Stability Analysis Of Numerical Methods For Several Classes Of Volterra Functional Differential Equations

Functional differential equations(FDEs) arise widely in the fields of control theory, biology, medicine, chemistry, economics and so on. It is meaningful to investigate the theory and application of numerical methods for FDEs. In recent 30 years, the theory of computational methods for Vloterra FDEs(VFDEs), especially for delay differential equations(DDEs), has been widely discussed by many authors and a great deal of results have been found. As to the linear stability analysis of numerical methods for DDEs, we refer to the works of Barwell, Watanabe, Zennaro, Spijker, in't Hout, Bellen, Jackiewicz, Liu mingzhu, Kuang jiaoxun, Tian hongjiong, Zhang chengjian, Hu guangda and so on. The main results can be found in the monograph of Bellen and Zennaro or Kuang. Nonlinear stability analysis of numerical methods for DDEs originated with Torelli(in 1989) and Bellen and Zennare(in 1992). In 1999, Huang and Li et al. pointed out that the requirements of Torelli's stability are so strong that only a few methods with low order satisfy the conditions and put forward a newly reasonable stability concept. On the basis of the new concept, the study of nonlinear numerical stabilty for DDEs has been developed vigorously. Though the study limited to constant delay, fixed step, linear interpolation and negative Lipschitz constant at that time, the research object contained almost the commonly used numerical methods for DDEs and a great deal of new stability results had been obtained. Convergence of numerical methods for DDEs is another important issue and lots of results based on the classical Lipschitz condition can be found. As to the case of VFDEs, we refer to the momograph of Li(in 1997). For the case of DDEs, we refer to the papers of Oberle, Pesch, Bellen, Zennaro, Tavernini, Arndt, Enright, Feldstein, Neves, Karoui, Vaillancourt, Baker and Paul etc. However, the above results are only suitable to nonstiff DDEs, however, which are not suitable to stiff DDEs. Convergence analysis of numerical methods for stiff DDEs originated from the works of Zhang et al. in 1997. They introduced the concept of D-convergence and proved that several implicit...