A fully implicit finite difference scheme based on extended cubic Bsplines for time fractional advection–diffusion equation
 Syed Tauseef MohyudDin^{1, 2}Email author,
 Tayyaba Akram^{3},
 Muhammad Abbas^{4},
 Ahmad Izani Ismail^{3} and
 Norhashidah H. M. Ali^{3}
https://doi.org/10.1186/s1366201815377
© The Author(s) 2018
Received: 22 November 2017
Accepted: 23 February 2018
Published: 27 March 2018
Abstract
In this paper, we investigate a fully implicit finite difference scheme for solving the time fractional advection–diffusion equation. The time fractional derivative is estimated using Caputo’s formulation, and the spatial derivatives are discretized using extended cubic Bspline functions. The convergence and stability of the fully implicit scheme are analyzed. Numerical experiments conducted indicate that the scheme is feasible and accurate.
Keywords
1 Introduction
Over the past few decades, several physical models have been developed in the form of fractional differential equations. Fractional differential equations have been found to be appropriate models for certain phenomena in astrophysics, fractal networks, signal processing, chaotic dynamics, turbulent flow, continuum mechanics, and wave propagation [1–7]. These models admit nonlocal memory effects in the mathematical formulation and thus overcome certain shortcomings in integerbased models.
An important fractional partial differential equation is the fractional advection–diffusion equation. It is important to solve this equation for a better understanding of advection and diffusion phenomena in a fractional setting, and for this purpose, numerical and approximate analytical methods are usually required. The finite element method was constructed for the space fractional advection–diffusion equation by Zheng et al. [8]. Wang and Wang [9] developed a fast characteristic finite difference scheme for space fractional advection–diffusion equation. For the space–time fractional advection–diffusions, explicit and implicit difference approximations were developed by Shen et al. [10]. Jiang et al. [11] presented analytical solutions for the multiterm time–space Caputo–Riesz fractional advection–diffusion equations on a finite domain with Dirichlet nonhomogeneous boundary conditions. In [11], the spectral representation of the fractional Laplacian operator was used to derive the analytical solution. A scheme based on the finite volume method for the solution of space fractional diffusion equation was investigated by Liu et al. [12]. A finite element multigrid method was developed for multiterm time fractional advection–diffusion equations by Bu et al. [13]. Parvizi et al. [14] presented a Jacobi collocation method for numerical solution of classical fractional advection–diffusion equation with a nonlinear source term. Rubab et al. [15] discussed analytical solutions to the time fractional advection–diffusion equation with timedependent pulses on the boundary. In [15], the Laplace and Fourier transforms were utilized to determine the analytical solutions of fractional advection–diffusion equation with time fractional Caputo–Fabrizio derivative. Povstenko and Kyrylych [16] discussed two approaches to obtaining the space–time fractional advection–diffusion equations. In this paper, Caputo time fractional derivative and Riesz fractional Laplacian were used.
Many researchers used a spline function for solving fractional differential equations. Bspline functions can give good approximation due to their small, compact support and continuity of order 2 [17, 18]. However, there is relatively not much work on the use of Bsplines for solving fractional advection–diffusion equation. Bspline collocation methods were proposed for the solutions of time fractional diffusion problems by Esen et al. [19, 20]. Sayevand et al. [21] solved anomalous time fractional diffusion problems in transport dynamic systems using a Bspline collocation scheme. In [21], the fractional derivative in Caputo sense was utilized to represent the time derivative. A cubic trigonometric Bspline collocation scheme for the time fractional diffusion problem was presented by Yaseen et al. [22]. In this paper, the Grunwald–Letnikov representation was used for Riemann–Liouville derivative, and the stability of the scheme (based on the finite difference method and cubic trigonometric Bspline) was discussed. Zhu and Nie [23] obtained a scheme based on exponential Bspline and wavelet operational matrix method for the time fractional convection–diffusion problem with variable coefficients. Yaseen et al. [24] constructed a finite difference method for solving time fractional diffusion problem via trigonometric Bspline. Zhu et al. [25] derived an efficient differential quadrature scheme based on modified trigonometric cubic Bspline for the solution of 1D and 2D time fractional advection–diffusion equations. Yuan and Chen [26] presented an expanded mixed finite element method for the twosided timedependent fractional diffusion problem with twosided Riemann–Liouville fractional derivatives.
In this paper, a fully implicit finite difference scheme using extended cubic Bspline is formulated for the numerical solution of time fractional advection–diffusion equation. A finite difference scheme, with Caputo’s formula, is applied to discretize the temporal derivative, while extended cubic Bspline is employed to discretize the spatial derivatives.
2 Extended cubic Bspline functions
3 Description of the scheme based on extended cubic Bspline
Lemma 3.1

\(b_{0}=1\);

\(b_{0}>b_{1}>b_{2}>\cdots>b_{s}\), \(b_{s}\rightarrow 0\) as \(s\rightarrow \infty \);

\(b_{s}>0\) for \(s=0,1,\ldots,n\);

\(\sum^{n}_{s=0}(b_{s}b_{s+1})+b_{n+1}=(1b_{1})+\sum^{n1}_{s=1}(b_{s}b_{s+1})+b_{n}=1\).
3.1 Fully implicit scheme
4 Initial state \(C^{0}\)

\((u^{0}_{i})_{x}=\frac{d}{dx}(\omega (x_{i}))\), \(i=0,N\);

\(u^{0}_{i}=u(x_{i},0)=\sum^{N+1}_{i=1}C^{0}_{i}(0)\phi (x_{i})\), \(i=0,1,\ldots,N\).
5 Stability
5.1 Stability for a fully implicit scheme
Proposition 5.1
Proof
Theorem 1
The implicit scheme (12) is unconditionally stable.
Proof
6 Convergence
In this section, we follow Kadalbajoo and Arora’s [29] technique to examine the convergence of the proposed method.
Theorem 2
Lemma 6.1
Proof
Theorem 3
Proof
7 Illustrative examples and discussions
7.1 Problem 1
A comparison of maximum error \((\Vert \cdot \Vert _{\infty })\) and Euclidean norm \((\Vert \cdot \Vert _{2})\) at \(T=1\) for problem1
N  τ = 1.0 × 10^{−2},γ = 0.2  

MCTBDQM [25]  Proposed method  
\(\Vert \cdot \Vert _{\infty }\)  \(\Vert \cdot \Vert _{2}\)  \(\Vert \cdot \Vert _{\infty }\)  \(\Vert \cdot \Vert _{2}\)  Order  CPU time  
08  1.4902e−02  1.0412e−02  7.0982e−04  5.2421e−04  …  0.09360 
16  3.8827e−03  2.6898e−03  6.9478e−05  5.0417e−05  3.35283  0.14040 
32  1.0156e−03  6.6522e−04  3.4560e−05  2.5203e−05  1.00747  0.26520 
64  2.5720e−04  1.4842e−04  1.7410e−06  1.2739e−06  4.31108  0.73321 
128  6.3504e−05  2.2129e−05  3.8083e−07  1.9860e−07  2.19272  2.07481 
A comparison of maximum error \((\Vert \cdot \Vert _{\infty })\) and Euclidean norm \((\Vert \cdot \Vert _{2})\) at \(T=1\) for problem1
N  τ = 1.0 × 10^{−2},γ = 0.5  

MCTBDQM [25]  Proposed method  
\(\Vert \cdot \Vert _{\infty }\)  \(\Vert \cdot \Vert _{2}\)  \(\Vert \cdot \Vert _{\infty }\)  \(\Vert \cdot \Vert _{2}\)  Order  CPU time  
08  6.3092e−03  4.4047e−03  1.9311e−04  1.4246e−04  …  0.062400 
16  1.6452e−03  1.1394e−03  7.0386e−05  5.1038e−05  1.45609  0.10920 
32  4.3121e−04  2.8317e−04  2.6417e−05  1.9079e−05  1.41382  0.23400 
64  1.0956e−04  6.4521e−05  5.4923e−06  3.8494e−06  2.26599  0.63960 
128  2.7227e−05  1.0443e−05  5.7211e−07  3.0277e−07  3.26304  2.01241 
A comparison of maximum error \((\Vert \cdot \Vert _{\infty })\) and Euclidean norm \((\Vert \cdot \Vert _{2})\) at \(T=1\) for problem1
N  τ = 1.0 × 10^{−2},γ = 0.8  

MCTBDQM [25]  Proposed method  
\(\Vert \cdot \Vert _{\infty }\)  \(\Vert \cdot \Vert _{2}\)  \(\Vert \cdot \Vert _{\infty }\)  \(\Vert \cdot \Vert _{2}\)  Order  CPU time  
08  4.1559e−03  2.9052e−03  2.6864e−05  1.4246e−04  …  0.07800 
16  1.0852e−03  7.5335e−04  1.0356e−05  7.4698e−06  1.37526  0.14040 
32  2.8491e−04  1.8911e−04  1.1811e−06  7.8660e−07  3.13225  0.24960 
64  7.2683e−05  4.4967e−05  5.3813e−07  3.2574e−07  1.13407  0.65520 
128  1.8220e−05  8.7572e−06  2.4165e−07  1.5278e−07  1.15502  2.01241 
A comparison of exact solution and approximated solution at \(T=1\) for problem1
τ = 1.0 × 10^{−3},γ = 0.5,λ = 0.00001,N = 100  

x  Exact solution  Approximated solution  Error 
0.1  5.53577911  5.53577900  1.1363e−07 
0.2  6.11798209  6.11798186  2.2948e−07 
0.3  6.76141588  6.76141555  3.3011e−07 
0.4  7.47252019  7.47251979  4.0100e−07 
0.5  8.25841200  8.25841157  4.3141e−07 
0.6  9.12695678  9.12695636  4.1533e−07 
0.7  10.0868472  10.0868468  3.5263e−07 
0.8  11.1476902  11.1476900  2.5034e−07 
0.9  12.3201030  12.3201029  1.2413e−07 
A comparison of maximum error \((\Vert \cdot \Vert _{\infty })\) and Euclidean norm \((\Vert \cdot \Vert _{2})\) at \(T=1\) for problem2
N  τ = 1.25 × 10^{−3},γ = 0.3  

CBSCM [21]  MCTBDQM [25]  Proposed method  
\(\Vert \cdot \Vert _{\infty }\)  \(\Vert \cdot \Vert _{2}\)  \(\Vert \cdot \Vert _{\infty }\)  \(\Vert \cdot \Vert _{2}\)  \(\Vert \cdot \Vert _{\infty }\)  \(\Vert \cdot \Vert _{2}\)  Order  CPU time  
08  4.8273e−02  3.4134e−02  1.5762e−02  9.4300e−03  2.2761e−05  5.6903e−06  …  6.08404 
16  1.2351e−02  8.7334e−03  2.1670e−03  1.1924e−03  7.4956e−06  1.3251e−06  1.60246  8.01845 
32  3.1048e−03  2.1955e−03  2.8541e−04  1.5040e−04  1.7463e−06  2.1829e−07  2.1017  15.3193 
64  7.7721e−04  5.4957e−04  3.6701e−05  1.8925e−05  1.3761e−07  1.2163e−08  3.6656  34.5386 
128  1.9430e−04  1.3739e−04  4.6559e−06  2.3752e−06  2.2313e−08  1.3945e−09  2.62468  93.4914 
7.2 Problem 2
A comparison of exact values and approximated values at different knots
x  Exact solution  Approximated solution  Absolute error 

0.1  0.58778525  0.58778972  4.4636e−06 
0.2  0.95105652  0.95106374  7.2223e−06 
0.3  0.95105652  0.95106374  7.2223e−06 
0.4  0.58778525  0.58778972  4.4636e−06 
0.5  0.00000000  2.96897923  2.9690e−14 
0.6  −0.5877853  −0.5877897  4.4636e−06 
0.7  −0.9510565  −0.9510637  7.2223e−06 
0.8  −0.9510565  −0.9510637  7.2223e−06 
0.9  −0.5877853  −0.5877897  4.4636e−06 
A comparison of exact values and approximated values at different knots at time \(T=10\)
x  Exact solution  Approximated solution  Absolute error 

0.1  58.7785252  58.7785251  1.2446e−07 
0.2  95.1056516  95.1056514  2.0138e−07 
0.3  95.1056516  95.1056514  2.0138e−07 
0.4  58.7785252  58.7785251  1.2446e−07 
0.5  0.00000000  −5.8308913  5.8309e−13 
0.6  −58.778525  −58.778525  1.2446e−07 
0.7  −95.105652  −95.105651  2.0138e−07 
0.8  −95.105652  −95.105651  2.0138e−07 
0.9  −58.778525  −58.778525  1.2446e−07 
7.3 Conclusion
A fully implicit finite difference scheme based on extended cubic Bspline has been formulated to solve the time fractional advection–diffusion equation. The proposed technique was examined and found to be unconditionally stable and convergent with \(O(\tau +h^{2})\). This technique was tested on two test problems, and the results indicated that the method is feasible and accurate.
Declarations
Acknowledgements
The authors are indebted to the anonymous reviewers for their helpful, valuable comments and suggestions in the improvement of this manuscript. This study was financially supported by Universiti Sains Malaysia Bridging Grant No. 304.PMATHS.6316011. Moreover, the first author Syed Tauseef MohyudDin is also thankful to Chairman of Bahria Town/ Patron and Chairman of FAIRE; Chief Executive FAIRE; Administration of University of Islamabad (a project of Bahria Town) for the establishment of Center for Research (CFR) and the provision of conducive research environment.
Authors’ contributions
All authors contributed equally to writing of this paper. 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
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