Convergence and stability of extended block boundary value methods for Volterra delay integro-differential equations

In this paper, we construct a class of extended block boundary value methods (B₂VMs) for Volterra delay integro-differential equations and analyze the convergence and stability of the methods. It is proven under the classical Lipschitz condition that an extended B₂VM is convergent of order p if the underlying boundary value methods (BVM) has consistent order p. The analysis shows that a B₂VM extended by an A-stable BVM can preserve the delay-independent stability of the underlying linear systems. Moreover, under some suitable conditions, the extended B₂VMs can also keep the delay-dependent stability of the underlying linear systems. In the end, we test the computational effectiveness by applying the introduced methods to the Volterra delay dynamical model of two interacting species, where the theoretical precision of the methods is further verified.