Waveform Relaxation Method For Fractional Delay Integral And Partial Differential Algebraic Equations

Fractional(partial)calculus is widely used in many scientific and engineering problems,such as simulating abnormal transport phenomena,acoustic attenuation phenomena,integrated circuits,medicine,material time-varying behavior,disordered semiconductor transmission,etc,so it has aroused the research interest of many scholars and obtained a lot of theoretical results.However,the analytical solutions of the fractional calculus equation are complex and even difficult to obtain analytical solutions,which makes the numerical method of fractional calculus equations become a research hotspot.Among them,the waveform relaxation method is characterized by high efficiency and easy parallelism,and has been widely used in solving ordinary differential equations and partial differential equations.However,due to the effects of time lag,memory and algebraic constraints,the numerical calculations and theoretical analysis of fractional partial(delayed)differential(integral)algebraic equations have been hindered.Therefore,this paper mainly studies discrete wave relaxation methods for fractional delay integral differential algebraic equations and fractional partial differential algebraic equations.In the first chapter,we introduce the research status of fractional(partial)differential equations,expound the main research progress of waveform relaxation methods at home and abroad,and introduce the main research work of this paper.In the second chapter,the Caputo fractional delay integral differential algebraic equation is split.After the time domain is segmented by constrained mesh,the discrete waveform relaxation iteration scheme of the system is constructed.The convergence of the waveform relaxation method for solving this problem is proved by the right-end function of the equation satisfying the classical Lipschitz condition,in which Caputo fractional derivative is discretized with Gr¨nwald-Letnikov format,and the integral term is approximated by complexized trapezoidal formula.Finally,the effectiveness of the discrete waveform relaxation method is illustrated by numerical experiments.In the third chapter,we first use the 2- order implicit difference scheme to discretize the Caputo fractional partial derivatives in the fractional semilinear partial differential algebraic equations with initial boundary conditions,and then use the first-order and second-order center differences to discretize first-order and second-order partial derivatives separately,and the discrete iterative scheme of the waveform relaxation method for fractional partial differential algebraic equations is obtained.Next,the vector form is used to simplify multiple discrete iterative systems.In this way,the convergence conditions of the discrete waveform relaxation method of the system are analyzed.Finally,the effectiveness of the theory is illustrated by numerical experiments.