Integral equation

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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:

<math> f(x) = \int_a^b K(x,t)\,\phi(t)\,dt </math>

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:

<math> \phi(x) = f(x) + \lambda \int_a^b K(x,t)\,\phi(t)\,dt </math>

The parameter λ is an unknown factor, which plays the same role as the eigenvalue in linear algebra.

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:

<math> f(x) = \int_a^x K(x,t)\,\phi(t)\,dt </math>
<math> \phi(x) = f(x) + \lambda \int_a^x K(x,t)\,\phi(t)\,dt </math>

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.

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