Classical Theory of Linear Multistep Methods for Volterra Functional Differential Equations

Yunfei Li,Shoufu Li,Piergiulio Tempesta

doi:10.1155/2021/6633554

Abstract

Based on the linear multistep methods for ordinary differential equations (ODEs) and the canonical interpolation theory that was presented by Shoufu Li who is exactly the second author of this paper, we propose the linear multistep methods for general Volterra functional differential equations (VFDEs) and build the classical stability, consistency, and convergence theories of the methods. The methods and theories presented in this paper are applicable to nonneutral, nonstiff, and nonlinear initial value problems in ODEs, Volterra delay differential equations (VDDEs), Volterra integro-differential equations (VIDEs), Volterra delay integro-differential equations (VDIDEs), etc. At last, some numerical experiments verify the correctness of our theories.

Highlights

Volterra functional differential equations (VFDEs) contain many subtypes, such as Volterra delay differential equations (VDDEs), Volterra integro-differential equations (VIDEs), Volterra delay integro-differential equations (VDIDEs), etc
Li [30,31,32,33,34,35] has carried on systematic research for stiff general VFDEs and the numerical methods for them
Based on the linear multistep methods for ordinary differential equations (ODEs) and the canonical interpolation theory presented by Li [34], we propose the linear multistep methods for general VFDEs and build the classical stability, consistency, and convergence theories of the methods. e methods and theories presented in this paper are applicable to nonneutral, nonstiff, and nonlinear initial value problems in ODEs, VDDEs, VIDEs, VDIDEs, etc., and are interesting companions to the methods and theories in [36]

Summary

Introduction

VFDEs contain many subtypes, such as VDDEs, VIDEs, VDIDEs, etc. Certainly ODEs are a subtype of them. In 2014, Li [36] established the classical stability and convergence theories of Runge–Kutta methods for nonstiff, nonlinear general VFDEs. It is well known that in solving ODEs, linear multistep methods have significant advantages in the aspects of format simplification and computation cost, and experts have presented many famous linear multistep methods such as Backward Differentiation Formula (BDF) methods and Adams methods. Based on the linear multistep methods for ODEs and the canonical interpolation theory presented by Li [34], we propose the linear multistep methods for general VFDEs and build the classical stability, consistency, and convergence theories of the methods. E methods and theories presented in this paper are applicable to nonneutral, nonstiff, and nonlinear initial value problems in ODEs, VDDEs, VIDEs, VDIDEs, etc., and are interesting companions to the methods and theories in [36].

Derivation of the Numerical Methods

Stability Analysis

Consistency and Convergence Analyses

Numerical Examples