Stability Analysis Of Numerical Methods For Nonlinear Delay Integro-differential Equations

Delay integro-differential equations (DIDEs) arise widely in the fields of physics, biology, chemistry, medicine, economics, control theory and so on. It is meaningful to investigate the theory and application of numerical methods for DIDEs. In recent 30 years, the theory of computational methods for DIDEs has been widely discussed by many authors and a great deal of results have been found. The papers are mainly focused on the linear systems of DIDEs, and a few papers study the numerical stability of nonlinear DIDEs. The present paper is devoted to the stability analysis of numerical methods for nonlinear DIDEs. The main results obtained are as follows.(1) One-leg methods are adapted for solving a class of nonlinear DIDEs. It is proved that an A-stable one-leg methods is numerically stable and a strongly A-stable one-leg methods is asymptotically stable under the suitable conditions. Numerical tests are given to confirm the theoretical results.(2) Linear multistep methods are adapted for solving a class, of nonlinear DIDEs. It is proved that an A-stable linear multistep methods is asymptotically stable under the suitable conditions. Numerical tests are given to illustrate the theoretical results.