Mathematical Methods for Physics and Engineering - Matematica.NET

23.3 OPERATOR NOTATION AND THE EXISTENCE OF SOLUTIONSIn fact, we shall be concerned with various special cases of (23.4), which areknown by particular names. Firstly, if g(x) = 0 then the unknown function y(x)appears only under the integral sign, and (23.4) is called a linear integral equationof the first kind. Alternatively, if g(x) = 1, so that y(x) appears twice, once insidethe integral and once outside, then (23.4) is called a linear integral equation ofthe second kind. In either case, if f(x) = 0 the equation is called homogeneous,otherwise inhomogeneous.We can distinguish further between different types of integral equation by theform of the integration limits a and b. If these limits are fixed constants then theequation is called a Fredholm equation. If, however, the upper limit b = x (i.e. itis variable) then the equation is called a Volterra equation; such an equation isanalogous to one with fixed limits but for which the kernel K(x, z) =0forz>x.Finally, we note that any equation for which either (or both) of the integrationlimits is infinite, or for which K(x, z) becomes infinite in the range of integration,is called a singular integral equation.23.3 Operator notation and the existence of solutionsThere is a close correspondence between linear integral equations and the matrixequations discussed in chapter 8. However, the former involve linear, integral relationsbetween functions in an infinite-dimensional function space (see chapter 17),whereas the latter specify linear relations among vectors in a finite-dimensionalvector space.Since we are restricting our attention to linear integral equations, it will beconvenient to introduce the linear integral operator K, whose action on anarbitrary function y is given byKy =∫ baK(x, z)y(z) dz. (23.5)This is analogous to the introduction in chapters 16 and 17 of the notation L todescribe a linear differential operator. Furthermore, we may define the Hermitianconjugate K † byK † y =∫ baK ∗ (z,x)y(z) dz,where the asterisk denotes complex conjugation and we have reversed the orderof the arguments in the kernel.It is clear from (23.5) that K is indeed linear. Moreover, since K operates onthe infinite-dimensional space of (reasonable) functions, we may make an obviousanalogy with matrix equations and consider the action of K on a function f asthat of a matrix on a column vector (both of infinite dimension).When written in operator form, the integral equations discussed in the previoussection resemble equations familiar from linear algebra. For example, the805