Waveform Relaxation Method For Several Classes Of Differential-Algebraic Equations

The fractional order calculus has extended the traditional integer order calculusover300years of history. After the difcult course of development, the fractional dif-ferential equation theory has become more complete during the past several decadesand been widely applied in many areas, such as quantum mechanics, stochastic difu-sion, control theory, finance, etc. It has shown that the mathematical models basedon the fractional diferential equations can describe some application problems moreaccurately than which based on the integer-order diferential equations. The delaydiferential-algebraic equations and the fractional-order (delay) diferential-algebraicequations have been successfully applied to biology, automatic control, electromag-netism, electric system, etc. As the delay diferential-algebraic equations and thefractional-order (delay) diferential-algebraic equations have the characteristics ofdelay, memory principle and constraint conditions. This brings intrinsic difcultiesinto the theoretical analysis and numerical computation. Recently, some scholarshave proposed several iterative algorithms to obtain approximate analytical solu-tions of many problems. Especially, waveform relaxation method has been widelyused in various fields of science and engineering because of its parallelism merit.We are devoted to solving two classes of equations by the waveform relaxationmethod in this paper: the linear delay diferential-algebraic equations and the lin-ear Caputo fractional-order (delay) diferential-algebraic equations. In chapter1,we mainly introduce the background and the current research situation of the de-lay diferential-algebraic equations and the fractional-order diferential equations.In chapter2, we introduce several classes of the waveform relaxation method forsolving diferential equations. In chapter3, firstly, we construct the discrete wave-form relaxation method with the technology of splitting for solving linear delaydiferential-algebraic equations, where the constrain grids are used and the deriva-tives are discretized by BDF methods, moreover, its convergence is proved. Thenwe still solve the linear delay diferential-algebraic equations by the discrete wave-form relaxation method without limitations on the step sizes, where the delay itemare handled by the linear interpolation. At last, the results of numerical examplessupport the results from theoretical analysis. In chapter4, we try to solve the linearCaputo fractional-order (delay) diferential-algebraic equations by the discrete wave-form relaxation method, here, the fractional-order derivative is discretized by theGr¨ nwald-Letnikov formula. At this time, we also prove its convergence. Further-more, the results of numerical examples support the results in theoretical analysis. At the end of the article, we summarize the work of this paper and put our nextwork on the prospect.