Singularly perturbed Volterra integral equations with weakly singular kernels

Abstract

We consider finding asymptotic solutions of the singularly perturbed linear Volterra integral equations with weakly singular kernels. An interesting aspect of these problems is that the discontinuity of the kernel causes layer solutions to decay algebraically rather than exponentially within the initial (boundary) layer. To analyse this phenomenon, the paper demonstrates the similarity that these solutions have to a special function called the Mittag-Leffler function.

Error bound analysis and singularly perturbed Abel‐Volterraequations

“…This result agrees with that obtained in [3] for f (x,s) = x using the explicit solution in terms of the Mittag-Leffler function.…”

“…To date, apart from the present paper, there have been no efforts to prove the validity of the formal approximation to nonlinear singularly perturbed Volterra integral equations with weakly singular kernels. The linear version of (1.1) has been studied in [3], where it is shown that the formal asymptotic solution is the sum of a slowly varying function and a rapidly varying function, described in terms of the Mittag-Leffler function. The Mittag-Leffler function (a special function) is an example of functions which decay algebraically at infinity, as pointed in the previous paragraph.…”

Asymptotics of integrodifferential models with integrable kernels

“…Moreover, the case g(0) ≠ 0 is not covered in the analysis presented in [1]. Recently, [2] studied, rigorously, asymptotic solutions of singularly perturbed linear Volterra integral equations with weakly singular kernels using the additive decomposition method. It is shown in [2] that the inner layer solution can explicitly be written in terms of the Mittag-Leffler function which tends to zero algebraically at infinity.…”

“…Recently, [2] studied, rigorously, asymptotic solutions of singularly perturbed linear Volterra integral equations with weakly singular kernels using the additive decomposition method. It is shown in [2] that the inner layer solution can explicitly be written in terms of the Mittag-Leffler function which tends to zero algebraically at infinity. Adopting the techniques used in [2] here, the formal asymptotic solution for (1.1) is derived by revealing the structure of its contents and proof of asymptotic correctness presented.…”

“…It is shown in [2] that the inner layer solution can explicitly be written in terms of the Mittag-Leffler function which tends to zero algebraically at infinity. Adopting the techniques used in [2] here, the formal asymptotic solution for (1.1) is derived by revealing the structure of its contents and proof of asymptotic correctness presented. Hence, this shows the essential difference between the results in this paper and the results in [1].…”