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In this paper, we present a second order parameter-uniform numerical method for a singularly perturbed Volterra delay-integro-differential equation on a Bakhvalov-type mesh. The equation is discretized by using the variable two-step backward differentiation formula of the first derivative term and the trapezoidal formula of the integral term. The stability and convergence of the numerical method in the discrete maximum norm are proved. Finally, the theoretical results are verified by some numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, a second finite difference method on a graded grid is proposed for a Volterra integro-differential equation with a weakly singular kernel. The proposed scheme is obtained by using the two-step backward differentiation formula (BDF2) to discretize the first derivative term and the first-order interpolation scheme to approximate the integral term. The analysis of stability is proved and used to prove the convergence of our presented numerical method in the discrete maximum norm. Finally, Numerical experiments are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

A singularly perturbed Volterra integro-differential problem is considered. The variable two-step backward differentiation formula is used to approximate the first-order derivative term and the trapezoidal formula is used to discretize the integral term. Then, the stability and convergence analysis of the proposed numerical method are proved. It is shown that the proposed scheme is second-order uniformly convergent with respect to perturbation parameter ε in the discrete maximum norm. Finally, a numerical experiment verifies the theoretical results. A singularly perturbed Volterra integro-differential problem is considered. The variable two-step backward differentiation formula is used to approximate the first-order derivative term and the trapezoidal formula is used to discretize the integral term. Then, the stability and convergence analysis of the proposed numerical method are proved. It is shown that the proposed scheme is second-order uniformly convergent with respect to perturbation parameter in the discrete maximum norm. Finally, a numerical experiment verifies the theoretical results. [ABSTRACT FROM AUTHOR]

A novel parameter-uniform finite difference scheme on a Shishkin-type mesh for a singularly perturbed Volterra integro-differential equation is studied. The problem is discretized by the variable two-step backward differentiation formula (BDF2) for the first-order derivative term and the trapezoidal formula for the integral term. The stability of the proposed numerical method is carried out. It is shown from the convergence analysis that our presented method is almost second-order uniformly convergent with respect to the perturbation parameter ε in the discrete maximum norm. Numerical results are given to support our theoretical result. [ABSTRACT FROM AUTHOR]

In this paper, an adaptive grid method for singularly perturbed Volterra integro-differential equations exhibiting a boundary layer is studied. At first, based on an arbitrary nonuniform mesh, we use the classical backward Euler formula to discrete the first-order derivative part and the left rectangle formula to approximate the integral part, respectively. Meanwhile, it is shown from the mesh equidistribution principle and the truncation error analysis that the presented adaptive grid method is first-order uniformly convergent with respect to the perturbation parameter. Furthermore, the Richardson extrapolation technique is utilized to improve the uniform accuracy of the presented adaptive grid method in the discrete supremum norm form O (N − 1) to O (N − 2) , where N is the mesh parameter. Finally, some numerical results are given to support the theoretical results. [ABSTRACT FROM AUTHOR]

A singularly perturbed Volterra integro-differential equation with an integrable singularity in the integral term is considered. The upwind difference method is used to discretize the singularly perturbed Volterra integro-differential equation. The stability properties and a posteriori error analysis for the discrete scheme are given. Since a priori information of the exact solution may be difficult to obtain, a posteriori adaptive algorithm based on a posteriori error analysis is established by equidistributing arc-length monitor function which attempts to automatically detect the singular behavior of the solution. Numerical experiments support this theoretical result. [ABSTRACT FROM AUTHOR]

A singularly perturbed Volterra integro-differential problem is considered. At first, this problem is discretized by using the variable two-step backward differentiation formula (BDF2) and the trapezoidal formula on a Bakhvalov-type mesh to approximate the first-order derivative term and the integral term, respectively. Then, the stability and convergence analysis of the proposed numerical method are carried out. It is shown that the proposed numerical method is second-order uniformly convergent with respect to perturbation parameter ɛ in the discrete maximum norm. Finally, the theoretical find is illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, a system of singularly perturbed Volterra integro-differential equations is considered. The backward Euler formula is used to discretize the differential part and the right-hand rectangle rule is applied to approximate the integral term. The stabilities of the continuous and discrete solutions are carried out using the Grönwall's inequality, respectively. The a posterior error bounds are given to design an adaptive grid generation algorithm. Numerical results complement the theoretical results. [ABSTRACT FROM AUTHOR]

A two-point boundary value problem whose highest order term is a Riemann–Liouville fractional derivative of order 2 − δ with 0 < δ < 1 is considered. Such problem is reformulated as a Volterra integral equation of the second kind. An integral discrete scheme is developed for this Volterra integral equation on an adaptive grid that is constructed adaptively from a knowledge of the exact solution. It is shown from a rigorous priori error analysis that the discrete solutions are uniformly convergent with respect to the parameter δ. Besides, in order to establish the parameter of the Volterra integral equation, we construct a nonlinear optimization problem, which is solved by the Nelder–Mead simplex method. Numerical results are given to demonstrate the performance of presented method. [ABSTRACT FROM AUTHOR]

A Riemann–Liouville two-point boundary value problem is considered. An integral discretization scheme is developed to approximate the Volterra integral equation transformed from the Riemann–Liouville boundary value problem. The stability results and a posteriori error analysis are given. A solution-adaptive algorithm based on a posteriori error analysis is designed by equidistributing arc-length monitor function. Numerical experiments are presented to support the theoretical result. [ABSTRACT FROM AUTHOR]