Discrete Waveform Relaxation Method for Linear Fractional Delay Differential-Algebraic Equations

Fractional order delay differential-algebraic equations have the characteristics of time lag and memory and constraint limit. These yield some difficulties in the theoretical analysis and numerical computation. In this paper, we are devoted to solving them by the waveform relaxation method. The corresponding convergence results are obtained, and some numerical examples show the efficiency of the method.


Introduction
Fractional delay differential-algebraic equations are composed of fractional delay differential 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 model is widely applied in many fields of electromagnetism, electrical systems, and biological materials.Nowadays, for the fractional differential equations, the main works focus on obtaining the analytic solution, approximately analytical solution, and numerical solution.For example, the Laplace transform method [1,2], Fourier transform method [3,4], and power method [5] are used to obtain the analytic solution.Furthermore, approximate analytical methods have been developed, such as variational iteration method [6,7], Adomian decomposition method [8], homotopy analysis method [9], and multiple time scale method [10].As for the studies of numerical methods, scholars discuss the linear multistep method [11], predictor corrector method [12,13], Adomian method [14][15][16], spectrum method, and the finite element method [17,18] for solving the fractional differential equations.
The fractional differential-algebraic equations have received much attention; nevertheless, the numerical methods in this field are still young; a few studies have been considered on the convergence of the numerical methods, such as variational iteration method, Adomian decomposition method, and fractional differential transform method [33,34].Ding and Jiang applied the WR method to solve fractional differential-algebraic equations and obtain the convergence results [35].But it is difficult to verify convergence conditions.The variational iteration method for the fractional delay integrodifferential-algebraic equations has been studied in [36], but it is not suitable for long-time numerical calculation.In order to overcome this limitation, according to the characteristics of the problems, the discrete WR method is employed to solve the linear fractional delay differential-algebraic equations by constructing the efficient iterative method and obtain the convergence results which is much easier to achieve.

Convergence
Consider the  fractional delay differential-algebraic system
Theorem 1.For the small step ℎ and  < , if then the discrete waveform relaxation iteration method ( 4) is convergent.
Proof.The iteration process (4) convergence is decided by  1 .Firstly, the left coefficient matrix of ( 7) can be written as because of the following equation: where and we can get where Based on the matrix spectral radius definition of  1 expression in (15), if and ℎ is small enough, we have ( 1 ) < 1.Therefore, when  < , the convergence is proved.
Proof.The iteration process (18) convergence is decided by  2 .Firstly, the left coefficient matrix of ( 18) can be written as and because where where where  2 =  2 + L1 ; we have where 1  2 , and because where , therefore, we have from ( 8), we have Based on the matrix spectral radius definition and combining  2 expression of ( 28) and if and ℎ is enough small, we have ( 2 ) < 1.
Therefore, the convergence is proved.

Illustrative Examples
In this section, some illustrative examples are given to show the efficiency of the WR method for solving the linear fractional delay differential-algebraic equations.
where  = 0.5, () = ( 1 (),  2 (),  3 ())  ; introduce x() = ( 1 (),  2 ())  , () =  3 (); system (33) can be written as  Splitting the coefficient matrix, we can construct the  waveform relaxation method and deduce According to the iterative format (4), we take time step ℎ = 0.1 and use the  waveform relaxation method to solve the initial value problem (33).Then, we take time step ℎ = 0.005 and take the numerical solution of (33) obtained by the implicit Gr ü nwald-Letnikov method as the approximate true solution.Then, the error of the numerical solution obtained by WR method is shown in Table 1.
The numerical solutions are plotted in Figure 1, where * , ⬦, and I, respectively, are numerical solutions of x1 (), x2 (), and  1 () by using the waveform relaxation.
Table 1 and Figure 1 show that the waveform relaxation method for obtaining the solution is accurate.
Table 2 shows that the iterative allowable error is smaller, and the number of corresponding iterations is higher.If the iteration tolerance is the same, then the condition of iterations number is relatively stable.
The numerical solutions are plotted in Figure 2, where * , ⬦, and I, respectively, are numerical solutions of x1 (), x2 (), and  1 () by using the waveform relaxation.
Table 3 and Figure 2 show that the waveform relaxation method for obtaining the solution is accurate.
Table 4 shows that the iterative allowable error is smaller, and the number of corresponding iterations is higher.If the iteration tolerance is the same, then the condition of iterations number is relatively stable.

Conclusion
In this paper, the discrete waveform relaxation method has been extended to solve linear fractional delay differential equations.Some illustrative examples show that this method gives very good approximation to the exact solution.The discrete waveform relaxation method is a promising method to handle linear fractional delay differential-algebraic equations.

Figure 2 :
Figure2: Numerical solutions by using the two methods for solving(34).

Table 1 :
The error of numerical solutions.

Table 3 :
The error of numerical solutions.

Table 4 :
Between the iteration allowable error and the number of iterations.