Quadrature rules are a common numerical tool for approximating definite integrals. While most classical rules are based on polynomial interpolation, recent results reveal the efficiency and effectiveness of quadrature rules based on linear barycentric rational interpolants. In this paper, we derive new quadrature rules from barycentric rational Hermite interpolants and prove their convergence orders. We then use the proposed quadrature rules to construct direct quadrature methods for solving Volterra integral equations. We provide several numerical experiments that validate our theoretical results and illustrate the efficiency of our new quadrature rules and methods.

Backward differential formulae (BDF) are the basis of the highly efficient schemes for the numerical solution of stiff ordinary differen- tial equations for decades. An alternative multistep schemes (RBDF) based on barycentric rational interpolation is proposed. Specifically, robust new methods of orders 1 to 5 are derived. The local trun- cation error is analyzed for variable stepsizes in order to implement a variable order, variable stepsize prototype in Matlab. Aspects of the implementation are addressed in detail. Numerical experiments illustrate that the RBDF code compares well with Matlab’s ode15s.

For their high accuracy and good stability properties, implicit numerical methods are widely used for solving Volterra integral equations, while, in order to save computational effort, explicit algorithms are preferred in the case of non-stiff problems. In this paper, highly accurate explicit methods based on the Floater–Hormann family of linear barycentric rational interpolants are presented. The order of convergence is obtained in terms of the parameters of the methods. Moreover, the linear stability properties with respect to both the basic and convolution test equations are analyzed in detail. Numerical experiments are discussed in order to validate the theoretical results and illustrate the efficiency and power of the methods applied to non-stiff and mildly stiff problems.

The Floater–Hormann family of the barycentric rational interpolants has recently gained popularity because of its excellent stability properties and highly order of convergence. The purpose of this paper is to design highly accurate and stable schemes based on this family of interpolants for the numerical solution of stiff Volterra integral equations of the second kind.

It is our purpose to introduce a simple multistep method based on linear barycentric rational interpolation for solving ordinary differential equations. Also, we design an adaptive version having one free parameter which is used to improve the stability properties. Numerical experiments of the constructed methods on some well-known stiff problems indicate efficiency and capability of the methods in solving stiff problems.

In this paper, two applications of linear barycentric rational interpolation are used to derive a difference-quadrature scheme for solving a class of systems of Volterra integro-differential equations of the second kind. The convergence of the proposed method is proved, and the order of convergence is obtained in terms of the parameters of the method. Furthermore, the linear stability of the proposed method with respect to both the basic and convolution test equations is analyzed. All the obtained theoretical results are verified by several numerical experiments.

A method for solving delay Volterra integro-differential equations is introduced. It is based on two applications of linear barycentric rational interpolation, barycentric rational quadrature and barycentric rational finite differences. Its zero–stability and convergence are studied. Numerical tests demonstrate the excellent agreement of our implementation with the predicted convergence orders.

This paper deals with the stability analysis of the composite barycentric rational quadrature method (CBRQM) for the second kind Volterra integral equations through application to the standard and the convolution test equations. In each case, some theoretical results are achieved by providing corresponding recurrence relation and stability matrix. Verification of these theoretical results is obtained by some numerical experiments.

We introduce two direct quadrature methods based on linear rational interpolation for solving general Volterra integral equations of the second kind. The first, deduced by a direct application of linear barycentric rational quadrature given in former work, is shown to converge at the same rate as the rational quadrature rule but is costly on long integration intervals. The second, based on a composite version of this quadrature rule, loses one order of convergence but is much cheaper. Both require only a sample of the involved functions at equispaced nodes and yield an infinitely smooth solution of most classical examples with machine precision.