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 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 Finite difference (Crank–Nicolson) Finite element Finite volume Galerkin Integrating factor Integral transforms Perturbation theory Runge–Kutta Separation of variables Undetermined coefficients
People padding-top:0.5em Isaac Newton Leonhard Euler Émile Picard Józef Maria Hoene-Wroński Ernst Lindelöf Rudolf Lipschitz Augustin-Louis Cauchy John Crank Phyllis Nicolson Carl David Tolmé Runge Martin Wilhelm Kutta

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 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 \ge 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 first-order problem,

u'(x) + 2u(x) + 5\int_{0}^{x}u(t)\,dt = \left\{ \begin{array}{ll} 1, \qquad x \geq 0\\ 0, \qquad x < 0 \end{array} \right. \qquad \text{with} \qquad u(0)=0.

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,

s U(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) .

Applications

Integro-differential equations model many situations from science and engineering. A particularly rich source is electrical circuit analysis.

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.

See also

• Laplace transform
• Integrodifference equation

External links

• Interactive Mathematics
• Numerical solution of the example using Chebfun

References

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