Integral Equations

2 Notion and examples of integral equations (IEs). FredholmIEs of the first and second kindConsider an integral equation (IE)∫ bf(x) = φ(x) + λ K(x, y)φ(y)dy. (1)aThis is an IE of the 2nd kind. Here K(x, y) is a given function of two variables (the kernel)defined in the squareΠ = {(x, y) : a ≤ x ≤ b, a ≤ y ≤ b},f(x) is a given function, φ(x) is a sought-for (unknown) function, and λ is a parameter.We will often assume that f(x) is a continuous function.The IEf(x) =∫ baK(x, y)φ(y)dy (2)constitutes an example of an IE of the 1st kind.We will say that equation (1) (resp. (2)) is the Fredholm IE of the 2nd kind (resp. 1st kind)if the the kernel K(x, y) satisfies one of the following conditions:(i) K(x, y) is continuous as a function of variables (x, y) in the square Π;(ii) K(x, y) is maybe discontinuous for some (x, y) but the double integral∫ b ∫ baa|K 2 (x, y)|dxdy < ∞, (3)i.e., takes a finite value (converges in the square Π in the sense of the definition of convergentRiemann integrals).An IE∫ xf(x) = φ(x) + λ K(x, y)φ(y)dy, (4)0is called the Volterra IE of the 2nd kind. Here the kernel K(x, y) may be continuous in thesquare Π = {(x, y) : a ≤ x ≤ b, a ≤ y ≤ b} for a certain b > a or discontinuous and satisfyingcondition (3); f(x) is a given function; and λ is a parameter,Example 1 Consider a Fredholm IE of the 2nd kindx 2 = φ(x) −∫ 10(x 2 + y 2 )φ(y)dy, (5)where the kernel K(x, y) = (x 2 + y 2 ) is defined and continuous in the square Π 1 = {(x, y) : 0 ≤x ≤ 1, 0 ≤ y ≤ 1}, f(x) = x 2 is a given function, and the parameter λ = −1.Example 2 Consider an IE of the 2nd kindf(x) = φ(x) −∫ 10ln |x − y|φ(y)dy, (6)5