Asymptotic Solution to a Class of Nonlinear Volterra Integral Equations

Abstract

A recent investigation of the problem of determining the temperature in a nonlinearly radiating semi-infinite solid has been the motivation for the study of the integral equation qg(t) --(t s)-/2{f(s) qgn(s)) ds, >-_ 0, n => 1.The asymptotic behavior of p(t) is found in the limits + and 0+. The method of analysis utilizes some recent results on integral transforms.

Asymptotic behavior of temperature in a nonlinear radiating, linear absorbing rod of finite length

“…The problem (1.1)-(1.3) and its various generalization have been treated by several authors (cf. [1][2][3][4][5][6][7][8]). The recent work by Keller-Olmstead [5] and Hartka [4] studies the asymptotic behavior of the temperature in a semi-infinite rod for the case c < 0, u0 = 0.…”

“…Several estimates for insuring the diminishing property of u as t -* m are given in [3,5] for the case c -ft = 0 and in [4] for the case ft2 + c < 0. However, when the rod is finite, no matter how great its length may be, it is reasonable to expect that the temperature remains diminishing to zero for certain range of values of c, ft even if both constants are positive.…”

“…We also extend the above investigation to a heat-conduction problem in a two-dimensional rectangular region. The consideration of a finite domain instead of a semi-infinite domain makes it possible to sharpen the results obtained in [3][4][5] and leads to more information about the solution. In fact, we obtain explicit estimates for the rate of decay for the solution c. v. pao in terms of the physical parameters D, c, a, 13, n, I and the boundary source h(t).…”

Temperature of a semi-infinite rod which radiates both linearly and nonlinearly

“…This problem is a generalization of the one considered by Keller and Olmstead [ll and by Handelsman and Olmstead [2], Specifically, they considered this problem with b = h = 0. Friedman [3] discussed the existence, uniqueness, and certain other properties of the solutions of similar problems but with the requirement that an outward derivative at the surface be a strictly decreasing function of the dependent variable.…”

Diffusion systems reacting at the boundary

“…As found in [8,9], the asymptotic analysis of integral equations like (3.3) is often made easier by an Abel inversion of the integral operator to achieve the alternative form…”