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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'': : 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: : : 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 . SEE ALSO REFERENCES
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