Integro-differential equation

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
v t e

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

[math]\displaystyle{ \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. }[/math]

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,

[math]\displaystyle{ u'(x) + 2u(x) + 5\int_{0}^{x}u(t)\,dt = \theta(x) \qquad \text{with} \qquad u(0)=0, }[/math]

where

[math]\displaystyle{ \theta(x) = \left\{ \begin{array}{ll} 1, \qquad x \geq 0\\ 0, \qquad x \lt 0 \end{array} \right. }[/math]

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

[math]\displaystyle{ U(s) = \mathcal{L} \left\{u(x)\right\}=\int_0^{\infty} e^{-sx} u(x) \,dx. }[/math]

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,

[math]\displaystyle{ s U(s) - u(0) + 2U(s) + \frac{5}{s}U(s) = \frac{1}{s}. }[/math]

Thus,

[math]\displaystyle{ U(s) = \frac{1}{s^2 + 2s + 5} }[/math].

Inverting the Laplace transform using contour integral methods then gives

[math]\displaystyle{ u(x) = \frac{1}{2} e^{-x} \sin(2x) \theta(x) }[/math].

Alternatively, one can complete the square and use a table of Laplace transforms ("exponentially decaying sine wave") or recall from memory to proceed:

[math]\displaystyle{ 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

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 [math]\displaystyle{ E(t) }[/math]. (It is essentially an application of energy conservation.) An RLC circuit therefore obeys [math]\displaystyle{ L \frac{d}{dt}I(t) + RI(t) + \frac{1}{C} \int_{0}^{t} I(\tau) d\tau = E(t), }[/math] where [math]\displaystyle{ I(t) }[/math] is the current as a function of time, [math]\displaystyle{ R }[/math] is the resistance, [math]\displaystyle{ L }[/math] the inductance, and [math]\displaystyle{ C }[/math] 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

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

0.00 (0 votes) Original source: https://en.wikipedia.org/wiki/Integro-differential equation. Read more