Integro-differential equation

Short description: Equation involving both integrals and derivatives of a function

Differential equations

Navier–Stokes differential equations used to simulate airflow around an obstruction.

Scope

* Natural sciences
* Engineering

* Astronomy
* Physics
* Chemistry

Biology
* Geology

Applied mathematics

* Continuum mechanics
* Chaos theory
* Dynamical systems

Social sciences

* Economics
* Population dynamics

Classification

Types

* Ordinary
* Partial
* Differential-algebraic
* Integro-differential
* Fractional
* Linear
* Non-linear

By variable type

* Dependent and independent variables

* Autonomous
* Complex
* Coupled / Decoupled
* Exact
* Homogeneous / Nonhomogeneous

Features

* Order
* Operator

* Notation

Relation to processes

* Difference (discrete analogue)

* Stochastic
* Stochastic partial
* Delay

Solution

General topics

* Picard–Lindelöf theorem
* Wronskian

* Phase portrait
* Phase space

* 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
* Integrating factor
* Integral transforms
* Perturbation theory
* Runge–Kutta

* Separation of variables
* Undetermined coefficients
* Variation of parameters

People

* Isaac Newton
* Gottfried Leibniz
* Leonhard Euler
* Émile Picard
* Józef Maria Hoene-Wroński
* Ernst Lindelöf
* Rudolf Lipschitz
* Augustin-Louis Cauchy
* John Crank
* Phyllis Nicolson
* Carl David Tolmé Runge
* Martin Kutta

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In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function.

## Contents

* 1 General first order linear equations
* 1.1 Example
* 2 Applications
* 2.1 Epidemiology
* 3 See also
* 4 References
* 5 Further reading
* 6 External links

## 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

ddxu(x)+∫x0xf(t,u(t))dt=g(x,u(x)),u(x0)=u0,x0≥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∫0xu(t)dt=θ(x)withu(0)=0,

where

θ(x)={1,x≥00,x<0

is the Heaviside step function. The Laplace transform is defined by,

U(s)=ℒ{u(x)}=∫0∞e−sxu(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)+5sU(s)=1s.

Thus,

U(s)=1s2+2s+5.

Inverting the Laplace transform using contour integral methods then gives

u(x)=12e−xsin⁡(2x)θ(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)=1s2+2s+5=122(s+1)2+4⇒u(x)=ℒ−1{U(s)}=12e−xsin⁡(2x)θ(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 LddtI(t)+RI(t)+1C∫0tI(τ)dτ=E(t), where I(t) is the current as a function of time, R is the resistance, L the inductance, and C the capacitance.[1]

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.[2]

### Epidemiology

Integro-differential equations have found applications in epidemiology, the mathematical modeling of epidemics, particularly when the models contain age-structure[3] or describe spatial epidemics.[4] The 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

1. ↑ 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. ISBN 978-1-111-82706-9. Chapter 7 concerns the Laplace transform.
2. ↑ Whitham, G.B. (1974). Linear and Nonlinear Waves. New York: Wiley. ISBN 0-471-94090-9.
3. ↑ Brauer, Fred; van den Driessche, Pauline; Wu, Jianhong, eds (2008). Mathematical Epidemiology. Lecture Notes in Mathematics. 1945. pp. 205–227. doi:10.1007/978-3-540-78911-6. ISBN 978-3-540-78910-9.
4. ↑ Medlock, Jan (March 16, 2005). "Integro-differential-Equation Models for Infectious Disease". http://people.oregonstate.edu/~medlockj/other/IDE.pdf.

## Further reading

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

## External links

* Interactive Mathematics
* Numerical solution of the example using Chebfun

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Differential equations

Classification|

| Operations|

* Differential operator
* Notation for differentiation
* Ordinary
* Partial
* Differential-algebraic
* Integro-differential
* Fractional
* Linear
* Non-linear
* Holonomic

|

Attributes of variables|

* Dependent and independent variables
* Homogeneous
* Nonhomogeneous
* Coupled
* Decoupled
* Order
* Degree
* Autonomous
* Exact differential equation
* Complex differential equation

Relation to processes|

* Difference (discrete analogue)
* Stochastic
* Stochastic partial
* Delay

Solutions|

| Solution topics|

* Picard–Lindelöf theorem (existence and uniqueness)
* Wronskian
* Phase portrait
* Phase space
* Lyapunov stability
* Asymptotic stability
* Exponential stability
* Rate of convergence
* Series solutions
* Integral solutions
* Numerical integration
* Dirac delta function

|

Solution methods|

* Inspection
* Separation of variables
* Method of undetermined coefficients
* Variation of parameters
* Integrating factor
* Integral transforms
* Euler method
* Finite difference method
* Crank–Nicolson method
* Runge–Kutta methods
* Finite element method
* Finite volume method
* Galerkin method
* Perturbation theory

Applications|

* List of named differential equations

Mathematicians|

* Isaac Newton
* Gottfried Wilhelm Leibniz
* Leonhard Euler
* Jacob Bernoulli
* Émile Picard
* Józef Maria Hoene-Wroński
* Ernst Lindelöf
* Rudolf Lipschitz
* Joseph-Louis Lagrange
* Augustin-Louis Cauchy
* John Crank
* Phyllis Nicolson
* Carl David Tolmé Runge
* Martin Kutta

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