Stability analysis of parabolic partial differential equations with piecewise continuous arguments

Abstract

This article deals with the analytic and numerical stability of numerical methods for a parabolic partial differential equation with piecewise continuous arguments of alternately retarded and advanced type. First, application of the theory of separation of variables in matrix form and the Fourier method, the necessary and sufficient condition under which the analytic solution is asymptotically stable is derived. Then, the θ‐methods are applied to solve the corresponding initial value problem, the sufficient conditions for the asymptotic stability of numerical methods are obtained. Finally, several numerical examples are presented to support the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 531–545, 2017

Convergence and Stability of Galerkin Finite Element Method for Hyperbolic Partial Differential Equation With Piecewise Continuous Arguments of Advanced Type

“…However, the literatures mentioned above only focus on the EPCA in case of ordinary differential equations. To the best of our knowledge, there are few publications concerning partial differential equation with piecewise continuous arguments (PEPCA) solved by numerical methods except for [14,15,22,23,25]. Liang et al investigated PEPCA with the θ-methods [14] and Galerkin finite element method [15], numerical stability was analyzed, respectively.…”