Integral equation

An equation in which an unknown function appears inside an integral. Its solutions are functions satisfying the integral relation, sometimes alongside other conditions.

Integral: A mathematical operation that accumulates a quantity over an interval, region, or other domain. The integral is where the unknown function appears in the defining relation.
Integral operator: A linear or nonlinear operator that maps a function to an integral expression. An integral equation can be written as an equation involving an integral operator.
Green's function: A function that represents a linear differential operator's response to a point source. Green's functions convert many boundary-value problems into integral equations.
Differential equation: An equation relating an unknown function to one or more of its derivatives. It uses local derivative relations rather than placing the unknown inside an integral.
Niels Henrik Abel: A Norwegian mathematician whose work included an integral equation for the tautochrone problem. His 1823 work produced the equation now associated with his name.
Kernel (integral operator): A function that weights contributions from points in an integral operator. The kernel determines how values of the unknown function at different points are coupled.
Neumann series: A series expansion for the inverse of an operator, when the series converges. It constructs solutions by repeatedly applying the integral operator.
Boundary element method: A numerical method that solves boundary-value problems using integral equations on domain boundaries. It uses boundary integral formulations to reduce the dimension of many computations.
Integro-differential equation: An equation containing both derivatives and integrals of an unknown function. It combines the differential and integral structures kept separate in a pure integral equation.
Vito Volterra: An Italian mathematician who developed foundational work on integral equations and functional analysis. His studies established the class of equations bearing his name.

Linked from 12 pages