Linear Multistep Methods for Volterra Integral and Integro-Differential Equations

Abstract

Abstract. A general class of linear multistep methods is presented for numerically solving firstand second-kind Volterra integral equations, and Volterra integro-differential equations. These so-called VLM methods, which include the well-known direct quadrature methods, allow for a unified treatment of the problems of consistency and convergence, and have an analogue in linear multistep methods for ODEs, as treated in any textbook on computational methods in ordinary differential equations.General consistency and convergence results are presented (and proved in an Appendix), together with results of numerical experiments which support the theory.

Fractional collocation boundary value methods for the second kind Volterra equations with weakly singular kernels

“…for some constant B independent of i, j . Noting that the kernel function K(t, s) is bounded in the region D,we arrive at the estimate, for (20) where the constant B does not depend on i, j , and n. Now we consider the bound of…”

“…Here u(t) is unknown, and K(t, t) = 0, ∀t ∈ I . Volterra integral equations can be solved by various numerical methods, such as collocation methods [8][9][10], spectral methods [11][12][13], Nyström method [14][15][16], Runge-Kutta methods [17][18][19], and linear multistep methods [20][21][22]. Specially, collocation methods have been extensively studied and theoretical results and numerical practice have indicated that this class of algorithms are able to deal with many problems.…”

Jacobi spectral collocation method for the approximate solution of multidimensional nonlinear Volterra integral equation

“…We observe that there are many numerical approaches for solving one-dimensional Volterra integral equation, such as Runge–Kutta method (Brunner 1984 ; Yuan and Tang 1990 ), polynomial collocation method (Brunner 1986 ; Brunner et al. 2001 ; Brunner and Tang 1989 ), multistep method (Mckee 1979 ; Houwen and Riele 1985 ), hp-discontinuous Galerkin method (Brunner and Schötzau 2006 ) and Taylor series method (Goldfine 1977 ). The spectral collocation method is the most popular form of the spectral methods among practitioners.…”

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.…”