Integral equation

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.

Related Topics:
Mathematics - Function - Integral - Differential - Maxwell's equations

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The most basic type of integral equation is a Fredholm equation of the first type:

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: f(x) = int_a^b K(x,t),phi(t),dt

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The notation follows Arfken.

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Here φ is an unknown function,

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f is a known function,

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and K is another known function of two variables,

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often called the kernel function.

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Note that the limits of integration are constant; this is what characterizes a Fredholm equation.

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

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: phi(x) = f(x) + lambda int_a^b K(x,t),phi(t),dt

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The parameter λ is an unknown factor,

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which plays the same role as the eigenvalue in linear algebra.

Related Topics:
Eigenvalue - Linear algebra

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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:

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: f(x) = int_a^x K(x,t),phi(t),dt

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: phi(x) = f(x) + lambda int_a^x K(x,t),phi(t),dt

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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.

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

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;Limits of integration

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: both fixed: Fredholm equation

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: one variable: Volterra equation

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;Placement of unknown function

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: only inside integral: first kind

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: both inside and outside integral: second kind

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;Nature of known function f

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: identically zero: homogeneous

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: not identically zero: inhomogeneous

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Integral equations are important in many applications.

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Problems in which integral equations are encountered include

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radiative energy transfer and the oscillation of a string, membrane, or axle.

Related Topics:
Radiative energy transfer - Oscillation

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Oscillation problems may also be solved as differential equations.

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