Abstract

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.

1. 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 [14–16], spectrum method, and the finite element method [17, 18] for solving the fractional differential equations.

The waveform relaxation (WR) method is proposed by Learsmee and so forth [19]. The main properties of the method are flexibility and good parallelism, which is used to solve differential equations [20, 21], fractional differential equations [22, 23], delay differential equations [24, 25], and differential-algebraic equations [26–28]. The waveform relaxation method based on multisplitting is proposed to improve parallelism [29, 30]. Furthermore, acceleration techniques are used to increase the speed of iteration, such as Krylov subspace acceleration, strange-type preconditioners, and windowing for the WR method [20, 21, 31, 32].

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.

2. Convergence

Consider the fractional delay differential-algebraic systemwhere denotes the derivative of order , , , are smooth vector functions, and , , , are nonsingular matrix, , , , , and , , , , are input function.

Applying the WR method for the coefficient matrix of (1), we get the iteration formwhere , , , , , , , , , and , are nonsingular matrix. We assume that the step is , , , , is approximating value of , and is approximating value of ,

Based on the fractional derivative of Grünwald-Letnikov discrete format where and , we obtain the iterative format wherewhere is the number of the time steps and , the initial value . Introduce Next, we will discuss two cases.

First Case. When , (4) can be written aswhere , andintroduce , where

Let

From (7), we have

Let the iterative matrix be in (10); the convergence condition of the iterative process (4) is . Moreover, we have the following theorem.

Theorem 1. For the small step and , if then the discrete waveform relaxation iteration method (4) is convergent.

Proof. The iteration process (4) convergence is decided by . Firstly, the left coefficient matrix of (7) can be written as because of the following equation: where ; then, and we can getwhere ; we haveBased on the matrix spectral radius definition of expression in (15), if and is small enough, we have .
Therefore, when , the convergence is proved.

Second Case. When , we assume ; then, system (4) can be written aswhere where introduce , where are

Let , and from (18) we have

Let the iterative matrix be in (22); then, the convergence condition of the iterative process (22) is ; we have the following theorem.

Theorem 2. For the small step , if it satisfies then the discrete waveform relaxation iteration method (18) is convergent.

Proof. The iteration process (18) convergence is decided by . Firstly, the left coefficient matrix of (18) can be written as and because where , hence where ; then, where ; we havewhere , and because where , therefore, we havefrom (8), we have Based on the matrix spectral radius definition and combining expression of (28) and if and is enough small, we have .
Therefore, the convergence is proved.

3. 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.

Example 1. Consider the initial value problem of the linear fractional delay differential-algebraic equationswhere , ; introduce ; system (33) can be written asand we have Splitting the coefficient matrix, we can construct the waveform relaxation method and deduce According to the iterative format (4), we take time step and use the waveform relaxation method to solve the initial value problem (33). Then, we take time step and take the numerical solution of (33) obtained by the implicit Grnwald-Letnikov method as the approximate true solution. Then, the error of the numerical solution obtained by WR method is shown in Table 1.