Polynomial spline collocation methods for second‐order Volterra integrodifferential equations

Abstract

We have presented a method for the construction of an approximation to the initial-value second-order Volterra integrodifferential equation (VIDE). The polynomial spline collocation methods described here give a superconvergence to the solution of the equation.2000 Mathematics Subject Classification: 34Bxx, 45L10, 65D05, 65D15.

Convergence Analysis of the Legendre Spectral Collocation Methods for Second Order Volterra Integro-Differential Equations

“…6. Comparison between polynomial spline collocation methods and the spectral methods Second order VIDEs ofthe form (1.1) have been solved numerically using polynomial spline spaces [9]. In order to describe these approximating polynomial spline spaces, let Ylfi : 0 = ÍQ < tj < • • • < t^ = r be the mesh for the interval /, and set that is, by a polynomial spline function of degree m + d which possesses the knots Zf, and is d times continuously differential on /.…”

“…For a survey of early results we refer the reader to [12,[19][20][21]25]. More recently, polynomial spline collocation methods were investigated in [9,23]. Bologna [22] found an asymptotic solution for first and second order VIDEs containing an arbitrary kernel.…”

Piecewise Spectral Collocation Method for Second Order Volterra Integro-Differential Equations with Nonvanishing Delay

“…For some early research results, such as the general linear method [7], the linear multistep method [8,9], the Runge-Kutta method [10,12]. Over the years, polynomial spline collocation method [13][14][15][16]27,28], spectral Galerkin method [20] and Bologna [29] have developed an asymptotic solution for first-order and second-order VIDEs containing arbitrary kernels. In [24,37], the collocation method is used to approximate second-order VIDEs.…”

A Spectral Method for Neutral Volterra Integro-Differential Equation with Weakly Singular Kernel

“…There are many existing numerical methods for solving VIDE, such as polynomial collocation method [3,7,23,26,27,33], Taylor series method [13], block-by-block method [18,19], multiStep method [21,35] and Runge-Kutta method [2,36]. However, very few works touched the spectral approximation to VIDE.…”