Integraaliyhtälöiksi

Integraaliyhtälöiksi, or integral equations in English, are mathematical equations where the unknown function appears under an integral sign. These equations are fundamental in various fields of science and engineering, including physics, mechanics, and electromagnetism. They arise when a problem is formulated in terms of accumulated quantities or effects, rather than instantaneous ones.

There are two main types of integral equations. Fredholm integral equations involve integration over a fixed interval, while Volterra integral equations have an integration limit that includes the variable itself. Both types can be further classified into linear and nonlinear forms, depending on whether the unknown function appears linearly. Additionally, they can be homogeneous or non-homogeneous, depending on the presence of a non-zero term outside the integral.

Solving integral equations can be challenging and often requires specialized techniques. Common methods include transforming the integral equation into an equivalent differential equation, using iterative approximations, or employing spectral methods. The theory of integral equations is closely related to the theory of linear operators and functional analysis, providing a rich framework for their analysis and solution. The study of integral equations has a long history, with significant contributions from mathematicians like Vito Volterra and Erik Ivar Fredholm.