Convergence And Stability Of General Linear Methods And Boundary Value Methods For Delay Differential Equations

Differential equations are one of the most important links between the classicalmathematics and the practical applications. The theory of differential equations hasimproved a lot after centuries of development. Especially, during the last century, thequantitative of differential equations obtained an unprecedented development, whichhas greatly promoted the development of other disciplines, due to the numericalsimulation.Apart from the ideal situation, it is necessary to use delay differential equations,due to the delay factors which can not be ignored in a number of practical problems.Also, the theories of the delay differential equations have certain difficulties, so it iswidely used in numerical methods. This paper mainly studies the convergence andstability of the general linear methods and the boundary value methods for solving delaydifferential equations.First of all, we introduce the research background of delay differential equations,and the basic thought of the general linear methods and the boundary value methods.Then, we give a simple collation of the achievements in this direction at home andabroad in the last few years, and pave the way for the later research.Then, the general linear methods are applied to solve the nonlinear delay integraldifferential equations, and we get the numerical scheme. Then, under the assumptions ofthe Lipschitz conditions and the theory of inequality, we give the sufficient conclusionof the stability. Also, we give numerical examples and the evaluation of theconvergence.After that, the boundary value methods are applied to solve the nonlinear delaydifferential equations, and we get the numerical scheme. Then, under the assumptions ofthe Lipschitz conditions, we give the definition of the globally contractive and theweakly globally contractive. Then, we prove the existence and uniqueness of thenumerical solution, and give the conclusion of the convergence and the stability. Then,some numerical examples are carried out by several common boundary value methods.