Unconditional stability analysis of Grünwald Letnikov method for fractional-order delay differential equations

Abstract

In this paper, we investigate the unconditional stability and the generally unconditional stability of the Grünwald Letnikov method for fractional-order delay differential equations (FDDEs), which is the generalization of P-stability and GP-stability for classical integer-order delay differential equations. Using the Z-transform, an equivalent form of the discrete Laplace transform, we first show the unconditional stability of the Grünwald Letnikov method for any delay and any constraint mesh. Secondly, we also derive the generally unconditional stability of the Grünwald Letnikov method with a linear interpolation for approximating the delay term under a general uniform mesh. It is shown that the Grünwald Letnikov method for FDDEs preserves the stability for the analytical solution and hence naturally inherits the α-dependence. Finally, two numerical examples for FDDEs and time fractional-order diffusion equations with delay are presented to demonstrate the validity and effectiveness of theoretical results.