Padé-approximant Bounds and Approximate Solutions for Kirkwood-Riseman Integral Equations

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Abstract
Two-point Padé approximants are used to calculate tight upper and lower bounds on the quantity <ø, f> associated with Kirkwood-Riseman integral equations (1+yL)ø=f, which arise in the diffusion theory of flexible macromolecules. The self-adjoint operator L is an integral operator on −1 ≤ x ≤ 1, with weakly singular kernel |x − x′|−½, and the two specific cases (i) f = 1, (ii) f = x2 are studied. In case (i) direct bounds on <ø, 1> are obtained; this quantity is inversely proportional to the translational diffusion constant. In case (ii) bounds on <ø, 1 > are found by a new technique involving combinations of bounds for the three cases f = 1, f = x2 and f = bx2±b−1. Various types of Pade and related approximants are compared, using the information <f, Lnf>, n = −2, −1, 0, 1, 2, 3 and λ (an upper bound on L) for several values of the positive parameter y. Padé-approximant-generating trial vectors are investigated and a convergence theorem is established. The vector consisting of an optimum linear combination of L−1f, f and Lf is found to be an accurate approximation to a numerical solution in case (ii), for all values of y and x. Specific analytical expressions are derived for the approximate solutions.

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