Boundary Value Problems for Integro-Differential Equations of Barbashin Type

A well-balanced scheme for kinetic models of chemotaxis derived from one-dimensional local forward–backward problems

“…Now it becomes necessary to quickly recall the main topics about the so-called well-balanced numerical approximations of linear kinetic equations of the type (2), sometimes referred to as of Barbashin type [4]. Such methods have already been used in the context of solving radiative transfer problems [26] and closely follow the classic method of Case's elementary solutions [1,12,13,37], further developed by Siewert and co-workers [6,7,46].…”

“…ϕ ν are the "generalized eigenfunctions", α, β and A(ν) are determined by boundary conditions (see also [4,24,34,37]). Later, this technique has been adapted to numerical computations by Siewert and co-workers who introduced the so-called Analytical Discrete-Ordinate (ADO) method [6,7,46].…”

“…Roughly speaking, it consists in discretizing the velocity variable v according to a Gaussian quadrature rule in order to be able to compute astutely a vector of normal modes ν to be used for computing approximately the integral term in (7). Altogether, these elements allow to build an efficient Riemann solver for (5) and thus a WB scheme for (4): it has been shown in [26,29] that this type of discretizations are feasible and moreover endowed with interesting stability properties.…”

On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order

“…Furthermore, shooting methods have also been designed to simulate these models successfully. Besides these cited works, few more contributions [21][22][23][24][25] have been made to the analytical and numerical study of the solutions of FBVPs and fractional integro-differential equations.…”

“…The texts by Agarwal, O'Regan and Wong [28] and by Guo and Lakshmikantham [29] are excellent resources for the use of fixed point theory in the study of existence of solutions to boundary value problems. While much attention has been focused on the Cauchy problem for fractional differential equations for both the Reimann-Liouville and Caputo differential operators, see [22][23][24][25][26][27] and references therein, not many papers are devoted to the study of fractional order boundary value problems.…”