In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way. See, for example, Maxwell's equations. For other meanings of mathematics or math, see mathematics (disambiguation). ... Partial plot of a function f. ... In calculus, the integral of a function is a generalization of area, mass, volume and total. ... Graph of a differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... Maxwells equations (sometimes called the Maxwell equations) are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ...

The most basic type of integral equation is a Fredholm equation of the first type: In mathematics, the Fredholm integral equation introduced by Ivar Fredholm gives rises to a Fredholm operator. ...

The notation follows Arfken. Here φ is an unknown function, f is a known function, and K is another known function of two variables, often called the kernel function. Note that the limits of integration are constant; this is what characterizes a Fredholm equation. In analysis, consider an integral transform T which transforms a function f into a function Tf given by the integral formula The function k(x,y) that appears in this formula is the kernel of the operator T. See also: Dirichlet kernel convolution kernel trick Categories: Stub | Mathematical analysis ...

If the unknown function occurs both inside and outside of the integral, it is known as a Fredholm equation of the second type:

The parameter λ is an unknown factor, which plays the same role as the eigenvalue in linear algebra. In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ...

If one limit of integration is variable, it is called a Volterra equation. Thus Volterra equations of the first and second types, respectively, would appear as: In Mathematics, the Volterra Integral Equations are a special type of integral equations. ...

In all of the above, if the known function f is identically zero, it is called a homogeneous integral equation. If f is nonzero, it is called an inhomogeneous integral equation.

In summary, integral equations are classified according to three different dichotomies, creating eight different kinds:

Limits of integration

both fixed: Fredholm equation

one variable: Volterra equation

Placement of unknown function

only inside integral: first kind

both inside and outside integral: second kind

Nature of known function f

identically zero: homogeneous

not identically zero: inhomogeneous

Integral equations are important in many applications. Problems in which integral equations are encountered include radiative energy transfer and the oscillation of a string, membrane, or axle. Oscillation problems may also be solved as differential equations. Oscillation is the periodic variation, typically in time, of some measure as seen, for example, in a swinging pendulum. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

See also

• Hilbert-Schmidt operator

In mathematics, a Hilbert-Schmidt operator is a bounded operator A on a Hilbert space H such that there exists an orthonormal basis of H with the property If this is true for one orthonormal basis, it is true for any other orthonormal basis. ...

References

• George Arfken and Hans Weber. Mathematical Methods for Physicists. Harcourt/Academic Press, 2000.
• Andrei D. Polyanin and Alexander V. Manzhirov Handbook of Integral Equations. CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4
• Integral Equations: Exact Solutions at EqWorld: The World of Mathematical Equations.
• Integral Equations: Index at EqWorld: The World of Mathematical Equations.
• Integral equations at exampleproblems.com

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