Stability Of Symmetrical Boundary Value Methods For Two Classes Of Delay Differential Equations With Three Parameters

Delay differential equations are mathematical models which are more precisedescriptions for the realistic and scientific world when compared with ordinarydifferential equations. In traditional research, scientists move quite slow on theanalysis about the existence and uniqueness of analytical solutions, and hence theresults cannot meet the increasing and strict demands of engneering calculations.Due to the development of the computer science, how to obtain the numericalsolutions which are under certain restrictions is becoming a new branch ofmathematics—Computational Mathematics. Especially within recent years, themany branches of applied mathematics and many other discplines have united toform the so-called interdisciplines that can stimulate the development of each otherand as well have benefits when put into reality. Many scientists become more andmore eager for powerful methametical techniques to solve real problems. Then moreand more models come out and are waiting to be solved.Usually, Runge-Kutta methods, linear multisteps methods and spectral methodsare popular among all the numerical formulas for the theoretical study for them aresystemical and thorogh. And the boundary methods, the generalization of the initialmethods, are known to have comparatively great properties such as high accuracyand stablity and then become the new research direction. We need lots ofconclusions regarding the conditions to apply this methods widely.This paper aims at the delay-dependent stability analysis for two classes ofmodel equaions with three parameters. We use the symmetrical formula of theboundary methods to discrete them and obtain the necessary and sufficientconditions for the specific numerical methods to preserve the asymptotic stability ofthe analytical solutions. Particularly, we use the boundary locus technique toanalyze the roots of the characteristic equation and make the boundary locus of thestability regions clear and easy to observe other than tedious deductions. And whenwe decide which region is the desired stable one, we choose special reference pointsto determine directly the analytical stable region without analyze the roots of thecomplex characteristic equation which is actually a transcendental equation.