Everything about Integral equation
Equations with an unknown function under an integral sign
In mathematical analysis, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: where is an integral operator acting on u. Hence, integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals. A direct comparison can be seen with the mathematical form of the general integral equation above with the general form of a differential equation which may be expressed as follows:where may be viewed as a differential operator of order i. Due to this close connection between differential and integral equations, one can often convert between the two. For example, one method of solving a boundary value problem is by converting the differential equation with its boundary conditions into an integral equation and solving the integral equation. In addition, because one can convert between the two, differential equations in physics such as Maxwell's equations often have an analog integral and differential form. See also, for example, Green's function and Fredholm theory. Wikipedia →
🧠Start here: a 6-step reading path
- Read the overview: Integral equationA 5-minute orientation
- Skim a recent paper: Solving integral equations in free space with inverse-designed ultrathin optical metagratingsInfluential work since 2022
- Pick up a classic book: Integral Methods in Science and EngineeringThe most-reprinted book on the subject
- Read the landmark paper: A new integral equation formalism for the polarizable continuum model: Theoretical background and applications to isotropic and anisotropic dielectricsCited 6,824 times
- Go sideways: Volterra integral equationA closely related topic
- Go sideways: Fredholm theoryA closely related topic
đź“– Definitions
Noun An equation involving a function f(x) and integrals of that function to be solved for f(x)
🏆 Landmark papers (most cited)
A new integral equation formalism for the polarizable continuum model: Theoretical background and applications to isotropic and anisotropic dielectrics
An integrable shallow water equation with peaked solitons
Adaptive integral solution of combined field integral equation
On a Stochastic Integral Equation
An integral equation from phytology
Integral equation methods in electrostatics
Interval Abel integral equation
Analytic Feynman integral solutions of an integral equation related to the Schroedinger equation
🆕 Influential since 2022
Solving integral equations in free space with inverse-designed ultrathin optical metagratings
The interference wave and the bright and dark soliton for two integro-differential equation by using BNNM
Bilinear Bäcklund Transformations, as well as N-Soliton, Breather, Fission/Fusion and Hybrid Solutions for a (3+1)-Dimensional Integrable Wave Equation in a Fluid
On Ulam Stabilities of Delay Hammerstein Integral Equation
A new meshless local integral equation method
đź“š Books worth reading
Integral Methods in Science and EngineeringChristian Constanda, Bardo E.J. Bodmann · 2008
Cours d'analyse de l'École polytechniqueCamille Jordan · 1882
Lévy processes and stochastic calculusDavid Applebaum · 2004
Grundzüge einer allgemeinen Theorie der linearen IntegralgleichungenDavid Hilbert · 1912- 📕Handbook of Integral EquationsAndrei D. Polyanin, Alexander V. Manzhirov · 1998
Elements of the integral calculusByerly, William Elwood · 1881
Integral and integrodifferential equationsRavi P. Agarwal, Donal O'Regan · 2000- 📕Die Differential- und Integralgleichungen der Mechanik und PhysikPhilipp Frank · 1925
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
đź”— Related topics
- Volterra integral equation In mathematics, the Volterra integral equations are a special type of integral equations, named after Vito Volterra. They are divided into two groups referred…
- Fredholm theory In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm…
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