Mathematical Methods for Physics and Engineering - Matematica.NET

23.6 FREDHOLM THEORYcommon ratio λ/3. Thus, provided |λ| < 3, this infinite series converges to the valueλ/(3 − λ), and the solution to (23.39) isy(x) =x +λx3 − λ = 3x3 − λ . (23.40)Finally, we note that the requirement that |λ| < 3 may also be derived very easily fromthe condition (23.38). ◭23.6 Fredholm theoryIn the previous section, we found that a solution to the integral equation (23.34)can be obtained as a Neumann series of the form (23.36), where the resolventkernel R(x, z; λ) is written as an infinite power series in λ. This solution is validprovided the infinite series converges.A related, but more elegant, approach to the solution of integral equationsusing infinite series was found by Fredholm. We will not reproduce Fredholm’sanalysis here, but merely state the results we need. Essentially, Fredholm theoryprovides a formula for the resolvent kernel R(x, z; λ) in (23.36) in terms of theratio of two infinite series:D(x, z; λ)R(x, z; λ) = . (23.41)d(λ)The numerator and denominator in (23.41) are given byD(x, z; λ) =d(λ) =∞∑n=0∞∑n=0(−1) nD n (x, z)λ n , (23.42)n!(−1) nd n λ n , (23.43)n!where the functions D n (x, z) and the constants d n are found from recurrencerelations as follows. We start withD 0 (x, z) =K(x, z) and d 0 =1, (23.44)where K(x, z) is the kernel of the original integral equation (23.34). The higherordercoefficients of λ in (23.43) and (23.42) are then obtained from the tworecurrence relationsd n =∫ bD n (x, z) =K(x, z)d n − naD n−1 (x, x) dx, (23.45)∫ baK(x, z 1 )D n−1 (z 1 ,z) dz 1 . (23.46)Although the formulae for the resolvent kernel appear complicated, they areoften simple to apply. Moreover, for the Fredholm solution the power series(23.42) and (23.43) are both guaranteed to converge for all values of λ, unlike815