Uniform difference method for singularly perturbed Volterra integro-differential equations

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.

show abstract

“…Various approximating aspects for singularly perturbed VIDE's have also been investigated in [2,5,6,18,20,26,28,31].…”

Section: Introductionmentioning

confidence: 99%

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

Kudu¹,

Amirali²,

Amiraliyev³

2016

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

show abstract

“…For singularly perturbed problems, when the perturbed parameter approaches to zero, the width of the boundary layer becomes thinner. The behavior of u in the boundary layer is hard to simulate numerically, i.e., the solution of (1.1) for the singular perturbation parameter varies very rapidly in a thin layer near t = 0 compared to the solution of (1.2) (see, e.g., [9,10]). Traditional methods, such as finite difference or finite element method, do not work well for these problems because they often produce oscillatory solutions which are inaccurate when the perturbed parameter is small.…”

Section: Introductionmentioning

confidence: 99%

“…From the numerical experiments in [9], uniform convergence of the exponential finite difference method under a Shishkin mesh at nodes was almost second order. In [10], Amiraliyev and Sevgin constructed an exponentially fitted difference scheme and analyzed the first order uniform convergence property under uniform mesh in the discrete maximum norm. Cen and Li [11] studied a finite difference scheme based on trapezoidal integration under the Shishkin mesh, which is almost second-order uniformly convergent at nodes theoretically and numerically.…”

Section: Introductionmentioning

confidence: 99%

The coupled method for singularly perturbed Volterra integro-differential equations

Tao

Zhang

2019

Adv Differ Equ

View full text Add to dashboard Cite

In this work a coupled (LDG-CFEM) method for singularly perturbed Volterra integro-differential equations with a smooth kernel is implemented. The existence and uniqueness of the coupled solution is given, provided that the source function and the kernel function are sufficiently smooth. Furthermore, the coupled solution achieves the optimal convergence rate p + 1 in the L 2 norm and a superconvergence rate 2p at nodes for the numerical solutionÛ N with the one-sided flux inside the boundary layer region under layer-adapted meshes uniformly with respect to the singular perturbation parameter .

show abstract

“…These problems often arise in various areas, for instance, in models of population dynamics, epidemics, diffusion with nonlinear surface dissipation, synchronous control systems and nonlinear renewal processes, filament stretching, polymer rheology, nonlinear radiation heat transfer (see, e.g., [2,13] and references quoted).…”

Section: Introductionmentioning

confidence: 99%

“…They discussed the convergence of the method and showed that the proposed method is almost second convergence. On the other hand, Amiraliyev and Ş evgin [2] presented an exponentially fitted finite difference method to solve the same problem. The fitting factor was intoduced via the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with weight and remainder terms in integral form.…”

Section: Introductionmentioning

confidence: 99%

New Parameter-Uniform Discretisations of Singularly Perturbed Volterra Integro-Differential Equations

Iragi¹,

Munyakazi²

2018

Appl. Math. Inf. Sci

View full text Add to dashboard Cite

We design and analyse two numerical methods namely a fitted mesh and a fitted operator finite difference methods for solving singularly perturbed Volterra integro-differential equations. The fitted mesh method we propose is constructed using a finite difference operator to approximate the derivative part and some suitably chosen quadrature rules for the integral part. To obtain a parameter-uniform convergence, we use a piecewise-uniform mesh of Shishkin type. On the other hand, to construct the fitted operator method, the Volterra integro-differential equation is discretised by introducing a fitting factor via the method of integral identity with the use of exponential basis function along with interpolating quadrature rules [2]. The two methods are analysed for convergence and stability. We show that the two methods are robust with respect to the perturbation parameter. Two numerical examples are solved to show the applicability of the proposed schemes.

show abstract

Uniform difference method for singularly perturbed Volterra integro-differential equations

Cited by 20 publications

References 13 publications

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

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

The coupled method for singularly perturbed Volterra integro-differential equations

New Parameter-Uniform Discretisations of Singularly Perturbed Volterra Integro-Differential Equations

Contact Info

Product

Resources

About