On the Fredholm integral equation associated with pairs of dual integral equations

Abstract

Consider the Fredholm equation of the second kindwhereand Jv is the Bessel function of the first kind. Here ka(t) and h(x) are given, the unknown function is f(x), and the solution is required for large values of the real parameter a. Under reasonable conditions the solution of (1.1) is given by its Neumann series (a set of sufficient conditions on ka(t) for the convergence of this series is given in Section 4, Lemma 2). However, in many applications the convergence of the series becomes too slow as a→∞ for any useful results to be obtained from it, and it may even happen that f(x)→∞ as a→∞. It is the aim of the present investigation to consider this case, and to show how under fairly general conditions on ka(t) an approximate solution may be obtained for large a, the approximation being valid in the norm of L2(0, 1). The exact conditions on ka(t) and the main result are given in Section 4. Roughly, it is required that 1 -ka(at) should behave like tp(p>0) as t→0. For example, ka(at) might be exp ⌈-(t/ap)⌉.

Application to magnetohydrodynamic duct flows of a singular elliptic problem associated with a rectangle

“…We are not able, therefore, to make any direct application of known theory [16] in order to determine f 1 . Although there is a large parameter a present in the equation, it does not seem possible to use a method due to Hutson [6] whereby an approximation to / , may be calculated. We therefore adopt the following new approach.…”

“…(4.12) yields the following equation )iK*(a)£ 2 (er) = K V ) ,(4)(5)(6)(7)(8)(9)(10)(11)(12)(13) and we have defined the Fourier transformation of a function <p(z) to be <t>(z)e iz<T dz.on making use of the known function K*(z). We note that k 2 (z) may be written as a contour integral from which representation it is easy enough to deduce that / r VwiN for large z k 2 (z) is Ol exp ~z--I.…”