The Stability Analysis Of Block θ-Methods For Delay Differential Equations

As we all know that many problems in physics, engineering, biology and economicscan be modelled by systems of ordinary equations(ODEs). Actually, in many realisticmodels, it is useful to know some past states of these systems, so it is converted into delaydifferential equations(DDEs), which are also known as functional differential equationsor differential-difference equations. Delay differential equations play an important rolein the research of various applied sciences, such as control theory, population dynamics,electrical networks, environment science, biology, bioecology, and life science. And neutraldelay differential equations (NDDEs) are arised from various fields of applied sciences. Forexample, the energy loss in power networks can be modelled by a system of neutral delaydifferential equations.There are many effective numerical methods for solving delay differential equations.The blockθ-methods have good stability properties without requiring high order detivatives.They have great potential.The purpose of this paper is to study the stability behavior of numerical solution ofblockθ-method for DDEs and NDDEs. Based on Lagrange interpolation, for three differentmodels, we will show that the blockθ-method for DDEs with multiple delays is GP_m -stable if and only if it is A-stable for ODEs and it is GPL_m - stable if and only ifθ= 1.Moreover, we will give the sufficient and necessary conditions of asymptotic stability ofblockθ-methods for NDDEs with many delays and will also show that the blockθ-methodsfor NDDEs is NGP_m-stable if and only if it is A-stable for ODEs and it is NGPL_m-stableif and only ifθ= 1.