integraldifferential

An integral-differential equation, often called an integro-differential equation, is an equation that involves both integrals and derivatives of an unknown function. These equations generalize ordinary differential equations by including integral terms that represent accumulated effects or memory.

Common linear forms include Volterra-type equations with a variable upper limit, such as y'(t) + ∫_a^t K(t,s) y(s) ds = g(t), and Fredholm-type equations with fixed limits, such as y'(t) + ∫_a^b K(t,s) y(s) ds = g(t). IDEs can also present as higher-order derivatives or systems, for example d^2y/dt^2 + ∫_0^t (t-s) y(s) ds = g(t).

These equations are studied for existence and uniqueness of solutions, stability, and long-term behavior. Linear IDEs often admit analytic techniques such as Laplace or Fourier transforms, the construction of resolvent kernels, or reduction to a higher-order differential equation. For nonlinear IDEs, qualitative methods and fixed-point theorems are commonly employed, especially for Volterra-type problems where the integral term depends only on past values.

Numerical methods augment analytic approaches by discretizing the integral term, converting the IDE into a large system of equations. Techniques include quadrature-based schemes, the method of steps for time-domain problems, and collocation methods. The choice of method depends on the kernel, the integration limits, and the desired accuracy.

Applications appear in fields that model memory or hereditary effects, such as viscoelasticity, heat conduction with memory, population dynamics, control theory, and certain neural or physiological models. IDEs thus provide a flexible framework for systems where instantaneous change depends on past or accumulated states.