Algorithms for the solution of second order Volterra integro-differential equations

Legendre spectral method for solving integral and integro-differential equations

“…The methods produce an accurate solution at small number of nodes. The comparison of the maximum absolute error resulting from the proposed method and those obtained by EL-Gendi [4]; El-Kady [6]; Garey and Shaw [8]; Kelly [lo] Kelley and Northrup [Ill show favorable agreement and always it is more accurate than these treatments.…”

“…When we apply Algorithm 5.1 and Table 5.1. The results of the proposed method is compared with those obtained by Garey and Shaw [8], they obtain a maximum absolute error 1.0 x lop3 at n = 20. …”

“…The approximation in these treatments was made by a direct substitution of the approximating expansion in the given equation and supplementary conditions. Garey and Shaw [8] considered a class of second order Volterra integrodifferential equations. They defined one-step methods of the Runge-Kutta type to provide the required starting values and identified a fourth order method to solve the problem.…”

Linear barycentric rational collocation method for solving second-order Volterra integro-differential equation

“…To get the equidistant node of the barycentric formula, Floater et al (2007Floater et al ( , 2012aFloater et al ( , 2012b have proposed a rational interpolation scheme which has high numerical stability and interpolation accuracy on both equidistant and special distributed nodes. In Garey and Shaw (1991), one-step methods of the Runge-Kutta type methods are presented for a class of second-order Volterra integro-differential equations in reference Abdi and Hossseint (2019), linear barycentric rational interpolation is used to derive a differencequadrature scheme for solving first-order Volterra integro-differential equations. In recent papers, Wang et al (2012Wang et al ( , 2015 successfully applied the collocation method to solve initial value problems, plane elasticity problems, incompressible plane problems and non-linear problems which have expanded the application fields of the collocation method.…”