Rigorous Results on the Asymptotic Solutions of Singularly Perturbed Nonlinear Volterra Integral Equations

Efficient Algorithm for Solving Fuzzy Singularly Perturbed Volterra Integro-Differential Equation

“…Many physical and biological problems include the perturbed Volterra integro-differential and integral equation (see, e.g., [7], [8], [9]). In [10] , the authors provide a survey of singularly perturbed Volterra integral and integro-differential equations.…”

Asymptotics of integrodifferential modelswith integrable kernels II

“…This implies that the singularly perturbed equation (1.1) possesses an initial layer width of order (ε 1/(2−β) ), ε → 0, meaning that the solution y(t; ε) of (1.1) is slowly varying for O(ε 1/(2−β) ) ≤ t ≤ T as ε → 0, but changes rapidly on a small interval 0 ≤ t ≤ O(ε 1/(2−β) ). Therefore, the initial layer region for problem (1.1) is thicker compared with similar equations with continuous kernels (see [2,3,5]) and integral equations with weakly singular kernels (see [4]).…”

“…It is demonstrated in [6] that the linear scalar singularly perturbed Volterra integrodifferential equation has a wider initial layer width, of order O(ε 1/(2−β) ), ε → 0, and that the formal approximate solution is an asymptotic solution up to the order of magnitude of O(ε 1/(2−β) ), ε → 0. The rescaling of the initial layer width and magnitude (of the solution) in [2,3,4,5,6] reveals differences in the order of magnitudes of the initial layer thickness and magnitudes of the solution in the layer region for the cases β = 0 and β ≠ 0. This, in fact, suggests different starting asymptotic approximations.…”

The coupled method for singularly perturbed Volterra integro-differential equations

“…which is a Volterra integral equation of the second kind. Singularly perturbed Volterra integro-differential equations arise in many physical and biological problems (see, e.g., [1][2][3][4][5][6][7]). A survey of singularly perturbed Volterra integral and integro-differential equations is provided in [8].…”