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.
The most basic type of integral equation is a Fredholm equation of the first type:
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.
If the unknown function occurs both inside and outside of the integral, it is known as a Fredholm equation of the second type:
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 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.
- Integral Equations: Exact Solutions at EqWorld: The World of Mathematical Equations.
- Integral Equations: Index at EqWorld: The World of Mathematical Equations.
- Integral Equations: Methods at EqWorld: The World of Mathematical Equations.
- Andrei D. Polyanin and Alexander V. Manzhirov Handbook of Integral Equations. CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4
- George Arfken and Hans Weber. Mathematical Methods for Physicists. Harcourt/Academic Press, 2000.