1. Introduction
Consider a class of generalized fractional diffusion equations (GFDE)
with the initial values
,
and zero boundary conditions
,
, where the parameters
,
,
, and
are the weighting function for
with
. This equation arises from the continuous time random walks (CTRWs) model, with some complicated power-law waiting time distributions WTDs [
1,
2,
3]. The weight function
are of significant importance in the CTRW model, where biological particles have a finite lifespan. In such cases, it is more reasonable to employ the tempered power-law waiting time distribution,
, instead of the divergent power-law distribution,
. This selection allows the model to describe the gradual transitions from subdiffusion to normal diffusion and, finally, to superdiffusion. These characteristics of the model have numerous potential applications in physical, biological, and chemical processes. For further details, please refer to [
4,
5]. The desired function
represents the concentration of a particle plume undergoing anomalous diffusion with a diffusion coefficient
, and the forcing function
denotes the source or sink term. Throughout the paper, we assume that the function
is separable (or decoupled) with respect to
x on
and
t on
, that is,
The GFDE (
1) reduces to the space fractional diffusion equation (SFDE) when
. Robust numerical schemes for SFDE have been studied extensively, as outlined in [
6,
7,
8,
9,
10] and references therein. For GFDE, the time fractional derivative is the
-order generalized Caputo fractional derivative [
1,
11], defined as
while the left-handed and the right-handed space fractional derivatives are the
-order Riemann–Liouville (R-L) fractional derivatives of the form [
12]
There are various ways to solve mathematical models that involve fractional order derivatives. One such approach is to use cubic splines, which are useful in modeling anomalous diffusion. In this technique, piece-wise polynomial functions are used to interpolate the data points, allowing for the diffusion coefficient to vary with time or space [
13]. Another approach is to adapt the finite element method to include fractional order derivatives, which has been applied to determine the rheological properties of biomaterials that exhibit fractal structures. This method has been useful in studying the viscoelastic behavior of collagen and elastin [
14]. The Galerkin method is yet another technique used to obtain numerical solutions for fractional differential equations. This method approximates the solution as a linear combination of basis functions and derives a system of algebraic equations, which can then be numerically solved using the Galerkin orthogonality [
15].
To obtain an unconditionally stable difference scheme, the implicit difference scheme can be developed for Equation (
1), which inherits
-order temporal and 2-order spatial convergence [
11]. The corresponding Toeplitz linear system is then solved efficiently by using preconditioned Krylov subspace solvers with fast Fourier transformation (FFT), costing about
flops and
memory for each temporal node, where
is the number of spatial nodes. However, the derivation of the entire
temporal nodes requires about
flops, which is not suitable for large-scale computations.
In this paper, we observe that the discretized coefficient matrices in the linear system are the nonsingular Toeplitz
M-matrix, fitting well with the frame of the
M-matrix Sylvester equation. This allows us to present a doubling Smith method [
16,
17] to deal with GFDE (
1). The main contributions of this paper include the following aspects:
Some notations and definitions are required in this paper. Let be the set of all integrable functions in real space. Symbols and are the real plane and the real matrices, respectively. For matrices A, , we write if their respective elements satisfy for all i, j. A real square matrix A is called a Z-matrix if all its off-diagonal elements are nonpositive. It is clear that any Z-matrix A can be written as with . A Z-matrix with is called an M-matrix if , where denotes the spectral radius. It is called a singular M-matrix if and a nonsingular M-matrix if . The (non)symmetric Toeplitz matrix is denoted by with vectors c and r being its first column and row, respectively. The matrix is numerically low-ranked if there is a constant independent of n such that .
The following results about
M-matrix are well known (see [
19] (Section 3.5), [
20] (Lem. 2.2) for an example).
Lemma 1. For a Z-matrix A, the following statements are equivalent:
- (a)
A is a nonsingular M-matrix.
- (b)
A is nonsingular and satisfies .
- (c)
for some vector .
- (d)
All eigenvalues of A have positive real parts.
Lemma 2. Suppose that A is an M-matrix and B is a Z-matrix.
- (a)
If , then B is an M-matrix. Particularly, is an M-matrix for and a nonsingular M-matrix for .
- (b)
The one with the smallest absolute value among all eigenvalues of A, denoted by , is nonnegative, and .
4. Numerical Examples
In this section, we will illustrate the effectiveness of the low-ranked DS method in computing the solution of large-scale GFDE using the implicit difference scheme. We compare the DS method with the Bi-CGSTAB solver [
18] (Alg. 3.6.3) (referred to as “ST”), which is used to solve the sequence of Toeplitz linear systems (
9) at each temporal node. The same solver is also used to construct
and
in (
13) for the DS method. Additionally, both algorithms employ the Gohber–Semecul formula [
24,
26] to solve their respective Toeplitz systems, and the algorithm terminates when the relative residual of the Toeplitz system is less than
.
In the DS method, we use the technique of truncation and compression of the economic QR decomposition [
25] (Section 2.2) (see also in [
16,
17] with a tolerance of
) to reduce the columns of
and
as much as possible. We also set the upper bound of the truncated maximal number of columns to
. The DS method stops either when the number of iterations exceeds six or when the low-ranked residual form [
16,
17] of the Stein Equation (
13) is less than
. We implemented both algorithms using MATLAB 2019a on a 64-bit PC with a 3.0 GHz Intel Core i5 processor and 32G RAM, with the machine error
eps =
.
To assess the accuracy of both algorithms, we calculate their errors as
and record the convergent rate as
where
and
are different step-lengths in two consecutively temporal nodes.
Example 1. Consider the GFDE (1) with the diffusion coefficient and . The weight function is , and the forcing function on the RHS of GFDE isThe initial-boundary value conditions for this problem are and . It is not difficult to see the exact solution function of this problem is (see Appendix A). To test the numerical performance of the two algorithms, we take
and test the values of
at 1.1, 1.5, and 1.9. The obtained results are listed in
Table 1, where the column labeled “
h” indicates the spatial step-length (corresponding to the number of nodes
). The columns “
” and “
” represent the elapsed CPU time for the sequence solver with the Bi-CGSTAB method (abbreviated as “ST”) and our DS method, respectively. The letters “It.” behind “
” represent the required number of the DS iteration. The columns labeled “
” and “
” represent the calculated errors of the ST method and our DS method, respectively, after termination. As both algorithms reach similar error levels, we only report the convergent rate of the DS method at various scales in the “Rate_DS” column.
We can see from
Table 1 that both methods efficiently compute the solution for various values of
with errors ranging from
to
. Our DS method requires only 4–5 iterations for middle-scale problems (
) and 5–6 iterations for large-scale problems (
= 16,384 to 32,768) to achieve the prescribed residual level of the Stein equation, resulting in similar error levels as the ST method. Although the Rate_DS column shows that the convergence rate gradually decreases with increasing scale, the DS method still requires less CPU time than the ST method for all different
, especially at
= 32,768. At this scale, the DS method takes only 1/14 of the CPU time required by the ST method to obtain a solution of almost the same order
.
We also carried out further numerical experiments to validate the efficacy of our DS method, with the aim of comparing the error surfaces of the DS and ST methods, and observe any differences in their respective performances.
Figure 1 presents the results of our experiments. The figure displays the error surfaces of the DS and ST methods, labeled as “D” and “S” respectively, at specific values of
and
. The variables
t and
s in
Figure 1 represent the discretized temporal and spatial values, respectively, within the interval [0, 1] in the error functions. Our analysis of the figure revealed that the error surface of the DS method encompasses that of the ST method, but the errors for both methods are at the level of
, which is a relatively low level of error. This result reinforces the effectiveness and reliability of our proposed DS method.
We subsequently increase the value of
for different
and compare the numerical performance of the DS method with that of the ST method. The obtained results are listed in
Table 2. We can see that for different
, the DS method requires 6 iterations to reach the prescribed residual level. For middle-scale problems (
), the ST method is faster than the DS method in terms of CPU time. However, with increasing scale (from
to 32,768), the DS method gradually becomes faster than the ST method, albeit sacrificing some accuracy. In particular, at the scale of
= 32,768, the DS method takes only about 1/14 of the CPU time required by the ST method to obtain a solution of the order
, indicating that the DS method is more suitable for dealing with large-scale problems. Furthermore, we conducted numerical experiments with a value of
. The results, as shown in
Table 3, indicate that while the DS method may sacrifice some accuracy, it still outperforms the ST method in terms of CPU time when
is no less than 8192.
Example 2. Consider the GFDE (1) with the diffusion coefficient and the weight function with [11]. The source term is The corresponding initial-boundary value conditions are
,
. It can be verified that the exact solution function is
( see
Appendix A) with
We chose values of
and
and implemented both algorithms for the discretized Stein equation from Example 2. The results obtained are displayed in
Table 4, which demonstrates that both methods are capable of efficiently computing the solution with an error range of approximately between
to
for different values of
. With the exception of when
and
, our DS method required six iterations to achieve the desired residual level. In addition, for
, the DS method was less time-consuming than the ST method but achieved almost the same level of accuracy. For cases where
and
, our DS method took more CPU time to compute the solution for middle-scale cases where
was 2048 and 4098. However, as the scale increased, the DS method required less CPU time than the ST method, with only a slight decrease in accuracy.
Additionally, we generated error surfaces for both methods and denoted them as “D” for the DS method and “S” for the ST method. These surfaces were plotted at
and
, and the results are shown in
Figure 2. The figure indicates that the error surface of the DS method covers that of the ST method, while both methods have similar error levels of approximately
.
We also raised the parameter
to 0.5 and executed both algorithms once more.
Table 5 illustrates the results, indicating that our DS method produces similar error levels to the ST method. Moreover, when
, our DS method performs better than the ST method in terms of CPU time. This tendency becomes increasingly apparent as the scale increases, demonstrating that the complexity of the DS method is roughly
and that it is more suitable for larger-scale problems. In
Table 6, we resumed conducting numerical experiments with
and similarly found that while the DS method may compromise some accuracy, it still outperforms the ST method in terms of CPU time when
.