Neumann
Robin
Cauchy problem
Wronskian
Phase portrait
Lyapunov / Asymptotic / Exponential stability
Rate of convergence
Series / Integral solutions
Numerical integration
Dirac delta function :
Solution methods
Inspection
Method of characteristics
Euler
Exponential response formula
Finite difference (Crank–Nicolson)
Finite element
Infinite element
Finite volume
Galerkin
Petrov–Galerkin
Green's function
Integrating factor
Integral transforms
Perturbation theory
Runge–Kutta
Separation of variables
Undetermined coefficients
Variation of parameters :
People :
List
Isaac Newton
Gottfried Leibniz
Jacob Bernoulli
Leonhard Euler
Joseph-Louis Lagrange
Józef Maria Hoene-Wroński
Joseph Fourier
Augustin-Louis Cauchy
George Green
Carl David Tolmé Runge
Martin Kutta
Rudolf Lipschitz
Ernst Lindelöf
Émile Picard
Phyllis Nicolson
John Crank :
vte :
In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function.

## 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
    ${\frac {d}{dx}}u(x)+\int _{x_{0}}^{x}f(t,u(t))\,dt=g(x,u(x)),\qquad u(x_{0})=u_{0},\qquad x_{0}\geq 0.$
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,
    $u'(x)+2u(x)+5\int _{0}^{x}u(t)\,dt=\theta (x)\qquad {\text{with}}\qquad u(0)=0,$
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,
    $sU(s)-u(0)+2U(s)+{\frac {5}{s}}U(s)={\frac {1}{s}}.$
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 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 and 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.

### Epidemiology

Integro-differential equations have found applications in epidemiology, the mathematical modeling of epidemics, particularly when the models contain age-structure or describe spatial epidemics. The Kermack-McKendrick theory of infectious disease transmission is one particular example where age-structure in the population is incorporated into the modeling framework.