A finite-difference method for a singularly perturbed delay integro-differential equation

Abstract

We consider the singularly perturbed initial value problem for a linear first order Volterra integro-differential equation with delay. Our purpose is to construct and analyse a numerical method with uniform convergence in the perturbation parameter. The numerical solution of this problem is discretised using an implicit difference rules for differential part and the composite numerical quadrature rules for integral part. On a layer-adapted mesh error estimations for the approximate solution are established. Numerical examples supporting the theory are presented.

On the Volterra Delay-Integro-Differential equation with layer behavior and its numerical solution

“…followed by the application appropriate quadrature rules. Namely, applying right side rectangle rule in (2.1), analogous to [13] we get…”

“…Amiraliyev and Yilmaz [2] gave an exponentially fitted difference method on a uniform mesh for (1.1)-(1.2) except for a delay term in differential part and shown that the method is first-order convergent uniformly in ". A useful discussion of uniform convergence on a fitted mesh, for another form of SPVDIDEs have been investigated in [13].…”

A Computational Method for Volterra Integro-Differential Equation

“…The authors in Hoppensteadt et al (2007); Fazeli and Hojjati (2015); Okayama (2018) are studied the numerical solutions of VIDEs by using various methods, such as collocation method, Runge-Kutta method, Sinc-Nyström method. Nevertheless, authors in Amiraliyev and Sevgin (2006); Amiraliyev and Yilmaz (2014); Kudu et al (2016) are suggested numerical solutions of VIDEs by using fitted difference method. In this paper, we are present a novel finite-difference scheme on a uniform mesh to approximate (1.1-(1.2).…”

A parameter uniform difference scheme for the parameterized singularly perturbed problem with integral boundary condition

“…Therefore, boundary value problems involving integral boundary conditions have been studied by many authors [6,7,[26][27][28][29][30][31] (see also references therein). Some approximating aspects of this kind of problems in the regular cases, i.e., in the absence of layers, were investigated in [7,26,32].…”

“…A hybrid scheme, which is second order convergent on Shishkin mesh, was discussed in [30] (see also [20,33]). For the numerical methods concerning second order singularly perturbed differential equations with integral boundary conditions, one can see, e.g., [28,29,31].…”