- Research
- Open Access
An exponential B-spline collocation method for the fractional sub-diffusion equation
- Xiaogang Zhu^{1}Email author,
- Yufeng Nie^{1},
- Zhanbin Yuan^{1},
- Jungang Wang^{1} and
- Zongze Yang^{1}
https://doi.org/10.1186/s13662-017-1328-6
© The Author(s) 2017
- Received: 24 April 2017
- Accepted: 25 August 2017
- Published: 13 September 2017
Abstract
In this article, we propose an exponential B-spline approach to obtain approximate solutions for the fractional sub-diffusion equation of Caputo type. The presented method is established via a uniform nodal collocation strategy by using an exponential B-spline based interpolation in conjunction with an effective finite difference scheme in time. The unique solvability is rigorously proved. The unconditional stability is well illustrated via a procedure closely resembling the classic von Neumann technique. A series of numerical examples are carried out, and by contrast to other algorithms available in the open literature, numerical results confirm the validity and superiority of our method.
Keywords
- fractional sub-diffusion equation
- exponential B-spline collocation method
- unique solvability
- unconditional stability
1 Introduction
The basic concept of anomalous diffusion dates back to Richardson’s treatise on atmospheric diffusion in 1926 [1]. It has increasingly got recognition since the late 1960s within transport theory. In contrast to a typical diffusion, such a process no longer follows Gaussian statistics, then the classic Fick’s law fails to apply. Its most striking characteristic is the temporal power-law pattern dependence of the mean squared displacement [2], i.e., \(\chi^{2}(t)\sim\kappa t^{\alpha}\), for sub-diffusion, \(\alpha<1\), while \(\alpha>1\) for super-diffusion. Anomalous transport behavior is ubiquitous in physical scenarios, and due to its universal mutuality, formidable challenges are introduced. In recent decades, fractional partial differential equations (PDEs) have entered public vision; they compare favorably with the usual models to characterize such transport motions in heterogeneous aquifer and the medium with fractal geometry [3, 4]. An explosive interest has been gained to investigate the theoretical properties, analytic techniques, and numerical algorithms for fractional PDEs [5–12].
There have been some works dedicated to developing numerical algorithms to obtain the solutions of Eqs. (1.1)-(1.3) apart from a few analytic techniques that are not always available for general situations. Zhang and Liu derived an implicit difference scheme and proved that it is unconditionally stable [17]. Yuste and Acedo studied an explicit difference scheme based on the Grünwald-Letnikov formula [18]. Along the same line, a group of weighted average difference schemes were then obtained [19]. In [20], Cui raised a high-order compact difference scheme and its convergence was detailedly discussed; another similar approach was the compact scheme stated in [21] for the fractional sub-diffusion equation with the Neumann boundary condition. In [22], an effective spectral method was constructed by using the common L1 formula in time and a Legendre spectral approximation in space. Later, this method was extended to the time-space case [23]. The finite element method was considered by Jiang and Ma [24]. The semi-discrete lump finite element method was studied by Jin et al. for a time-fractional model with a nonsmooth right-hand function [25]. Liu et al. described an implicit RBF meshless approach for the time-fractional diffusion equation [26]. Li et al. suggested an adomian decomposition algorithm for the equations of the same type [27]. In [28], the authors solved such equations by employing a fully discrete direct discontinuous Galerkin method. Gao et al. proposed a new effective difference scheme with the Caputo derivative discretized by the L1-2 formula [29]. Recently, Luo et al. established a quadratic spline collocation method for the fractional sub-diffusion equation [30], where the convergence under \(L^{\infty}\)-norm was analyzed. Sayevand et al. gave a cubic B-spline collocation method [31], whose stability was provided. A cubic trigonometric B-spline collocation approach was conducted in [32], and a wavelet Galerkin method was studied in [33]. In [34], a Sinc-Haar collocation method which uses the Haar operational matrix to convert the original problem into a set of linear algebraic equations via expanding the approximation as a truncated series based on Sinc and Haar functions was proposed.
In the present work, regarding the current interest in numerical algorithms for the fractional PDEs, we showcase a robust collocation method based on exponential B-spline trial functions to solve Eqs. (1.1)-(1.3). The resultant algebraic system is proved to be strictly diagonally dominant, and therefore the unique solvability is ensured. A von Neumann like procedure is proceeded, and the system is shown to be unconditionally stable. Its codes are tested on five numerical examples and studied in contrast to other algorithms. The proposed method is highly accurate and calls for a lower cost to implement. This may make sense to treat the equations as the model we consider here with a long time range. The outline is as follows. In Section 2, we give a concise description of exponential B-spline trial basis, which will be useful hereinafter. In Section 3, we construct a fully discrete exponential B-spline approach on uniform meshes to discretize the model and prove that it is stable. The initial vector, which we require to start our method, is addressed in Section 4. To evaluate its accuracy, numerical examples are covered in Section 5.
2 Description of exponential spline functions
The set of \(B_{j}(x)\in C^{2}(\mathbb{R})\), \(j=-1,0,\ldots,M+1\), are linearly independent and form an exponential spline space on the interval \([a,b]\). The non-negative free p is termed as ‘tension’ parameter and \(p\rightarrow0\) yields cubic spline, whereas \(p\rightarrow\infty\) corresponds to the linear spline. The cubic spline interpolation causes extraneous inflexion points, while the exponential splines can produce co-convex interpolation and allow to remedy this issue.
3 An exponential B-spline collocation method
Let \(t_{n}=n\tau\), \(n=0,1,\ldots,N\), \(T=\tau N\), \(N\in\mathbb{Z}^{+}\), and \(x_{j}=a+jh\), \(j=-1,0,\ldots,M+1\), \(h=(b-a)/M\), \(M\in\mathbb{Z}^{+}\). On this time-space lattice, we set about deriving the desired exponential B-spline collocation method for Eqs. (1.1)-(1.3).
3.1 Discretization of Caputo derivative
Lemma 3.1
- (a)
\(\omega^{\alpha}_{0}=1\), \(\omega^{\alpha}_{k}< 0\), \(\forall k\geq1\),
- (b)
\(\sum_{k=0}^{\infty}\omega^{\alpha}_{k}=0\), \(\sum_{k=0}^{n-1}\omega^{\alpha}_{k}>0\).
3.2 A fully discrete exponential B-spline based scheme
The unknown weights \(\boldsymbol{\alpha}^{n}\) depend on \(\boldsymbol {\alpha}^{n-k}\), \(k=0,1,\ldots,n\), at their previous time levels and are found via a recursive style; once \(\boldsymbol{\alpha}^{n}\) is obtained, \(\alpha ^{n}_{-1}\), \(\alpha^{n}_{M+1}\) can further be determined with the help of Eqs. (3.9)-(3.10). On the other hand, A is an \((M+1)\times(M+1)\) tri-diagonal matrix, therefore the system can be performed by the well-known Thomas algorithm, which simply needs the arithmetic operation cost \(O(M+1)\).
4 Initial state
In the same fashion, K is an \((M+1)\times(M+1)\) tri-diagonal matrix, so the solution of Eq. (4.1) can also be computed by the Thomas algorithm.
5 Stability and solvability
Lemma 5.1
System (3.11)-(4.1) is uniquely solvable since its coefficient matrices A, K are strictly diagonally dominant.
Proof
The stability analysis is proceeded as follows.
Proof
6 Numerical experiments
Example 6.1
The numerical results at \(\pmb{t=1}\) with \(\pmb{\alpha=0.9}\) , \(\pmb{p=5.16}\) , and \(\pmb{N=5\mbox{,}000}\) for Example 6.1
M | \(\boldsymbol{\|u-u_{N}\|_{L^{2}}}\) | Cov. rate | \(\boldsymbol{\|u-u_{N}\|_{L^{\infty}}}\) | Cov. rate |
---|---|---|---|---|
10 | 8.6951e-4 | - | 1.3969e-3 | - |
20 | 2.2384e-4 | 1.9578 | 3.6042e-4 | 1.9545 |
40 | 5.7089e-5 | 1.9712 | 9.2262e-5 | 1.9659 |
80 | 1.5162e-5 | 1.9127 | 2.4710e-5 | 1.9007 |
Example 6.2
The numerical results at \(\pmb{t=1}\) in time with \(\pmb{p=1.52}\) and \(\pmb{M=300}\) for Example 6.2
N | \(\boldsymbol{\|u-u_{N}\|_{L^{2}}}\) | Cov. rate | \(\boldsymbol{\|u-u_{N}\|_{L^{\infty}}}\) | Cov. rate |
---|---|---|---|---|
10 | 6.8812e-4 | - | 9.6064e-4 | - |
20 | 3.2075e-4 | 1.1012 | 4.4793e-4 | 1.1007 |
40 | 1.5101e-4 | 1.0868 | 2.1097e-4 | 1.0862 |
80 | 7.1577e-5 | 1.0771 | 1.0003e-4 | 1.0765 |
The numerical results at \(\pmb{t=1}\) in space with \(\pmb{p=1.52}\) and \(\pmb{N=5\mbox{,}000}\) for Example 6.2
M | \(\boldsymbol{\|u-u_{N}\|_{L^{2}}}\) | Cov. rate | \(\boldsymbol{\|u-u_{N}\|_{L^{\infty}}}\) | Cov. rate |
---|---|---|---|---|
10 | 2.3131e-4 | - | 3.2118e-4 | - |
20 | 5.7206e-5 | 2.0156 | 7.9627e-5 | 2.0121 |
40 | 1.3598e-5 | 2.0727 | 1.8958e-5 | 2.0704 |
80 | 2.6917e-6 | 2.3368 | 3.7520e-6 | 2.3371 |
Example 6.3
The global errors at different time with \(\pmb{p=0.01}\) and various M , N for Example 6.3
M , N | \(\boldsymbol{\|u-u_{N}\|_{L^{2}}}\) | \(\boldsymbol{\|u-u_{N}\|_{L^{\infty}}}\) | ||||
---|---|---|---|---|---|---|
t = 1 | t = 2 | t = 3 | t = 1 | t = 2 | t = 3 | |
32, 4,000 | 5.4324e-5 | 3.8735e-5 | 3.1716e-5 | 8.8587e-5 | 6.3211e-5 | 5.1768e-5 |
64, 4,000 | 1.3203e-5 | 9.6132e-6 | 7.9258e-6 | 2.1878e-5 | 1.5795e-5 | 1.2985e-5 |
128, 9,000 | 3.0826e-6 | 2.3273e-6 | 1.9418e-6 | 5.2449e-6 | 3.8791e-6 | 3.2140e-6 |
256, 9,000 | 5.3117e-7 | 4.7773e-7 | 4.2641e-7 | 9.5837e-7 | 8.4372e-7 | 7.3492e-7 |
1,024, 250 | 5.9928e-6 | 2.1116e-6 | 1.1412e-6 | 9.4652e-6 | 3.3298e-6 | 1.7970e-6 |
1,024, 500 | 3.6171e-6 | 1.2685e-6 | 6.8189e-7 | 5.6050e-6 | 1.9593e-6 | 1.0500e-6 |
2,048, 1,000 | 2.2589e-6 | 7.9847e-7 | 4.3255e-7 | 3.4336e-6 | 1.2092e-6 | 6.5273e-7 |
2,048, 2,000 | 1.4133e-6 | 4.9781e-7 | 2.6867e-7 | 2.1044e-6 | 7.3642e-7 | 3.9486e-7 |
Example 6.4
The comparison of absolute errors between CBSCM and our method when \(\pmb{p=2.53}\)
x | M = 25, N = 625 | M = 50, N = 2,500 | ||
---|---|---|---|---|
CBSCM | Our method | CBSCM | Our method | |
0.1 | 7.4297e-5 | 1.7521e-5 | 2.2881e-5 | 5.2238e-6 |
0.2 | 1.7128e-4 | 3.1447e-5 | 4.2725e-5 | 7.8796e-6 |
0.3 | 2.2488e-4 | 3.3028e-5 | 5.9053e-5 | 8.1580e-6 |
0.4 | 2.8563e-4 | 2.5425e-5 | 7.1249e-5 | 6.3822e-6 |
0.5 | 3.1076e-4 | 1.5134e-5 | 7.8544e-5 | 3.0497e-6 |
0.6 | 3.2060e-4 | 4.5617e-6 | 7.9982e-5 | 1.1163e-6 |
0.7 | 3.0518e-4 | 1.7614e-5 | 7.4401e-5 | 5.1068e-6 |
0.8 | 2.4201e-4 | 3.0270e-5 | 6.0392e-5 | 7.5532e-6 |
0.9 | 1.6825e-4 | 2.8820e-5 | 3.6264e-5 | 6.6400e-6 |
Example 6.5
The numerical results at \(\pmb{t=1}\) in time with \(\pmb{p=2}\) and \(\pmb{M=2\mbox{,}000}\) for Example 6.5
N | \(\boldsymbol{\|u-u_{N}\|_{L^{2}}}\) | Cov. rate | \(\boldsymbol{\|u-u_{N}\|_{L^{\infty}}}\) | Cov. rate | |
---|---|---|---|---|---|
q = 2 | 5 | 3.5140e-3 | - | 5.1249e-3 | - |
10 | 1.0602e-3 | 1.7288 | 1.5457e-3 | 1.7293 | |
20 | 2.9124e-4 | 1.8641 | 4.2449e-4 | 1.8644 | |
30 | 1.3357e-4 | 1.9224 | 1.9467e-4 | 1.9227 | |
q = 3 | 5 | 1.2790e-3 | - | 1.8610e-3 | - |
10 | 1.9799e-4 | 2.6915 | 2.8777e-4 | 2.6931 | |
20 | 2.7289e-5 | 2.8590 | 3.9643e-5 | 2.8598 | |
30 | 8.3666e-6 | 2.9158 | 1.2150e-5 | 2.9167 | |
q = 4 | 5 | 3.7550e-4 | - | 5.4383e-4 | - |
10 | 2.5907e-5 | 3.8574 | 3.7487e-5 | 3.8587 | |
20 | 1.7273e-6 | 3.9067 | 2.4951e-6 | 3.9092 | |
30 | 3.7751e-7 | 3.7506 | 5.4254e-7 | 3.7631 |
The numerical results at \(\pmb{t=1}\) in space with \(\pmb{p=2}\) , \(\pmb{q=3}\) , and \(\pmb{N=1\mbox{,}000}\) for Example 6.5
M | \(\boldsymbol{\|u-u_{N}\|_{L^{2}}}\) | Cov. rate | \(\boldsymbol{\|u-u_{N}\|_{L^{\infty}}}\) | Cov. rate |
---|---|---|---|---|
10 | 1.5963e-3 | - | 2.2245e-3 | - |
20 | 3.9985e-4 | 1.9972 | 5.5719e-4 | 1.9973 |
40 | 1.0001e-4 | 1.9993 | 1.3940e-4 | 1.9990 |
80 | 2.5006e-5 | 1.9998 | 3.4874e-5 | 1.9990 |
Example 6.6
7 Conclusion
In this research, an effective exponential B-spline collocation method is proposed to simulate the diffusion equation with a time-fractional derivative in the Caputo sense. The resultant algebraic system is tri-diagonal and it can rapidly be solved by the Thomas algorithm with low computing cost and storage. The unique solvability and unconditional stability of the fully discrete scheme with the temporal first-order accuracy are rigorously discussed. The codes are studied on several numerical examples, and the reported results validate that this method is capable of dealing with these equations. The comparisons with CBSCM and the implicit difference scheme manifest its practicability and advantages. In addition, the method is easy and economical to implement, so it can serve as a good alternative to model other complex fractional problems.
Declarations
Acknowledgements
The authors are thankful to the referees for the constructive comments and suggestions. This research was supported by the National Natural Science Foundation of China (Nos. 11471262 and 11501450).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Richardson, LF: Atmospheric diffusion shown on a distance-neighbour graph. Proc. R. Soc. A, Math. Phys. Eng. Sci. 110, 709-737 (1926) View ArticleGoogle Scholar
- Metzler, R, Klafter, J: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1-77 (2000) MathSciNetView ArticleMATHGoogle Scholar
- Adams, EE, Gelhar, LW: Field study of dispersion in a heterogeneous aquifer: 2. Spatial moments analysis. Water Resour. Res. 28, 3293-3307 (1992) View ArticleGoogle Scholar
- Nigmatullin, RR: The realization of the generalized transfer equation in a medium with fractal geometry. Phys. Status Solidi B 133, 425-430 (1986) View ArticleGoogle Scholar
- Barkai, E: CTRW pathways to the fractional diffusion equation. Chem. Phys. 284, 13-27 (2002) View ArticleGoogle Scholar
- Deng, WH: Numerical algorithm for the time fractional Fokker-Planck equation. J. Comput. Phys. 227(2), 1510-1522 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Gorenflo, R, Mainardi, F: Random walk models for space-fractional diffusion processes. Fract. Calc. Appl. Anal. 1, 167-191 (1998) MathSciNetMATHGoogle Scholar
- Mainardi, F: The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9(6), 23-28 (1996) MathSciNetView ArticleMATHGoogle Scholar
- Meerschaert, MM, Tadjeran, C: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65-77 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Momani, S, Odibat, Z: Numerical comparison of methods for solving linear differential equations of fractional order. Chaos Solitons Fractals 31(5), 1248-1255 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Povstenko, Y: Signaling problem for time-fractional diffusion-wave equation in a half-space in the case of angular symmetry. Nonlinear Dyn. 59(4), 593-605 (2010) View ArticleMATHGoogle Scholar
- Zhuang, P, Liu, F, Anh, V, Turner, I: New solution and analytical techniques of the implicit numerical method for the sub-diffusion equation. SIAM J. Numer. Anal. 46, 1079-1095 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Datsko, B, Gafiychuk, V, Podlubny, I: Solitary travelling auto-waves in fractional reaction-diffusion systems. Commun. Nonlinear Sci. Numer. Simul. 23(1), 378-387 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Kurzke, M: A nonlocal singular perturbation problem with periodic well potential. ESAIM Control Optim. Calc. Var. 12(1), 52-63 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999) MATHGoogle Scholar
- Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) MATHGoogle Scholar
- Zhuang, PH, Liu, FW: Implicit difference approximation for the time fractional diffusion equation. J. Comput. Appl. Math. 22(3), 87-99 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Yuste, SB, Acedo, L: An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42(5), 1862-1874 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Yuste, SB: Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 216, 264-274 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Cui, MR: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228(20), 7792-7804 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Ren, JC, Sun, ZZ, Zhao, X: Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions. J. Comput. Phys. 232(1), 456-467 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Lin, YM, Xu, CJ: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225(2), 1533-1552 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Li, XJ, Xu, CJ: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47, 2108-2131 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Jiang, YJ, Ma, JT: High-order finite element methods for time-fractional partial differential equations. J. Comput. Appl. Math. 235(11), 3285-3290 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Jin, BT, Lazarov, R, Pasciak, J, Zhou, Z: Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion. IMA J. Numer. Anal. 35, 561-582 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Liu, Q, Gu, YT, Zhuang, PH, Liu, FW, Nie, YF: An implicit RBF meshless approach for time fractional diffusion equations. Comput. Mech. 48(1), 1-12 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Li, CP, Wang, YH: Numerical algorithm based on Adomian decomposition for fractional differential equations. Comput. Math. Appl. 57, 1672-1681 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Huang, CB, Yu, XJ, Wang, C, Li, ZZ, An, N: A numerical method based on fully discrete direct discontinuous Galerkin method for the time fractional diffusion equation. Appl. Math. Comput. 264, 483-492 (2015) MathSciNetGoogle Scholar
- Gao, GH, Sun, ZZ, Zhang, HW: A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259(2), 33-50 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Luo, WH, Huang, TZ, Wu, GC, Gu, XM: Quadratic spline collocation method for the time fractional subdiffusion equation. Appl. Math. Comput. 276, 252-265 (2016) MathSciNetGoogle Scholar
- Sayevand, K, Yazdani, A, Arjang, F: Cubic B-spline collocation method and its application for anomalous fractional diffusion equations in transport dynamic systems. J. Vib. Control 22, 2173-2186 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Yaseen, M, Abbas, M, Ismail, AI, Nazir, T: A cubic trigonometric B-spline collocation approach for the fractional sub-diffusion equations. Appl. Math. Comput. 293, 311-319 (2017) MathSciNetGoogle Scholar
- Heydari, MH: Wavelets Galerkin method for the fractional subdiffusion equation. J. Comput. Nonlinear Dyn. 11(6), 061014 (2016) View ArticleGoogle Scholar
- Pirkhedri, A, Javadi, HHS: Solving the time-fractional diffusion equation via Sinc-Haar collocation method. Appl. Math. Comput. 257, 317-326 (2015) MathSciNetMATHGoogle Scholar
- McCartin, BJ: Theory, computation, and application of exponential splines. Courant Mathematics and Computing Laboratory Research and Development Report DOE/ER/03077-171, New York (1981) Google Scholar
- McCartin, BJ: Theory of exponential splines. J. Approx. Theory 66(1), 1-23 (1991) MathSciNetView ArticleMATHGoogle Scholar
- Chen, MH, Deng, WH: Fourth order difference approximations for space Riemann-Liouville derivatives based on weighted and shifted Lubich difference operators. Commun. Comput. Phys. 16(2), 516-540 (2014) MathSciNetView ArticleGoogle Scholar
- Gorenflo, R, Mainardi, F, Moretti, D, Paradisi, P: Time fractional diffusion: a discrete random walk approach. Nonlinear Dyn. 29(1-4), 129-143 (2002) MathSciNetView ArticleMATHGoogle Scholar
- Deng, WH, Chen, MH, Barkai, E: Numerical algorithms for the forward and backward fractional Feynman-Kac equations. J. Sci. Comput. 62(3), 718-746 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Brunner, H, Ling, L, Yamamoto, M: Numerical simulations of 2D fractional subdiffusion problems. J. Comput. Phys. 229(18), 6613-6622 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Murio, DA: Implicit finite difference approximation for time fractional diffusion equations. Comput. Math. Appl. 56(4), 1138-1145 (2008) MathSciNetView ArticleMATHGoogle Scholar