Normalized Bernstein polynomials in solving space-time fractional diffusion equation
- A Baseri^{1},
- E Babolian^{1} and
- S Abbasbandy^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-017-1401-1
© The Author(s) 2017
Received: 26 June 2017
Accepted: 15 October 2017
Published: 25 October 2017
Abstract
In this paper, we solve a time-space fractional diffusion equation. Our methods are based on normalized Bernstein polynomials. For the space domain, we use a set of normalized Bernstein polynomials and for the time domain, which is a semi-infinite domain, we offer an algebraic map to make the rational normalized Bernstein functions. This study uses Galerkin and collocation methods. The integrals in the Galerkin method are established with Chebyshev interpolation. We implemented the proposed methods for some examples that are presented to demonstrate the theoretical results. To confirm the accuracy, error analysis is carried out.
Keywords
1 Introduction
Fractional calculus allows mathematicians and engineers better modeling of a wide class of systems with anomalous dynamic behavior and better understanding of the facets of both physical phenomena and artificial processes. Hence the mathematical models derived from differential equations with noninteger/fractional order derivatives or integrals are becoming a fundamental research issue for scientists and engineers [1].
The fractional advection-diffusion equation is presented as a useful approach for the description of transport dynamics in complex systems. The time, space and time-space fractional advection-diffusion equations have been used to describe important physical phenomena that occur in amorphous, colloid, geophysical and geological processes [2].
Bernstein polynomials play an important role in many branches of mathematics, such as probability, approximation theory and computer-aided geometric design [3]. Also, in recent decades, the authors discovered some new analytic properties and some applications for these polynomials. For example, the rate of convergence of these polynomials derived by Cheng [4] for a certain class of functions. Farouki [5] showed that the Bernstein polynomial basis is an optimal stable basis among non-negative bases on the desired interval. Alshbool [6] approximated solutions of fractional-order differential equations with estimation error by using fractional Bernstein polynomials. Also, he applied a new modification of the Bernstein polynomial method called multistage Bernstein polynomials to solve fractional order stiff systems [7].
The structure of this paper is as follows. In Section 2, some definitions are presented. In Section 3, normalized Bernstein polynomials and their required properties are given. In Section 4, we introduce the rational Bernstein functions and also describe some useful properties of these basis functions. In Section 5, the relation between the Legendre polynomials and orthonormal Bernstein is demonstrated. In Section 6, to estimate the integrals, we introduce a rational Chebyshev interpolation. In Section 7, Galerkin and collocation methods to approximate the unknown function \(u (x, t) \) are applied. In Section 8, we offer error bounds for integer and fractional derivatives. To show the effectiveness of the proposed method, we report our numerical findings in Section 9; and finally, in Section 10, we give a brief conclusion.
2 Fundamentals of fractional calculus
There are various definitions for fractional derivative and integration. The widely-used definition for a fractional derivative is the Caputo sense and a fractional integration is the Riemann-Liouville definition.
Definition 2.1
([9])
Definition 2.2
([9])
For simplicity, we denote \(\frac{\partial^{\alpha_{i}}}{\partial z^{\alpha_{i}}} \) by \(D^{\alpha_{i}}_{z}\) which \(\alpha_{i}\) denotes the Caputo derivative with respect to z, note that \(\alpha_{i}\) may be β, γ or α and z can be x or t.
According to Definition 2.2, the Caputo time and space fractional derivatives of the function u are given as follows.
Definition 2.3
([11])
3 Bernstein polynomials
4 Rational normalized Bernstein functions
5 Relation between Legendre and normalized Bernstein polynomials
6 Rational Chebyshev interpolation approximation
7 Function approximation by normalized Bernstein polynomials and rational normalized Bernstein functions
7.1 Collocation method
7.2 Galerkin method
8 Error estimates
We begin this section with the basic error bound for an integer derivative, and then we refer to fractional derivatives, which are important for the main result (see [22] for more details). So, for fractional time-space fractional diffusion equations like Eq. (1), an approach to the convergence analysis of the Bernstein method is presented.
Definition 8.1
Theorem 8.1
Proof
It is shown that a valid projection property for any basis of the space, and in fact any element, means that they are basis-free. So, we do not present the proof for other theorems.
Through examining the temporal domain in the interval \(\Lambda= (0, +\infty)\), we also have definitions and theorems where they are fundamental results with the mapped orthonormal Bernstein approximations as follows.
Definition 8.2
Theorem 8.2
([15])
Definition 8.3
Theorem 8.3
Proof
We have also error bounds for the fractional derivatives as follows.
Theorem 8.4
Proof
So when \(N,M \rightarrow\infty\), \((\|D^{\alpha}_{x}(e_{N,M})\|=\| D^{\alpha}_{x} u - D^{\alpha}_{x}(\Pi_{N,M}u) \|) \rightarrow0 \).
Theorem 8.5
Proof
The proof is similar to Theorem 8.4. □
This theorem shows that, when \(N,M \rightarrow\infty\), \((\|D^{\beta }_{t}(e_{N,M})\|=\|D^{\beta}_{t} u - D^{\beta}_{t}(\Pi_{N,M}u) \| )\rightarrow0 \).
For the proposed method, the error assessment relying on the residual error function is presented [24].
9 Numerical examples
Example 9.1
([25])
The maximum absolute error with \(\pmb{\alpha=1.4}\) for Example 9.1 , the collocation method
γ | T = 2 | T = 10 | T = 100 | CPU time (s) | \(\boldsymbol{R_{\infty}}\) |
---|---|---|---|---|---|
0.2 | 1.15249 × 10^{−4} | 1.74967 × 10^{−5} | 1.71026 × 10^{−7} | 17.209 | 7.48674 × 10^{−6} |
0.4 | 1.18497 × 10^{−4} | 2.16571 × 10^{−5} | 3.2801 × 10^{−7} | 15.085 | 9.01317 × 10^{−6} |
0.6 | 1.171774 × 10^{−4} | 1.47567 × 10^{−5} | 2.97227 × 10^{−7} | 17.471 | 7.26192 × 10^{−6} |
0.8 | 2.22216 × 10^{−4} | 1.93125 × 10^{−5} | 1.94697 × 10^{−7} | 16.473 | 8.93887 × 10^{−6} |
The maximum absolute error with \(\pmb{\gamma=0.4}\) for Example 9.1 , the Galerkin method
α | T = 2 | T = 10 | T = 100 | CPU time (s) | \(\boldsymbol{R_{\infty}}\) |
---|---|---|---|---|---|
1.2 | 1.18738 × 10^{−4} | 2.17028 × 10^{−5} | 2.28411 × 10^{−7} | 17.082 | 5.4121 × 10^{−5} |
1.4 | 1.18497 × 10^{−4} | 2.16751 × 10^{−5} | 2.2801 × 10^{−7} | 16.331 | 5.41498 × 10^{−5} |
1.6 | 1.18139 × 10^{−4} | 2.15888 × 10^{−5} | 2.27469 × 10^{−7} | 16.472 | 5.42155 × 10^{−5} |
1.8 | 1.17613 × 10^{−4} | 2.14884 × 10^{−5} | 2.26757 × 10^{−7} | 16.256 | 5.43212 × 10^{−5} |
Example 9.2
The \(\pmb{L_{\infty}}\) error for Example 9.2 with \(\pmb{\alpha=1.8}\) and \(\pmb{\beta=1}\) , the collocation method
n | T = 2 | T = 10 | T = 100 | CPU time (s) | \(\boldsymbol{R_{\infty}}\) |
---|---|---|---|---|---|
5 | 5.0 × 10^{−4} | 2.5 × 10^{−5} | 2.0 × 10^{−7} | 4.37 | 7.82322 × 10^{−5} |
8 | 1.5 × 10^{−4} | 1.5 × 10^{−4} | 1.2 × 10^{−7} | 12.917 | 3.52693 × 10^{−8} |
10 | 6.0 × 10^{−6} | 6.0 × 10^{−6} | 1.2 × 10^{−7} | 21.669 | 1.60277 × 10^{−10} |
16 | 4.0 × 10^{−6} | 3.0 × 10^{−6} | 3.5 × 10^{−10} | 74.458 | 2.32527 × 10^{−17} |
The \(\pmb{L_{\infty}}\) error for Example 9.2 with \(\pmb{\alpha=1.8}\) and \(\pmb{\beta=1}\) , the Galerkin method
n | T = 2 | T = 10 | T = 100 | CPU time (s) | \(\boldsymbol{R_{\infty}}\) |
---|---|---|---|---|---|
5 | 3.0 × 10^{−3} | 1.0 × 10^{−3} | 5.0 × 10^{−4} | 4.868 | 2.26905 × 10^{−3} |
8 | 8.0 × 10^{−4} | 2.0 × 10^{−4} | 5.0 × 10^{−5} | 21.416 | 4.78888 × 10^{−6} |
10 | 1.5 × 10^{−4} | 1.5 × 10^{−5} | 1.2 × 10^{−5} | 53.039 | 3.13802 × 10^{−8} |
16 | 1.5 × 10^{−6} | 1.5 × 10^{−6} | 8.0 × 10^{−7} | 636.265 | 3.69793 × 10^{−15} |
Maximum residual errors for \(\pmb{n=8}\) for Example 9.2 , the collocation method
β | α | T = 2 | T = 10 | T = 50 | T = 100 | \(\boldsymbol{R_{\infty}}\) |
---|---|---|---|---|---|---|
0.4 | 2 | 8.0 × 10^{−3} | 1.4 × 10^{−2} | 7.0 × 10^{−3} | 5.0 × 10^{−3} | 2.64548 × 10^{−8} |
0.6 | 3.0 × 10^{−3} | 6.0 × 10^{−3} | 2.0 × 10^{−3} | 1.5 × 10^{−3} | 2.57232 × 10^{−8} | |
0.8 | 5.0 × 10^{−3} | 2.0 × 10^{−3} | 5.0 × 10^{−4} | 3.0 × 10^{−4} | 2.85507 × 10^{−8} | |
1 | 1.4 | 6.0 × 10^{−2} | 1.5 × 10^{−2} | 2.5 × 10^{−3} | 5.0 × 10^{−5} | 9.5567 × 10^{−8} |
1.6 | 2.0 × 10^{−2} | 2.5 × 10^{−3} | 4.0 × 10^{−4} | 5.0 × 10^{−6} | 4.49503 × 10^{−8} | |
1.8 | 1.5 × 10^{−4} | 1.5 × 10^{−4} | 2.0 × 10^{−5} | 1.2 × 10^{−7} | 3.52396 × 10^{−8} |
The absolute error at \(\pmb{t=1}\) with \(\pmb{\alpha=1.8}\) , \(\pmb{\beta =1}\) and \(\pmb{n=5}\) for Example 9.2
x | Proposed method (collocation) | Proposed method (Galerkin) | Method in [ 21 ] |
---|---|---|---|
0.1 | 2.40083 × 10^{−8} | 5.84791 × 10^{−7} | 3.050 × 10^{−6} |
0.2 | 2.97841 × 10^{−8} | 8.63544 × 10^{−7} | 4.260 × 10^{−6} |
0.3 | 2.54578 × 10^{−8} | 9.29233 × 10^{−7} | 5.995 × 10^{−6} |
0.4 | 1.72148 × 10^{−8} | 8.58172 × 10^{−7} | 6.942 × 10^{−6} |
0.5 | 9.34788 × 10^{−9} | 7.11067 × 10^{−7} | 4.575 × 10^{−6} |
0.6 | 4.30953 × 10^{−9} | 5.34071 × 10^{−7} | 2.370 × 10^{−6} |
0.7 | 2.76392 × 10^{−9} | 3.59841 × 10^{−7} | 1.142 × 10^{−5} |
0.8 | 3.63934 × 10^{−9} | 2.08591 × 10^{−7} | 1.393 × 10^{−5} |
0.9 | 4.18034 × 10^{−9} | 8.91471 × 10^{−8} | 7.452 × 10^{−6} |
Example 9.3
([26])
The \(\pmb{L_{\infty}}\) error for Example 9.3 with \(\pmb{\alpha =1.8}\) and \(\pmb{\beta=1}\) , the collocation method
n | T = 2 | T = 10 | T = 100 | CPU time (s) | \(\boldsymbol{R_{\infty}}\) |
---|---|---|---|---|---|
5 | 4.0 × 10^{−3} | 7.0 × 10^{−4} | 1.2 × 10^{−5} | 4.275 | 2.3301 × 10^{−4} |
8 | 8.0 × 10^{−4} | 1.0 × 10^{−3} | 5.0 × 10^{−6} | 13.009 | 8.91987 × 10^{−7} |
10 | 6.0 × 10^{−5} | 6.0 × 10^{−5} | 4.0 × 10^{−7} | 22.198 | 7.66056 × 10^{−9} |
16 | 2.0 × 10^{−5} | 2.0 × 10^{−5} | 2.0 × 10^{−7} | 76.581 | 9.82014 × 10^{−16} |
The \(\pmb{L_{\infty}}\) error for Example 9.3 with \(\pmb{\alpha =1.8}\) and \(\pmb{\beta=1}\) , the Galerkin method
n | T = 2 | T = 10 | T = 100 | CPU time (s) | \(\boldsymbol{R_{\infty}}\) |
---|---|---|---|---|---|
5 | 1.5 × 10^{−2} | 7.0 × 10^{−3} | 2.5 × 10^{−3} | 5.053 | 1.267 × 10^{−3} |
8 | 8.0 × 10^{−3} | 1.5 × 10^{−3} | 7.0 × 10^{−4} | 22.385 | 2.91652 × 10^{−6} |
10 | 2.5 × 10^{−3} | 4.0 × 10^{−4} | 3.0 × 10^{−4} | 53.508 | 1.52719 × 10^{−8} |
16 | 1.4 × 10^{−5} | 2.0 × 10^{−5} | 6.0 × 10^{−6} | 646.187 | 5.18714 × 10^{−16} |
Maximum residual errors with \(\pmb{n=8}\) for Example 9.3 , the Galerkin method
β | α | T = 2 | T = 10 | T = 50 | T = 100 | \(\boldsymbol{R_{\infty}}\) |
---|---|---|---|---|---|---|
0.4 | 1.8 | 1.0 × 10^{−1} | 1.0 × 10^{−1} | 6.0 × 10^{−2} | 5.0 × 10^{−2} | 1.12744 × 10^{−7} |
0.6 | 8.0 × 10^{−2} | 6.0 × 10^{−2} | 2.5 × 10^{−2} | 2.5 × 10^{−2} | 3.93491 × 10^{−7} | |
0.8 | 8.0 × 10^{−2} | 3.5 × 10^{−2} | 7.0 × 10^{−3} | 1.0 × 10^{−2} | 1.60004 × 10^{−6} | |
1 | 1.3 | 4.0 × 10^{−2} | 8.0 × 10^{−3} | 2.0 × 10^{−3} | 1.4 × 10^{−3} | 1.36568 × 10^{−5} |
1.5 | 6.0 × 10^{−2} | 8.0 × 10^{−3} | 4.0 × 10^{−3} | 8.0 × 10^{−3} | 2.85066 × 10^{−5} | |
1.8 | 8.0 × 10^{−3} | 1.5 × 10^{−3} | 4.0 × 10^{−4} | 7.0 × 10^{−4} | 2.91652 × 10^{−6} |
The absolute error at \(\pmb{t=1}\) with \(\pmb{\alpha=1.8}\) , \(\pmb{\beta =1}\) and \(\pmb{n=5}\) for Example 9.3
x | Proposed method (collocation) | Proposed method (Galerkin) | Method in [ 26 ] |
---|---|---|---|
0.1 | 2.20576 × 10^{−9} | 2.35702 × 10^{−8} | 1.930 × 10^{−6} |
0.2 | 3.09058 × 10^{−8} | 6.54584 × 10^{−7} | 1.434 × 10^{−7} |
0.3 | 8.8553 × 10^{−8} | 5.16048 × 10^{−7} | 2.644 × 10^{−6} |
0.4 | 1.60935 × 10^{−7} | 3.24496 × 10^{−6} | 3.862 × 10^{−6} |
0.5 | 2.25077 × 10^{−7} | 5.80895 × 10^{−6} | 3.466 × 10^{−6} |
0.6 | 2.57137 × 10^{−7} | 6.17752 × 10^{−6} | 2.012 × 10^{−6} |
0.7 | 2.40314 × 10^{−7} | 3.18557 × 10^{−6} | 4.571 × 10^{−7} |
0.8 | 1.72743 × 10^{−7} | 2.29409 × 10^{−6} | 3.391 × 10^{−7} |
0.9 | 7.54001 × 10^{−8} | 6.17811 × 10^{−6} | 9.767 × 10^{−8} |
Example 9.4
The \(\pmb{L_{\infty}}\) error for Example 9.4 , the collocation method
n | T = 2 | T = 10 | T = 100 | CPU time (s) | \(\boldsymbol{R_{\infty}}\) |
---|---|---|---|---|---|
5 | 4.0 × 10^{−3} | 3.0 × 10^{−3} | 2.0 × 10^{−7} | 15.257 | 7.35512 × 10^{−3} |
8 | 7.0 × 10^{−4} | 9.0 × 10^{−3} | 7.0 × 10^{−8} | 41.432 | 1.26371 × 10^{−6} |
10 | 1.0 × 10^{−3} | 3.4 × 10^{−4} | 4.0 × 10^{−9} | 58.343 | 2.17796 × 10^{−9} |
16 | 1.0 × 10^{−4} | 9.5 × 10^{−4} | 8.0 × 10^{−10} | 217.746 | 1.30039 × 10^{−16} |
The absolute error at \(\pmb{t=1}\) and \(\pmb{n=8}\) for Example 9.4
x | Galerkin method | Collocation method |
---|---|---|
0.1 | 2.42738 × 10^{−11} | 1.60766 × 10^{−12} |
0.3 | 1.20381 × 10^{−11} | 1.79642 × 10^{−11} |
0.5 | 1.69176 × 10^{−11} | 3.06019 × 10^{−11} |
0.7 | 6.62043 × 10^{−12} | 3.04411 × 10^{−11} |
0.9 | 3.01258 × 10^{−11} | 1.63679 × 10^{−11} |
10 Conclusion
In this article, we presented effective numerical methods for solving a space-time fractional diffusion equation with initial boundary conditions. For these problems defined in the unbounded time domain, we use the rational normalized Bernstein functions as basis functions to approximate the exact solution. We compared the execution of the collocation and Galerkin methods using normalized Bernstein basis for solving a given problem. We have presented some numerical experiments to confirm the theoretical analysis. Precision increases with the increase in the number of terms in the normalized Bernstein expansion. However, for the same number of terms, the collocation method yields relatively more accurate results in a comparatively shorter time compared with the Galerkin method. On the other hand, the collocation method is very sensitive to the collocation points. Generally, the most significant property of the collocation method is its fluency in the application; e.g., matrix elements of the given equation are evaluated directly rather than by numerical integration as in the Galerkin procedure. Generally, the results show that the proposed methods achieve better approximation accuracy than other methods, especially for the long time domain.
Declarations
Acknowledgements
The authors would like to thank the referee for his valuable comments and suggestions which improved the paper into its present form.
Authors’ contributions
Effective numerical methods for solving space-time fractional diffusion equation with initial boundary conditions are proposed. The rational normalized Bernstein functions as basis functions to approximate the exact solution are used in the unbounded time domain. Some numerical experiments to confirm the theoretical analysis are provided. In our examples we found that the collocation method yields relatively more accurate results in a comparatively shorter time compared with the Galerkin method; on the other hand, the collocation method is very sensitive to the collocation points. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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
- Elwakil, AS: Fractional-order circuits and systems: an emerging interdisciplinary research area. IEEE Circuits Syst. Mag. 10(4), 40-50 (2010) View ArticleGoogle Scholar
- Xing, Y, Wu, X, Xu, Z: Multiclass least squares auto-correlation wavelet support vector machines. ICIC Express Lett. 2(4), 345-350 (2008) Google Scholar
- Farin, GE, Hoschek, J, Kim, MS: Handbook of Computer Aided Geometric Design. Elsevier, Amsterdam (2002) MATHGoogle Scholar
- Farouki, RT, Rajan, VT: Algorithms for polynomials in Bernstein form. Comput. Aided Geom. Des. 5, 1-26 (1988) View ArticleMATHMathSciNetGoogle Scholar
- Farouki, RT, Goodman, TNT: On the optimal stability of the Bernstein basis. Math. Comput. 64, 1553-1566 (1996) View ArticleMATHMathSciNetGoogle Scholar
- Alshbool, MHT, Bataineh, AS, Hashim, I, Isik, OR: Solution of fractional-order differential equations based on the operational matrices of new fractional Bernstein functions. J. King Saud Univ., Sci. 29(1), 1-18 (2017) View ArticleGoogle Scholar
- Alshbool, MHT, Hashim, I: Multistage Bernstein polynomials for the solutions of the fractional order stiff systems. J. King Saud Univ., Sci. 28(4), 280-285 (2016) View ArticleGoogle Scholar
- Kemppainen, J: Existence and uniqueness of the solution for time-fractional diffusion equation with Robin boundary condition. Abstr. Appl. Anal. 2011, Article ID 321903 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Podlubny, I: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Academic Press, San Diego (1998) MATHGoogle Scholar
- Maleknejad, K, Basirat, B, Hashemizadeh, E: A Bernstein operational matrix approach for solving a system of high order linear Volterra-Fredholm integro-differential equations. Math. Comput. Model. 55, 1363-1372 (2012) View ArticleMATHMathSciNetGoogle Scholar
- Akinlar, MA, Secer, A, Bayram, M: Numerical solution of fractional Benney equation. Appl. Math. Inf. Sci. 8, 1-5 (2014) MathSciNetGoogle Scholar
- Tadjeran, C, Meerschaert, MM, Scheffer, HP: A second-order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213(1), 205-213 (2006) View ArticleMATHMathSciNetGoogle Scholar
- Alipour, M, Rostamy, D, Baleanu, D: Solving multi-dimensional FOCPs with inequality constraint by BPs operational matrices. J. Vib. Control 19(16), 2523-2540 (2013) View ArticleMATHMathSciNetGoogle Scholar
- Bellucci, MA: On the explicit representation of orthonormal Bernstein polynomials, Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Google Scholar
- Shen, J, Tang, T, Wang, L: Spectral Methods: Algorithms, Analysis and Applications. Springer, Berlin (2010) Google Scholar
- Rostamy, D, Karimi, K: Bernstein polynomials for solving fractional heat-and wave-like equations. Fract. Calc. Appl. Anal. 15(4), 556-571 (2012) View ArticleMATHMathSciNetGoogle Scholar
- Canuto, C, Hussaini, MY, Quarteroni, A, Zang, TA: Spectral Methods in Fluid Dynamic. Prentice Hall, Englewood Cliffs (1987) MATHGoogle Scholar
- Gottlieb, D, Hussaini, MY, Orszag, S: In: Voigt, R, Gottlieb, D, Hussaini, MY (eds.) Theory and Applications of Spectral Methods in Spectral Methods for Partial Differential Equations. SIAM, Philadelphia (1984) Google Scholar
- Guo, BY, Shen, J, Wang, ZQ: Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval. Int. J. Numer. Methods Eng. 53, 65-84 (2000) View ArticleMATHMathSciNetGoogle Scholar
- Ren, R, Li, H, Jiang, W, Song, M: An efficient Chebyshev-tau method for solving the space fractional diffusion equations. Appl. Math. Comput. 224, 259-267 (2013) MATHMathSciNetGoogle Scholar
- Alavizadeh, SR, Maalek Ghaini, FM: Numerical solution of fractional diffusion equation over a long time domain. Appl. Math. Comput. 263, 240-250 (2015) MathSciNetGoogle Scholar
- Bhrawy, AH, Zaky, MA, Van Gorder, RA: A Space-Time Legendre Spectral Tau Method for the Two-Sided Space-Time Caputo Fractional Diffusion-Wave Equation. Springer, New York (2015) MATHGoogle Scholar
- Bass, RF: Real Analysis for Graduate Students, 2nd edn. (2011) Google Scholar
- Bhrawy, AH, Zaky, MA: A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations. J. Comput. Phys. 281, 876-895 (2015) View ArticleMATHMathSciNetGoogle Scholar
- Javadi, S, Babolian, E, Jani, M: A numerical scheme for space-time fractional advection-dispersion equation (2015). arXiv:1512.06629v1 [math.NA]
- Doha, EH, Bhrawy, AH, Ezz-Eldien, SS: Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method. Cent. Eur. J. Phys. (2013). doi:10.2478/s11534-013-0264-7 MATHGoogle Scholar