# A Hahn computational operational method for variable order fractional mobile–immobile advection–dispersion equation

- 139 Downloads

## Abstract

In this paper, we consider the discrete Hahn polynomials \(\{H_n\}\) and investigate their application for numerical solutions of the time fractional variable order mobile–immobile advection–dispersion model which is advantageous for modeling dynamical systems. This paper presented the operational matrix of derivative of discrete Hahn polynomials. The main advantage of approximating a continuous function by Hahn polynomials is that they have a spectral accuracy in the interval [0, *N*]. Furthermore, for computing the coefficients of the expansion \(u(x)=\sum _{n=0}^{\infty }c_nH_n(x)\), we have to only compute a summation and the calculation of coefficients is exact. Also an upper bound for the error of the presented method, with equidistant nodes, is investigated. Illustrative examples are provided to show the accuracy and efficiency of the presented method. Using a small number of Hahn polynomials, significant results are achieved which are compared to other methods.

## Keywords

Variable order fractional derivatives Mobile–immobile advection–dispersion equation Hahn polynomials Operational matrix method## Introduction

Fractional calculus is the generalization of the ordinary calculus which studies integrals and derivatives of non-integer (real or complex) order. Some numerical methods have been used to solve functional equations containing fractional derivatives, such as [18, 22, 27, 31]. In a wide class of physical, dynamical and biological models and complex problems, the constant order fractional equation cannot describe the characterization of problems. In 1993, intellectual curiosity of Samko and Ross led to an extension of the classical fractional calculus [32]. They presented continuously varying order *q*(*x*) for differential and integral operators and defined these operators in two ways. The first way was a direct definition and the second way was based on Fourier transforms. Variable-order (VO) fractional calculus is good at representing the memory property which changes with time or location. Applications of the VO idea have been developed by several researchers such as Lorenzo and Hartley [25], Ingman and Suzdalnitsky [17], Coimbra [9] and others. VO fractional calculus as a powerful tool has recently been applied in the fluid dynamics and control of nonlinear viscoelasticity [29], mechanics [9], medical imaging [38], processing of geographical data [10], etc. The analytic results on the existence and uniqueness of the solutions for a generalized fractional differential equations with VO operators have been discussed in [30].

Solving VO fractional problems is quite difficult. Therefore, efficient numerical techniques are necessary to be developed. A finite difference technique is used for solving VO fractional integro-differential equations [37]. Recently, spectral methods using continuous orthogonal polynomials, such as Jacobi, Chebyshev and Legendre polynomials, have been developed for solving different kinds of VO fractional differential equations. Chen et al. proposed Legendre wavelets functions to solve a class of nonlinear VO fractional differential equations [8]. Bernstein polynomials are used to numerically solve the VO fractional partial differential equations by Wang et al. [35]. In [7], numerical solutions of the VO linear cable equation with adopted Bernstein polynomials basis on the interval [0, *R*] are presented. Although in most cases continuous orthogonal polynomials are used as basis functions for the approximate solution of equations, recently discrete orthogonal polynomials have been noticed for solving stochastic differential equations [36] and in numerical fluid dynamics problems [13] because of the behavior and property of these polynomials [15].

Discrete orthogonal polynomials are orthogonal with respect to a weighted discrete inner product. Classical orthogonal polynomial (continuous/discrete) are a class of the orthogonal polynomials associated with the Wiener–Askey polynomials including Hermite, Laguerre, Jacobi as continuous, and Charlier, Meixner, Krawtchouk and Hahn as discrete polynomials [24]. That in this paper we focus on Hahn polynomials.

Advection–dispersion equation and its extensions (such as the mobile–immobile equation) are a combination of the diffusion and advection equations. These equations are used to the modeling of transformation of pollutants, energy, subsurface water flows, deeper river flows, streams, and groundwater [4, 11, 12, 19, 23]. Recently, VO fractional diffusion equation is expanded to describe time-dependent anomalous diffusion and diffusion process in inhomogeneous porous media and for the description of complex dynamical problems [40].

In this work, we investigate the VO fractional problem arises in a mobile–immobile advection–dispersion equation (VOFMIE) [39]. This model is obtained from the standard advection–dispersion equation by adding the variable time-fractional derivative in the Caputo sense of order \(0< q(x,t)\le 1\). In [26], this model is used to simulate solute transport in watershed catchments and rivers. Some numerical methods have been developed for the solution of VOFMIE. Authors of [20] are used reproducing kernel theory and collocation method for solving the VOFMIE. An implicit Euler approximation for the VOFMIE has been explained in [39] Also, Chebyshev wavelets method [16] is employed to solve this VO equation. In [34], the VOFMIE in 2-D arbitrary domains is introduced and a meshless method based on the MLS approach is proposed to solve it. In [1], a method, based on shifted Jacobi Gauss–Lobatto and shifted Jacobi Gauss–Radau spectral methods, is presented for solving the mobile–immobile advection–dispersion model with a Coimbra time variable fractional derivative.

## Hahn polynomials

### Definition and properties of Hahn polynomials

First, we state some definitions.

## Definition 1

## Definition 2

*N*, the Hahn polynomials \(H_n(x;\alpha ,\beta ,N)\), \(n=0,1,\ldots ,N\), are defined in [3] by

*x*with degree exactly

*n*. The Hahn polynomials \(H_n(x;\alpha ,\beta ,N)\) are classical discrete orthogonal polynomials of degree

*n*on the interval \(I:=[0,N]\). They are orthogonal on

*I*with respect to the discrete inner product:

### Expansion of Hahn polynomials in terms of Taylor basis

*Remark 1*

### Function approximations by Hahn polynomials

*u*, by Hahn polynomials, converges pointwise, if the series expansion \(\sum _{i=0}^{N}d_{i}P_i(x)\) of the function

*u*, by Jacobi polynomials, converges pointwise and \(n^4/N \rightarrow 0\) for \(n,N\rightarrow {\infty }\). From Theorem 3.1 in [15], spectral accuracy follows directly for Hahn polynomials. Therefore, a continuous function

*u*(

*x*), of bounded variation, can be expanded by truncated Hahn series as follows:

*u*(

*x*) on [

*a*,

*b*], in terms of continuous orthogonal polynomials \(\phi _i(x)\), we use the expansion \(u(x)=\sum _{i=0}^{N}c_{i}\phi _i(x)\), where the coefficients \(u_{i}\) are computed by

## Operational matrix of derivatives for Hahn polynomials

Here, we present the ordinary and VO differentiation matrix of the Hahn polynomials in the Caputo sense.

**Lemma 1**

*The Caputo VO derivative of Hahn vector*\(\mathbf{H}(t)\)

*is*

*where*\(\mathbf{Q}^{q(x,t)}=\mathbf{A}\mathbf{M}^{q(x,t)}\mathbf{A}^{-1}\)

*is an*\((N+1)\times (N+1)\)

*matrix and*\(\mathbf{M}^{q(x,t)}\)

*is defined as*:

*Proof*

**Corollary 1**

*Since the ordinary derivative is a special case of fractional derivative, so from Lemma*1,

*we have*

## Description of the proposed method

*X*] and [0,

*T*] respectively. By discretizing Eq. (28), we get a matrix equation which is in the general form \(\mathbf{AUB+CUD=E}\) where \(\mathbf{A}\), \(\mathbf{B}\), \(\mathbf{C}\), \(\mathbf{D}\), and \(\mathbf{U}\) are multipliable matrices.

## Error bound

In this section, an upper bound for the error of the presented approximation scheme, with equidistant nodes, is obtained with a similar procedure as in [5].

*u*(

*x*,

*t*) is a smooth function, of bounded variation, in \(\Omega\). Consider

*u*(

*x*,

*t*) and the nodes \((x_i,t_j)\), for \(i= 0,1,\ldots ,I\) and \(j=0,\ldots ,J\) are equidistant points on [0,

*X*] and [0,

*T*], respectively. Let

*u*(

*x*,

*t*) at equidistant points \((x_i, t_j)\), for \(0\le i\le I\), \(0\le j\le J\). Then, we have

*u*(

*x*,

*t*) is a smooth function on \(\Omega\), then there exist the constants \(K_1\), \(K_2\), and \(K_3\) such that

## Numerical examples

*T*, via the following norm:

*N*. Also, to show the efficiency, we report the convergence order (CO) of our method for two last examples. CO is defined by [6]

*Example 1*

The absolute errors of the numerical solutions, at \(T=1,2,4\), and for \(\alpha =0.01\), \(\beta =0.01\), and \(N=5\) are shown in Table 1. A comparison of the numerical solutions is made by the results reported in [39] and [20] at \(T=1\), by Finite Difference Method (FD) and Reproducing Kernel Method (RKM), respectively. Also, the results of the presented method at \(T=2\) and \(T=4\) show that the method is accurate even for larger domains. Fig. 1 shows the results of the approximate and exact solutions at \(T=1,2,4\). The absolute error for this approach is shown in Fig. 2.

Error at \(T = 1,2,4\) for Eq. (40)

\(T=1\) | \(T=2\) | \(T=4\) | |||
---|---|---|---|---|---|

| FD (\(N=100\)) | RKM (\(N=20\)) | Hahn (\(N=5\)) | Hahn (\(N=5\)) | Hahn (\(N=5\)) |

0.1 | 1.562E−4 | 6.376E−5 | 3.750E−13 | 6.680E−11 | 5.010E−09 |

0.2 | 1.401E−3 | 4.904E−6 | 4.410E−13 | 1.349E−10 | 1.140E−08 |

0.3 | 2.975E−3 | 5.984E−5 | 4.420E−13 | 1.585E−10 | 1.407E−08 |

0.4 | 4.298E−3 | 7.781E−6 | 3.400E−14 | 1.377E−10 | 1.263E−08 |

0.5 | 4.972E−3 | 5.809E−5 | 3.350E−13 | 1.009E−10 | 9.610E−09 |

0.6 | 4.803E−3 | 8.298E−6 | 8.700E−14 | 8.780E−11 | 8.640E−09 |

0.7 | 3.815E−3 | 5.893E−5 | 1.051E−12 | 1.305E−10 | 1.268E−08 |

0.8 | 2.275E−3 | 5.946E−6 | 1.355E−12 | 2.353E−10 | 2.222E−08 |

0.9 | 7.208E−4 | 6.298E−5 | 6.400E−13 | 3.680E−10 | 3.349E−08 |

*Example 2*

The errors of the numerical solutions for \(\alpha =1\), \(\beta =1\), and \(N=10\), at \(T=1, 2, 4\), are shown in Table 2. Also, a comparison of the numerical solutions, at \(T=1\), is made by the results of RKM [20], in Table 2. In Fig. 3, the results of the approximate and the exact solutions at \(T=1, 2, 4\) are compared. The absolute errors for this approach are shown in Fig. 4.

Absolute error at \(T = 1, 2, 4\) for Eq. (42)

\(T=1\) | \(T=2\) | \(T=4\) | ||
---|---|---|---|---|

| RKM (\(N=13\)) | Hahn (\(N=10\)) | Hahn (\(N=10\)) | Hahn (\(N=10\)) |

0.1 | 0 | 7.7716E−16 | 3.5816E−09 | 2.0973E−05 |

0.2 | 2.2205E−16 | 8.8818E−16 | 2.3241E−09 | 1.4376E−05 |

0.3 | 4.4409E−16 | 8.8818E−16 | 1.7188E−09 | 1.1013E−05 |

0.4 | 0 | 4.4409E−16 | 1.0461E−09 | 7.2836E−06 |

0.5 | 0 | 0 | 6.6425E−10 | 4.9987E−06 |

0.6 | 4.4409E−16 | 0 | 6.3439E−10 | 4.9261E−06 |

0.7 | 0 | 4.4409E−16 | 1.0719E−09 | 7.8506E−06 |

0.8 | 0 | 0 | 2.1776E−09 | 1.4766E−05 |

0.9 | 6.6613E−16 | 5.5511E−16 | 3.2358E−09 | 2.0979E−05 |

*Example 3*

Figure 5 shows the numerical solution of Eq. (43) where it displays a typical mobile–immobile behavior.

*Example 4*

The errors of the numerical solutions for \(\alpha =\beta =1\) and different *N* are shown in Table 3. We investigate the convergence order of our method, and a comparison of the numerical solutions is made by the results of Chebyshev wavelets method (CWs) [16], in Table 4. The absolute errors for this approach, at \(T=1\), are shown in Fig. 6.

Absolute errors at \(T = 1\) and different *N*, for Eq. (44)

| \(N=2\) | \(N=4\) | \(N=6\) | \(N=8\) | \(N=10\) |
---|---|---|---|---|---|

0.1 | 4.5433E−02 | 7.7271E−05 | 2.9621E−07 | 5.7762E−10 | 1.7524E−12 |

0.2 | 7.7018E−02 | 5.5015E−05 | 1.6603E−07 | 3.2999E−10 | 2.3681E−12 |

0.3 | 9.5918E−02 | 9.8537E−06 | 5.7100E−08 | 1.5395E−10 | 2.7929E−12 |

0.4 | 1.0342E−01 | 2.4603E−05 | 2.6394E−08 | 3.4051E−11 | 2.9381E−12 |

0.5 | 1.0094E−01 | 4.4895E−05 | 1.2243E−07 | 2.0723E−10 | 2.7940E−12 |

0.6 | 9.0057E−02 | 6.3491E−05 | 2.1114E−07 | 3.6830E−10 | 2.3717E−12 |

0.7 | 7.2499E−02 | 9.3072E−05 | 2.7361E−07 | 5.1937E−10 | 1.7084E−12 |

0.8 | 5.0187E−02 | 1.2916E−04 | 3.5231E−07 | 6.3469E−10 | 8.7264E−13 |

0.9 | 4.5433E−02 | 7.7271E−05 | 2.9621E−07 | 5.7762E−10 | 1.7524E−12 |

Comparison of \(\Vert e_N\Vert _{\infty }\) at \(T = 1\), and convergence error (for our method) for Eq.(44)

\(\Vert e_N\Vert _{\infty }\) | |||
---|---|---|---|

| CWs [16] | Our method | CO |

2 | – | 1.0342E−01 | – |

4 | 2.022E−04 | 1.3888E−04 | 9.5405 |

6 | 1.838E−06 | 4.3929E−07 | 14.1966 |

8 | 5.188E−10 | 8.0734E−10 | 21.8963 |

10 | 1.515E−12 | 2.9381E−12 | 25.1676 |

*Example 5*

The errors of the numerical solutions for \(\alpha =\beta =1\) and different *N* are shown in Table 5. The convergence order of our method is investigated in Table 6. Also, the absolute errors for this approach, at \(T=0.5\), are shown in Fig. 7.

Absolute error at \(T=0.5\) and different *N*, for Eq. (45)

| \(N=2\) | \(N=4\) | \(N=6\) | \(N=8\) | \(N=10\) |
---|---|---|---|---|---|

1 | 1.2749E−02 | 4.3189E−04 | 5.1970E−06 | 2.4940E−08 | 6.6889E−12 |

2 | 1.3059E−02 | 1.2390E−04 | 1.0420E−06 | 9.2309E−09 | 7.9431E−13 |

3 | 7.7534E−03 | 5.8910E−05 | 4.1286E−07 | 2.8462E−09 | 1.5582E−13 |

4 | 2.2446E−03 | 3.9771E−05 | 2.4108E−07 | 5.5964E−10 | 4.7934E−14 |

5 | 9.0645E−06 | 2.0914E−07 | 1.8357E−09 | 8.9560E−12 | 2.8089E−14 |

6 | 2.2446E−03 | 3.9811E−05 | 2.4179E−07 | 5.6323E−10 | 4.6046E−14 |

7 | 7.7534E−03 | 5.8980E−05 | 4.1305E−07 | 2.8563E−09 | 1.4000E−13 |

8 | 1.3059E−02 | 1.2382E−04 | 1.0449E−06 | 9.2442E−09 | 6.7760E−13 |

9 | 1.2749E−02 | 4.3189E−04 | 5.1970E−06 | 2.4940E−08 | 6.6889E−12 |

Convergence error of Eq. (45)

| \(|E|_{\infty }\) | CO |
---|---|---|

2 | 1.4007E−02 | – |

4 | 4.5351E−04 | 4.9489 |

6 | 8.5058E−06 | 9.8067 |

8 | 9.8931E−08 | 15.4827 |

10 | 9.2145E−10 | 20.9561 |

## Discussion and conclusion

In this paper, we present an operational matrix method based on Hahn polynomials for solving the time VO fractional mobile–immobile advection–dispersion model. This method converts the VO fractional equation into an algebraic system which is solved by a technique of linear algebra. Using the discrete Hahn polynomials in a projection approach will not result in any numerical calculation errors.

As the main result, a new operational matrix of VO fractional derivative, in the Caputo sense, is derived for the Hahn polynomials. Then, it is used to obtain an approximate solution to the problem under study. Also, an upper bound for the error of the presented method, with equidistant nodes, is investigated. Numerical examples show the accuracy and the efficiency of the presented method. An advantage of using the Hahn polynomials is that the numerical results achieved only using a small number of bases are accurate in a larger intervals and significant results are achieved. A comparison of the numerical solutions by FDM [39] and RKM [20] shows that this technique is accurate enough to be known as a powerful device. This method can be applied to solve other types of VO fractional functional equations.

## Notes

## References

- 1.Abdelkawy, M.A., Zaky, M.A., Bhrawy, A.H., Baleanu, D.: Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model. Rom. Rep. Phys.
**67**(3), 773–791 (2015)Google Scholar - 2.Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972)MATHGoogle Scholar
- 3.Beals, Richard, Wong, Roderick: Special Functions: A Graduate Text, Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2010)CrossRefMATHGoogle Scholar
- 4.Benson, D.A., Schumer, R., Meerschaert, M.M., Wheatcraft, S.W.: Fractional dispersion, lévy motion, and the made tracer tests. Transp. Porous Media
**42**(1), 211–240 (2001)MathSciNetCrossRefGoogle Scholar - 5.Bhrawy, A.H., Zaky, M.A.: 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)MathSciNetCrossRefMATHGoogle Scholar - 6.Bhrawy, A.H., Zaky, M.A.: Numerical algorithm for the variable-order caputo fractional functional differential equation. Nonlinear Dyn.
**85**(3), 1815–1823 (2016)MathSciNetCrossRefMATHGoogle Scholar - 7.Chen, Y., Liu, L., Li, B., Sun, Y.: Numerical solution for the variable order linear cable equation with bernstein polynomials. Appl. Math. Comput.
**238**, 329–341 (2014)MathSciNetMATHGoogle Scholar - 8.Chen, Y.M., Wei, Y.Q., Liu, D.Y., Yu, H.: Numerical solution for a class of nonlinear variable order fractional differential equations with legendre wavelets. Appl. Math. Lett.
**46**, 83–88 (2015)MathSciNetCrossRefMATHGoogle Scholar - 9.Coimbra, C.F.M.: Mechanics with variable-order differential operators. Ann. Phys.
**12**(11–12), 692–703 (2003)MathSciNetCrossRefMATHGoogle Scholar - 10.Cooper, G.R.J., Cowan, D.R.: Filtering using variable order vertical derivatives. Comput. Geosci.
**30**(5), 455–459 (2004)CrossRefGoogle Scholar - 11.Deng, Z., Bengtsson, L., Singh, V.P.: Parameter estimation for fractional dispersion model for rivers. Environ. Fluid Mech.
**6**(5), 451–475 (2006)CrossRefGoogle Scholar - 12.Elbeleze, A.A., Kiliçman, A., Taib, B.M.: Application of homotopy perturbation and variational iteration methods for Fredholm integrodifferential equation of fractional order. Abstr. Appl. Anal.
**2012**, 14 (2012)MathSciNetCrossRefMATHGoogle Scholar - 13.Glaubitz, J., Öffner, P., Sonar, T.: Application of modal filtering to a spectral difference method (2016). arXiv:1604.00929Google Scholar
- 14.Goertz, R., Öffner, P.: On Hahn polynomial expansion of a continuous function of bounded variation (2016). arXiv:1610.06748Google Scholar
- 15.Goertz, R., Öffner, P.: Spectral accuracy for the Hahn polynomials (2016). arXiv:1609.07291Google Scholar
- 16.Heydari, M.H.: A new approach of the Chebyshev wavelets for the variable-order time fractional mobile-immobile advection-dispersion model (2016). arXiv:1605.06332Google Scholar
- 17.Ingman, D., Suzdalnitsky, J.: Control of damping oscillations by fractional differential operator with time-dependent order. Comput. Methods Appl. Mech. Eng.
**193**, 5585–5595 (2004)MathSciNetCrossRefMATHGoogle Scholar - 18.Ismaeelpour, T., Askari Hemmat, A., Saeedi, H.: B-spline operational matrix of fractional integration. Optik. Int. J. Light Electron. Opt.
**130**, 291–305 (2017)CrossRefMATHGoogle Scholar - 19.Jafari, H., Kadem, A., Baleanu, D., Yilmaz, T.: Solutions of the fractional davey-stewartson equations with variational iteration method. Rom. Rep. Phys.
**64**(2), 337–346 (2012)Google Scholar - 20.Jiang, W., Liu, N.: A numerical method for solving the time variable fractional order mobile-immobile advection-dispersion model. Appl. Numer. Math.
**119**, 18–32 (2017)MathSciNetCrossRefMATHGoogle Scholar - 21.Karlin, S., McGregor, J.: The hahn polynomials, formulas and an application. Scripta Math.
**26**, 33–46 (1961)MathSciNetMATHGoogle Scholar - 22.Khan, R.A., Khalil, H.: A new method based on legendre polynomials for solution of system of fractional order partial differential equations. Int. J. Comput. Math.
**91**(12), 2554–2567 (2014)MathSciNetCrossRefMATHGoogle Scholar - 23.Kim, S., Levent Kavvas, M.: Generalized Fickś law and fractional ade for pollution transport in a river: detailed derivation. J. Hydrol. Eng.
**11**(1), 80–83 (2006)CrossRefGoogle Scholar - 24.Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer Monographs in Mathematics. Springer, New York (2010)CrossRefMATHGoogle Scholar
- 25.Hartley, T.T., Lorenzo, C.F.: Initialization, conceptualization, and application in the generalized fractional calculus. Crit. Rev. Biomed. Eng.
**35**(6), 447–553 (2007)CrossRefGoogle Scholar - 26.Ma, Heping, Yang, Yubo: Jacobi spectral collocation method for the time variable-order fractional mobile-immobile advection-dispersion solute transport model. East Asian J. Appl. Math.
**6**(3), 337–352 (2016)MathSciNetCrossRefMATHGoogle Scholar - 27.Mohseni Moghadam, M., Saeedi, H., Mollahasani, N., Mollahasani, N.: A new operational method for solving nonlinear Volterra integro-differential equations with fractional order. J. Inf. Math. Sci.
**2**(1, 2), 95–107 (2010)MATHGoogle Scholar - 28.Muthukumar, P., Ganesh Priya, B.: Numerical solution of fractional delay differential equation by shifted Jacobi polynomials. Int. J. Comput. Math.
**94**(3), 471–492 (2017)MathSciNetCrossRefMATHGoogle Scholar - 29.Orosco, J., Coimbra, C.F.M.: On the control and stability of variable-order mechanical systems. Nonlinear Dyn.
**86**, 695–710 (2016)MathSciNetCrossRefGoogle Scholar - 30.Razminiaa, A., Feyz Dizajib, A., Johari Majd, V.: Solution existence for non-autonomous variable-order fractional differential equations. Math. Comput. Model.
**55**, 1106–1117 (2012)MathSciNetCrossRefMATHGoogle Scholar - 31.Saeedi, H.: The linear b-spline scaling function operational matrix of fractional integration and its applications in solving fractional-order differential equations. Iran. J. Sci. Technol. Trans. A Sci.
**41**(3), 723–733 (2017)Google Scholar - 32.Samko, S.G., Ross, B.: Integration and differentiation to a variable fractional order. Integral Transform Spec. Funct.
**1**(4), 277–300 (1993)MathSciNetCrossRefMATHGoogle Scholar - 33.Shores, T.S.: Applied Linear Algebra and Matrix Analysis, Undergraduate Texts in Mathematics. Springer, New York (2007)CrossRefGoogle Scholar
- 34.Tayebi, A., Shekari, Y., Heydari, M.H.: A meshless method for solving two-dimensional variable-order time fractional advectiondiffusion equation. J. Comput. Phys.
**340**, 655–669 (2017)MathSciNetCrossRefMATHGoogle Scholar - 35.Wang, J., Liu, L., Liu, L., Chen, Y.: Numerical solution for the variable order fractional partial differential equation with bernstein polynomials. Int. J. Adv. Comput. Technol.
**6**(3), 22–37 (2014)Google Scholar - 36.Xiu, Dongbin, Karniadakis, GeorgeEm: The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput.
**24**(2), 619–644 (2002)MathSciNetCrossRefMATHGoogle Scholar - 37.Xu, Y., Ertürk, V.S.: A finite difference technique for solving variable-order fractional integro-differential equations. Bull. Iran. Math. Soc.
**40**(3), 699–712 (2014)MathSciNetMATHGoogle Scholar - 38.Yu, Q., Vegh, V., Liu, F., Turner, I.: A variable order fractional differential-based texture enhancement algorithm with application in medical imaging. PLoS One
**10**(7), 1–35 (2015)Google Scholar - 39.Zhang, H., Liu, F., Phanikumar, M.S., Meerschaert, M.M.: A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model. Comput. Math. Appl.
**66**(5), 693–701 (2013)MathSciNetCrossRefMATHGoogle Scholar - 40.Zhang, X., Crawford, J.W., Deeks, L.K., Stutter, M.I., Bengough, A.G., Young, I.M.: A mass balance based numerical method for the fractional advection-dispersion equation: theory and application. Water Res 41(7) (2005)Google Scholar

## Copyright information

**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.