Stability Of Delay Differential Equations

Delay di?erential equation is an important branch of functional di?erential equa-tions. It plays an important role in physics, biology, economics, medical science,engineering cybernetics, aviation and so on, so the numerical analysis for the delaydi?erential equation is very important. Some papers have done a series of researchesregarding the numerical solution and the stability of the numerical method.The re-searches on that have got great progress. But the majority are directed at the lowerorder derivative methods, the researches about the higher order derivative methodsare still very few. This paper is devoted to a study of the asymptotic stability ofthe improved linear multiply method for delay di?erential equations with the singledelay and many delays.In the first chapter, we brie?y introduce some examples of delay di?erentialequation in di?erent areas, and the development of the stability for the theoreticaland numerical solution of delay di?erential equation.In the second chapter, we brie?y introduce the definition of delay di?erentialequations with the single delay and many delays, we review the asymptotic stabilityof theoretical solution, and introduce the definition of the linear multiply methodand A-stability. Studying the numerical stability of the linear multiply method, andobtain the su?cient and necessary conditions.In the third chapter, we introduce the improved linear multiply method anddiscuss the numerical stability of the improved linear multiply method for delaydi?erential equations with the single delay at integer points, and obtain the su?-cient and necessary conditions. Some numerical examples are given to prove theasymptotic stability of the improved method.In the fourth chapter, through the improved linear multiply method we discussthe numerical stability of the improved linear multiply method for delay di?erentialequations with the many delays at integer points, and obtain the su?cient andnecessary conditions. Some numerical examples are given to prove the asymptoticstability of the improved method.In the fifth chapter, according to the study above, we get the summary.