Waveform Relaxation Method For Nonlinear Fractional Differential-algebraic Equations

Fractional calculus has been widely applied to describing various engineering and physical problems, such as anomalous di?usion, medicine, soft matter, disorder media and signal processing, and so on. At the same time, the study of the fractional di?erential equations and their numerical methods have attracted considerable attention, and gradually become a research hot spot. Fractional(delay)di?erential-algebraic equations have the memory and binding properties so that it is di?cult to study theoretical analysis and numerical calculation. This article mainly aims at discussing the discrete waveform relaxation method for nonlinear fractional(delay) di?erential-algebraic equations and studying its convergence conditions. Details are as follows:The ?rst chapter introduces the research status of di?erential equations of fractional order at ?rst. Secondly, We describe the basic idea of the waveform relaxation method and classical iterative scheme of waveform relaxation method about several kinds of di?erential equations.In the second chapter, we ?rst give some basic de?nitions and properties of fractional order derivaties. Then the discrete iterative schemes of waveform relaxation method are given for nonlinear Caputo fractional di?erential algebraic equations. Among them, the Caputo derivative is discrete by Grunwald-Letnikovl(G-L) method. And then,we also analyze the convergence conditions of discrete iterative schemes of waveform relaxation method. At last, numerical examples are used to illustrate the e?ectiveness of the method.In the third chapter, we ?rst give the discrete iterative schemes of waveform relaxation method for nonlinear Caputo fractional delay di?erential-algebraic equations. Among them, the Caputo derivative is discrete by G-L method. Secondly, because of time delay phenomenon, we discuss the convergence condition of discrete waveform relaxation method by two cases. The convergence condition is strong in the second case while the applicable scope of the real problem is small.So the new division of function is used to get the weaker convergent conditions.At last, numerical examples are used to illustrate the e?ectiveness of the method.