Higher-order approximations
for space-fractional diffusion equation
Abstract.
Second-order and third-order finite difference approximations for fractional derivatives are derived from a recently proposed unified explicit form. The Crank-Nicholson schemes based on these approximations are applied to discretize the space-fractional diffusion equation. We theoretically analyze the convergence and stability of the Crank-Nicholson schemes, proving that they are unconditionally stable. These schemes exhibit unconditional stability and convergence for fractional derivatives of order
Key words and phrases:
Fractional derivatives, Grünwald approximation, unified explicit form, space fractional diffusion equation.2005 Mathematics Subject Classification:
26A33, 34B05, 34D20, 65L03, 65L20, 65L11.1. Introduction
Fractional derivatives, for example, the Riemann-Liouville, Caputo, and Grünwald-Letnikov derivatives, have found numerous applications across various applied fields, including physics [10, 13, 27, 28], biology[7, 29, 30], finance[25, 26], and engineering [11, 12, 15, 16, 17, 18]. Their unique characteristics such as non-locality give more suitable descriptions for various phenomena, including anomalous diffusion, population dynamics, fractional Brownian motion, etc. compared to traditional derivatives. However, the non-local nature of fractional derivatives often leads to complex formulas, making it difficult to solve fractional-order differential equations, such as fractional-order diffusion equations, using both analytical and approximate methods [6, 8, 24].
The Grünwald difference (GD) approximation presents a finite difference technique for approximating fractional derivatives. It utilizes an infinite sum of terms derived from the power series expansion of the generator
The shifted Grünwald approximation has served as a cornerstone for constructing higher-order finite difference approximations for fractional order differential equations. Meerchaert et al. [20] employed extrapolation technique on the Crank-Nicholson scheme of the Shifted Grünwald approximation for the fractional diffusion equation to obtain the second-order accuracy for the space discretization. Nasir et al. [19], from the shifted Grünwal approximations with a non-integer shift
Moreover, Lubich [5] proposed generators in the form of power or rational polynomials to construct higher-order approximations for fractional derivatives. While these generators provide coefficients for higher-order accuracy without shifts, their shifted forms only yield first-order approximations regardless of the original accuracy orders.
Nasir and Nafa [4] introduced polynomial-type generators for higher-order approximations with shifts and derived a second-order finite-difference scheme for the one-dimensional fractional diffusion equation. In construction of this work, Nasir and Nafa [3] and Gunarathna et al. [14] developed quasi-compact schemes with third-and fourth-order accuracy, respectively, both derived from the second-order approximation and applied them to the one-dimensional fractional diffusion equation.
The generators for the Nasir and Nafa [4] approximations are usually obtained manually by hand calculations, solving a resulting system of linear equations or by symbolic computations and these processes are specific to the problem at hand. To alleviate those difficulties, Gunarathna et al. [9] have obtained an explicit form for generators that gives approximations for fractional derivatives with shifts retaining their higher orders. This form generalizes the Lubich form with shift and hence the Lubich form becomes a special case with no shift. Gunarathna et al. [31] then extended the explicit form developed in [9] to a more general unified explicit form that gives more new approximations for fractional derivatives and various finite difference formulas for any classical derivative.
In this paper, we apply the unified explicit form in [9] to the space fractional diffusion equation given by (1). We consider a second-order approximation derived from this unified form. Subsequently, a new quasi-compact third-order approximation is derived from this second-order approximation. Using these approximations, second-and third -order Crank-Nicholson (C-N) schemes are constructed for the space fractional diffusion equation. Theoretical analyses of stability and convergence are established for both the C-N schemes.
(1) |
with the initial and boundary conditions:
(2) |
where
The remaining sections of this paper are structured as follows: Section 2 presents essential preliminaries and terminologies. Section 3 applies the unified form to obtain new second-and third-order approximations for fractional derivatives and obtain their Crank-Nicholson (C-N) schemes with order
2. Preliminaries and Terminologies
This section presents the requisite materials and definitions relevant to the subject of the paper.
Let
Definition 1 ([6]).
The left(-) and right(+) Riemann-Liouville (R-L) fractional derivatives of a real order
(3) |
and
(4) |
respectively, where
Definition 2 ([4]).
Let
Define a shifted difference formula with shift
(5) |
-
1)
is said to approximate the fractional derivatives if(6) -
2)
is said to approximate the fractional derivatives with order if(7)
Proposition 3 (Theorem 1 [4, 3]).
Let
(8) |
Moreover, if
(9) |
2.1. The unified explicit form
In this section, the unified form appearing in [31] is presented. This unified form extends the explicit form in [9] to a more general form that covers compact finite difference formulas for higher order classical derivatives as well as some new Lubich type generators for fractional derivatives.
For this, we introduce a base differential order
and consider approximating the fractional derivative by a Lubich type generator of the form
(10) |
where
Theorem 4.
With assumptions of Proposition 3,
the generator of the form
(11) |
where
Proof.
In view of Proposition 3, we have
where
(12) |
Since
where
(13) |
Expansion of
(14) |
Theorem 5.
3. Applications of the unified explicit form
This section applies the unified explicit form to derive a second and third order approximations.
3.1. Second-order approximation
To derive the second order approximation, the following generating function form:
(17) |
is considered.
The coefficients
3.2. Third-order approximation: Quasi-compact form
Now, we derive a new quasi-compact third-order approximation from the second-order approximation described in the Section 3.1. In view of Proposition 3, we have
where
3.3. Discretization of the space fractional diffusion equation
The discretization of the space fractional diffusion equation (1) in the domain
Furthermore, prior to the construction of the C-N schemes, it should be noted that the time derivative at
(22) | ||||
and | ||||
(23) |
3.4. Second-order Crank-Nicholson scheme
Using Equations (22) and (23)
the FDE at
(24) |
where
(25) |
for all
(26) |
for all
where
Now, let
(27) |
where
3.5. Third-order quasi-compact Crank-Nicholson scheme
The new order
(28) |
With the aid of the second-order approximations given by Equations (22) and (23),
the FDE at
(29) |
where
(30) |
Consequently, in matrix language, the C-N scheme (30) can be read as
(31) |
for
(32) |
where
4. Stability and convergence analysis
This section analyzes the stability and convergence of the C-N schemes presented in Section 3.4 and Section 3.5 for the fractional diffusion equation. The analysis also requires certain properties of definite matrices and equivalent norms, to which the reader is referred in the references [33, 34], in addition to the following useful results.
Lemma 6 ([33]).
Let
where
Definition 7 ([34]).
A function
Lemma 8 (Grenander-Szego theorem, [32]).
Let the generator
where
Lemma 9.
If
Proof.
Let
The result follows with
Lemma 10.
The generating functions of the matrices
Proof.
The matrix
The result follows with
Note that the two generating functions are mutually conjugate. Furthermore, the following results are also required.
Let
Define difference operators on the component of
Theorem 11.
Let
(33) |
provided
(34) |
Then,
where
Proof.
The negative definiteness of the operator
Equivalence of the two norms concludes the proof. ∎
Lemma 12.
Leading to the above inner products and norms, the following results :
-
(a)
The operator
is self adjoint on . -
(b)
for any . -
(c)
The operator
is selft–adjoint on , where is a given constant.
Proof.
-
(a)
Take any
. Then, we must show that We first note that and , since the vectors and have zero boundary values; thereby, we have:for all
. That is, is self adjoint on . -
(b)
From Part (a), we have:
-
(c)
Letting
and using Part (a), we get:for all
. Therefore, is a self–adjoint operator.
∎
4.1. Analysis of the second-order C–N scheme
This section gives the stability and convergence analysis of the C–N scheme presented in Section 3.4. First, Lemma 13 is presented along with its proof:
Lemma 13.
The matrices
Proof.
The generating function of matrix
where
and
The foregoing derivative assumes positive values over the interval
Now, for any non–zero vector
Remark 14.
Since the matrices
Theorem 15.
Proof.
We have from Equation (27) that the iteration matrix of the C–N scheme,
Theorem 16.
The Crank-Nicholson finite difference scheme (24) with given initial and boundary conditions converges with order
Proof.
Let
Also, let
Now,
Also, it is easy to see that, the error vector
(38) |
In comparison with Equation (33), in Equation (38),
where
So, we complete the proof. ∎
4.2. Analysis of the third–order C–N quasi-compact scheme
In this section, the analysis of the proposed third order quasi–compact approximation is presented.
Lemma 17.
The QCD operator of order
-
(a)
for . -
(b)
The operator
is self–adoint and for , where .
Proof.
(a) It is not hard to see that, the maximum of
(b) Take any
Now, using Part (b) of Lemma 12, we have:
Theorem 18.
The quasi compact Crank–Nicholson scheme (32) with the approximation from the generating function
Proof.
Consider the iteration matrix,
since when
Now,
Theorem 19.
5. Numerical results
In this section, numerical examples are given to demonstrate the unconditional stability, convergence order, and accuracy of each scheme derived in Section 3. The following test example is considered:
Example 20.
Let
Let, at a time final time
8 | 3.6245e-05 | – | 3.4278e-05 | – | 1.3912e-05 | – | |
---|---|---|---|---|---|---|---|
16 | 8.5169e-06 | 2.08 | 8.3193e-06 | 2.04 | 5.6605e-06 | 1.29 | |
32 | 2.0773e-06 | 2.03 | 2.0709e-06 | 2.00 | 1.4222e-06 | 1.99 | |
64 | 5.1597e-07 | 2.00 | 5.1907e-07 | 1.99 | 3.5503e-07 | 2.00 | |
128 | 1.2880e-07 | 2.00 | 1.3012e-07 | 1.99 | 8.8753e-08 | 2.00 | |
256 | 3.2192e-08 | 2.00 | 3.2587e-08 | 1.99 | 2.2192e-08 | 1.99 | |
512 | 8.0477e-09 | 2.00 | 8.1546e-09 | 1.99 | 5.5488e-09 | 1.99 | |
1024 | 2.0120e-09 | 1.99 | 2.0397e-09 | 1.99 | 1.3877e-09 | 1.99 | |
2048 | 5.0323e-10 | 1.99 | 5.1009e-10 | 1.99 | 3.4964e-10 | 1.98 |
M | ||||||||
---|---|---|---|---|---|---|---|---|
8 | 23 | 3.3799e-06 | – | 1.3300e-06 | – | 1.3815e-06 | – | |
16 | 65 | 1.1898e-07 | 4.82 | 2.1456e-07 | 2.63 | 3.0808e-08 | 5.48 | |
32 | 182 | 5.3530e-09 | 4.47 | 3.1650e-08 | 2.76 | 1.9968e-09 | 3.94 | |
64 | 513 | 2.8116e-10 | 4.25 | 4.2935e-09 | 2.88 | 4.6661e-10 | 2.09 | |
128 | 1449 | 3.6421e-11 | 2.94 | 5.5870e-10 | 2.94 | 7.2068e-11 | 2.69 | |
256 | 4097 | 6.5222e-12 | 2.48 | 7.1243e-11 | 2.97 | 9.8501e-12 | 2.87 | |
512 | 11586 | 9.2923e-13 | 2.81 | 8.9930e-12 | 2.98 | 1.2345e-12 | 2.99 |

Both Table 1 and Table 2 confirm convergence orders, unconditional stability, and the accuracy of the second and third schemes, respectively. Furthermore, Fig. 1 exhibits the surface plot of the exact solution of the fractional diffusion equation in Example 20 over the domain
6. Conclusion
In this paper, we present two new approximations for fractional derivatives, utilizing a recently developed unified explicit form. The first approximation achieves second-order accuracy, while the second approximation demonstrates third-order accuracy, derived from the former using a quasi-compact technique. These approximations were employed, together with the Crank-Nicholson method, to solve the space fractional diffusion equation. The unconditional stability and convergence of the resulting Crank-Nicholson schemes were established for fractional derivatives of order
References
- [1]
- [2] J. S. Jacob, J. H. Priya and A. Karthika, Applications of fractional calculus in science and engineering, J. Crit. Rev, 7 (2020) no. 13, pp. 4385–4394. https://www.jcreview.com/admin/Uploads/Files/624890460526e8.90194993.pdf
- [3] H. M. Nasir and K. Nafa, Algebraic construction of a third order difference approximation for fractional derivatives and applications, ANZIAM Journal, 59 (2018) no. EMAC2017, pp. C231–C245. https://doi.org/10.21914/anziamj.v59i0.12592
- [4] H. M. Nasir and K. Nafa, A new second order approximation for fractional derivatives with applications, SQU Journal of Science, 23 (2018) no. 1, pp. 43-55. https://doi.org/10.24200/squjs.vol23iss1pp43-55
- [5] C. Lubich, Discretized fractional calculus, SIAM J. Math. Anal., 17 (1986) no. 3, pp. 704–719. https://doi.org/10.1137/0517050
- [6] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1998.
- [7] A. Akgül and S. H. A. Khoshnaw , Application of fractional derivative on non-linear biochemical reaction models, Int. J. Intell. Netw., 1 (2020), pp. 52–58. https://doi.org/10.1016/j.ijin.2020.05.001
- [8] M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection–dispersion flow equations, J. Comput. Appl. Math., 172 (2004) no. 1, pp. 65–77. https://doi.org/10.1016/j.cam.2004.01.033
- [9] W.A. Gunarathna, H.M. Nasir and W.B. Daundasekera, An explicit form for higher order approximations of fractional derivatives, Appl. Numer. Math., 143 (2019), pp. 51–60. https://doi.org/10.1016/j.apnum.2019.03.017
- [10] F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Solitons Fractals, 7 (1996) no. 9, pp. 1461–1477. https://doi.org/10.1016/0960-0779(95)00125-5
- [11] R.L. Bagley and P.J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27 (1983) no. 3, pp. 201–210. https://doi.org/10.1122/1.549724
- [12] B.M. Vinagre, I. Podlubny, A. Hernandez and V Feliu, Some approximations of fractional order operators used in control theory and applications, Fractional calculus and applied analysis, 3 (2000) no. 3, pp. 231–248.
- [13] R. Metzler and J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A Math. Gen., 37 (2004) no. 31, pp. R161. http://dx.doi.org/10.1088/0305-4470/37/31/R01
- [14] W. A. Gunarathna, H. M. Nasir and W. B. Daundasekara, Quasi-compact fourth-order approximations for fractional derivatives and applications, Ceylon J. Sci., 51 (2022) no. 5, pp. 589-595. https://doi.org/10.4038/cjs.v51i5.8085
- [15] H.G. Sun, Y. Zhang, D. Baleanu, W. Chen and YQ Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64 (2018), pp. 213–231. https://doi.org/10.1016/j.cnsns.2018.04.019
- [16] Y. Zhang, H.G. Sun, H.H. Stowell, M. Zayernouri and S.E. Hansen, A review of applications of fractional calculus in Earth system dynamics, Chaos Solitons Fractals, 102 (2017), pp. 29–46. https://doi.org/10.1016/j.chaos.2017.03.051
- [17] A.D. Obembe, H.Y. Al-Yousef, M.E. Hossain and S.A. Abu-Khamsin, Fractional derivatives and their applications in reservoir engineering problems: A review, J. Pet. Sci. Eng., 157 (2017), pp. 312–327. https://doi.org/10.1016/j.petrol.2017.07.035
- [18] M. Khan, A. Rasheed, M.S. Anwar and S.T.H. Shah, Application of fractional derivatives in a Darcy medium natural convection flow of MHD nanofluid, Ain Shams Eng. J., 14 (2013) no. 9. p. 102093. https://doi.org/10.1016/j.asej.2022.102093
- [19] H.M. Nasir, B.L.K. Gunawardana and H.M.N.P. Abeyrathna A second order finite difference approximation for the fractional diffusion equation, Int. J. Appl. Comput. Math., 3 (2013) no. 4, pp. 237–243. https://doi.org/10.7763/ijapm.2013.v3.212
- [20] C. Tadjeran, M.M. Meerschaert and H.P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys., 213 (2006) no. 1, pp. 205–213. https://doi.org/10.1016/j.jcp.2005.08.008
- [21] W.Y. Tian, H. Zhou and W. Deng, A class of second order difference approximations for solving space fractional diffusion equations, Math. Comp., 84 (2015) no. 294, pp. 1703–1727. https://doi.org/10.1090/S0025-5718-2015-02917-2
- [22] Z. Hao, Z. Sun and W. Cao, A fourth-order approximation of fractional derivatives with its applications, J. Comput. Phys., 281 (2015), pp. 787–805. https://doi.org/10.1016/j.jcp.2014.10.053
- [23] C. Li and W. Deng, A new family of difference schemes for space fractional advection diffusion equation, Adv. Appl. Math. Mech., 9 (2017) no. 2, pp. 282–306. https://doi.org/10.4208/aamm.2015.m1069
- [24] N.J. Ford and A.C. Simpson, The numerical solution of fractional differential equations: speed versus accuracy, Numer. Algorithms, 26 (2001), pp. 333–346.https://doi.org/10.1023/A:1016601312158
- [25] N. Laskin, Fractional market dynamics, Phys. A, 287 (2000) nod. 3-4, pp. 482–492. https://doi.org/10.1016/S0378-4371(00)00387-3
- [26] E.K. Akgül, A. Akgül and M. Yavuz, New illustrative applications of integral transforms to financial models with different fractional derivatives, Chaos Solitons Fractals, 146 (2021), pp. 110877. https://doi.org/10.1016/j.chaos.2021.110877
- [27] R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000) no. 1, pp. 1–77. https://doi.org/10.1016/S0370-1573(00)00070-3
- [28] R. Hilfer, Applications of fractional calculus in physics, World scientific, 2000. https://doi.org/10.1142/9789812817747
- [29] A. Boukhouima, K. Hattaf, E.M. Lotfi, M. Mahrouf, D.F.M. Torres and N. Yousfi, Lyapunov functions for fractional-order systems in biology: Methods and applications, Chaos Solitons Fractals, 140 (2020), pp. 110224. https://doi.org/10.1016/j.chaos.2020.110224
- [30] E.F.D. Goufo, M. Mbehou and M.M.K. Pene, A peculiar application of Atangana–Baleanu fractional derivative in neuroscience: Chaotic burst dynamics, Chaos Solitons Fractals, 115 (2018), pp. 170–176. https://doi.org/10.1016/j.chaos.2018.08.003
- [31] W.A. Gunarathna, H.M. Nasir and W.B. Daundasekera, A unified explicit form for difference formulas for fractional and classical derivatives and applications, Comput. Methods Differ. Equ., 2024. https://doi.org/10.22034/cmde.2023.58229.2459
- [32] R.H. Chan, Toeplitz preconditioners for Toeplitz systems with nonnegative generating functions, IMA J. Numer. Anal., 11 (1991) no. 3 pp. 333–345. https://doi.org/10.1093/imanum/11.3.333
- [33] A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, Springer Science & Business Media, 37 (2010).
- [34] L. Zhao and W. Deng, A series of high-order quasi-compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives, Numer. Methods Partial Differential Equations, 31 (2015) no. 5, pp. 1345–1381. https://doi.org/10.1002/num.21947