Variational Iteration Method For Several Classes Of Delay Differential-Integro-Algebraic Equations

Delay differential (integro) algebraic equations are composed of delay differential (in-tegro) equations and algebraic equations, and they are more accurate in describing some scientific and engineering problems with memory function and algebraic restrictions. This kind of mathematical models are widely applied in many fields of biology, automatic control,electromagnetic waves, signal processing, system identification, and multi-body dynamics.Delay differential (integro) algebraic equations have time lag, memory, non-locality and con-straint limit. These yield some difficulties in numerical computation. German mathematician G.W.Leibniz first proposed the idea of fractional calculus theory. After that, some practice has proved that they have more advantages in describing some scientific and engineering problems with memory function. In recent years, fractional delay differential (integro) al-gebraic equations arise in practical problems, have received much attention. The iterative methods of obtaining the solution have been developed, such as wavelet relaxation method,homotopy perturbation method and variational iterative method. Among them, variation it-erative method because of the advantages of high efficiency and less storage, has been widely used in solving linear and nonlinear problems. Therefore, the variational iterative method is used to obtain the approximate analytic solution of delay differential-integro-algebraic equations and fractional delay differential-integro-algebraic equations. This is good choice.In this paper, we use variational iterative method to solve the several classes of de-lay differential-integro-algebraic equations. In chapter 1, we introduce the research back-ground of (delay) differential-algebraic equations and fractional (delay) differential (integro)algebraic equations. In chapter 2, we introduce the variational iteration method. In chap-ter 3, the 2-index delay differential-integro-algebraic-equations become the 1-index delay differential-integro-algebraic equations by using the way of reduce the index, then we use the variational iteration method to obtain the approximate analytic solution and the con-vergence of the method. The results of numerical examples support the results in theoret-ical analysis. In chapter 5, the variational iteration method is used to solve delay partial differential-integro-algebraic equations, and the convergence results are confirmed by the numerical examples. Finally, we summarize the full article and put our next work on the prospect. In chapter 5, we use the variational iteration method to solve fractional delay differential-integro-algebraic equations, and obtain the convergence, and some numerical ex-amples illustrate the efficiency of method.