# Space-Time Spectral Collocation Algorithm for the Variable-Order Galilei Invariant Advection Diffusion Equations with a Nonlinear Source Term.

1 IntroductionIn recent years, spectral methods (see [4, 10, 21]) are often efficient and highly accurate schemes when compared with the local methods. The speed of convergence is one of the great advantages of spectral methods. Besides, spectral methods have exponential rates of convergence; they also have high level of accuracy. The main idea of all versions of spectral methods is to express the approximate solution of the problem as a finite sum of certain basis functions (orthogonal polynomials or combination of them) and then choose the coefficients in order to minimize the difference between the exact and approximate solutions as well as possible. The spectral collocation method is a specific type of spectral methods, that is more applicable and widely used to solve almost types of differential equations [2, 3].

Fractional calculus [1, 19, 28] is a branch of calculus theory, which makes partial differential equations (PDEs) more convenient to describe many phenomena in several fields such as fluid mechanics, chemistry [12, 15], biology [19], viscoelasticity [20], engineering, finance and physics [14] fields. The concept of variable-order fractional allows the power of the fractional operator to be a function of the independent variable. The early studies of variable-order fractional was firstly introduced by Samko and Ross [22] and Lorenzo and Hartley [17, 18]. Several phenomena can be more accurately described using variable-order fractional operators. Mechanical [9], diffusion [5, 26, 34], FIR filters [29] multifractional Gaussian noises [25] and physical [13] models can be more accurately described by variable order derivatives mathematical models. Few numerical methods have been introduced and discussed to solve the variable-order fractional problems. Stability and convergence of explicit finite-difference method has been studied in [16] for solving the variable-order nonlinear fractional diffusion equation. Using Fourier analysis, Chen [6] obtained the numerical solutions for two-dimensional variable-order modified diffusion equations. Numerical methods based on finite difference techniques [8, 24, 31, 32] have been proposed by Liu, Shen, Zhang et al. for the numerical treatment of variable-order fractional partial differential equations. Zhao et al. [33], introduced two second-order approximation algorithms for the variable-order fractional time derivatives. Also, finite difference method has been applied by Xu and Erturk [30] to solve the fractional integro-differential equations with variable order. Moreover, finite difference schemes [27] have been introduced to solve variable-order time fractional diffusion equation.

The fractional advection-diffusion equation [11, 23] can be considered as a generalized version of the classical advection-diffusion equation. The fractional advection-diffusion equation is used to model many physical phenomena such as amorphous, colloid, the transport dynamics in complex systems, fractals and percolation clusters, biological systems, glassy and porous media, comb structures, dielectrics and semiconductors, polymers, random and disordered media, geophysical and geological processes, the transport of passive tracers carried by fluid flow in a porous medium. Here, we focus on the application of SJ-GL-C and SJ-GR-C schemes to numerically solve the VO-NGIADE in one and two dimensional space. The proposed collocation scheme is investigated for both temporal and spatial discretizations. The SJ-GL-C and SJ-GR-C are proposed, with a suitable modification for treating the boundary and initial conditions, for spatial and temporal discretizations. This treatment, for the conditions, improves the accuracy of the scheme greatly. Therefore, the VO-NGIADE with its conditions is reduced to system of nonlinear algebraic equations which is far easier to be solved. In addition, this algorithm is developed to numerically solve the two-dimensional VO-NGIADE. Thus, we introduce a fully spectral collocation approach to numerically treat the multi-dimensional VO-NGIADE. Moreover, there are no numerical results on the spectral collocation method for solving the VO-NGIADE. Finally, several numerical examples with com parisons lighting the high accuracy and effectiveness of the proposed algorithm are presented.

This paper is organized as follows. Few facts of shifted Jacobi polynomials are listed in Section 2. In Section 3, we introduce a new collocation method for the one-dimensional space VO-NGIADE. In Section 4, the proposed scheme is successfully extended to solve the two-dimensional space VO-NGIADE. Section 5 is customized to solve several problems. Conclusions are given in the last section.

2 Properties of shifted Jacobi polynomials

Some few properties of shifted Jacobi polynomials are presented in this section. In the following, few relations related to Jacobi polynomials are listed:

[mathematical expression not reproducible], (2.1)

where [alpha], [beta] > -1, x [member of] [-1,1] and

[mathematical expression not reproducible].

Moreover, the rth derivative (r is an intger) of [P.sup.([alpha],[beta]).sub.j](x), may be obtained from

[mathematical expression not reproducible].

For the shifted Jacobi polynomial [P.sup.([alpha],[beta]).sub.L,k](x) = [P.sup.([alpha],[beta]).sub.k](2x/L - 1), L > 0, the explicit analytic form is written as

[mathematical expression not reproducible].

Thus, we can derive the following properties

[mathematical expression not reproducible]. (2.2)

We used [x.sup.([alpha], [beta]).sub.N, j], and [[bar.w].sup.([alpha],[beta]).sub.N,j], 0 [less than or equal to] j [less than or equal to] N, as the nodes and Christoffel numbers of the standard Jacobi-Gauss interpolation on the interval [-1, 1].

The corresponding nodes and Christoffel numbers of the shifted Jacobi-Gauss interpolation on the interval [0, L] can be given by

[mathematical expression not reproducible].

3 One-dimensional space of VO-NGIADE

In this section, we introduce a numerical algorithm based on the SJ-GR-C and SJ-GL-C methods for solving numerically one-dimensional VO-NGIADE. The collocation points are selected at the SJ-GR and SJ-GL interpolation nodes for temporal and spatial variables, respectively. The core of the proposed method consists of discretizing the one-dimensional VO-NGIADE to create a system of nonlinear algebraic equations of the unknown coefficients. This system can be then easily solved with a standard numerical scheme. In particular, we consider the following VO-NGIADE

[mathematical expression not reproducible], (3.1)

given in (x, t) [member of] [0, L] x [0, T] with the initial-boundary conditions

u(x, 0) = [g.sub.1](x), u(0, t) = [g.sub.2](t), u(L, t) = [g.sub.3](t), (x, t) [member of] [0, L] x [0, T],

where H(u(x, t), x, t), [g.sub.1](x), [g.sub.2](t) and [g.sub.3](t) are given functions, 0 < [gamma](x, t) < 1 and [D.sup.1-[gamma](x, t).sub.t]u(x, t) is the temporal fractional derivative of variable order 1 - [gamma](x, t) in the Riemann-Liouville sense [7, 16, 34]:

[mathematical expression not reproducible]

We are interested in using the SJ-GL-C and SJ-GR-C methods to transform the previous VO-NGIADE into a system of nonlinear algebraic equations. In order to do this, we approximate the independent space variable x using the SJ-GL-C method at the [mathematical expression not reproducible] nodes, while the independent temporal variable t was approximated by the SJ-GR-C methods. The nodes are the set of points in a specified domain where the dependent variable values are to be approximated. In general, the choice of the location of the nodes is optional. However, taking the roots of the shifted Jacobi orthogonal polynomials, referred to as shifted Jacobi collocation points, gives particularly accurate solutions for the spectral methods.

Now, we outline the main steps of the mixed SJ-GL-C and SJ-GR-C methods for solving the one-dimensional space VO-NGIADE. We choose the approximate solution to be of the form

[mathematical expression not reproducible], (3.2)

where [mathematical expression not reproducible]. Then the spatial partial derivatives [partial derivative]u(x,t)/[partial derivative]x and [[partial derivative].sup.2]u(x, t)/[partial derivative][x.sup.2] were computed as

[mathematical expression not reproducible],

dependence on Eq. (2.2), we obtain

[mathematical expression not reproducible].

Furthermore, the temporal derivative [partial derivative]u(x, t)/[partial derivative]t is evaluated as

[mathematical expression not reproducible],

dependence on Eq. (2.2), we obtain

[mathematical expression not reproducible].

Moreover, the Riemann-Liouville fractional partial derivative of variable order [D.sup.1-[gamma](x,t).sub.t] [[partial derivative].sup.2]u(x,t)/[partial derivative][x.sup.2] is given by

[mathematical expression not reproducible], (3.3)

dependence on Eq. (2.2), we obtain

[mathematical expression not reproducible].

Now, adopting (3.2)-(3.3), enable one to write (3.1) in the form:

[mathematical expression not reproducible]. (3.4)

The initial condition immediately gives

u(x, 0) [N.summation over (i=0)] [M.summation over (j=0)] [a.sub.i,j] [f.sup.i,j.sub.0] (x, 0) = [g.sub.1](x) (3.5)

while the numerical treatments of the boundary conditions are

[mathematical expression not reproducible]. (3.6)

In the proposed mixed SJ-GL-C and SJ-GR-C methods, the residual of (3.4) is set to zero at M (N - 1) of SJ-GL and SJ-GR points. Consequently, we find

[mathematical expression not reproducible], (37)

for r = 1, ..., N - 1, s = 1, ..., M, where,

[mathematical expression not reproducible].

Dependence on Eqs. (3.5) and (3.6), we obtain

[mathematical expression not reproducible], (3.8)

[mathematical expression not reproducible] (3.9)

[mathematical expression not reproducible]. (3.10)

Combining Eqs. (3.7), (3.8), (3.9) and (3.10), we obtain for r = 1, ..., N - 1, s = 1, ..., M,

[mathematical expression not reproducible],

the previous system of nonlinear algebraic equations can be easily solved. After the coefficients [a.sub.i,j] are determined, it is straightforward to compute the approximate solution [u.sub.N,M](x,t) at any value of (x, t) in the given domain from the following equation

[mathematical expression not reproducible].

4 Two-dimensional space of VO-NGIADE

In the present section, we extend the previous algorithm to numerically solve the two-dimensional space VO-NGIADE in the following form

[mathematical expression not reproducible], (4.1)

subject to the initial-boundary conditions

[mathematical expression not reproducible],

where H(u(x, y, t), x, y, t), [g.sub.0](x,y), [g.sub.1](y,t), [g.sub.2](y,t), [g.sub.3](x,t) and [g.sub.4](x,t) are given real valued functions and u(x,y,t) is an unknown function. Therefore, the SJ-GL-C and SJ-GR-C methods will be applied to transform the previous two-dimensional VO-NGIADE into system of nonlinear algebraic equations. The SJ-GL-C and SJ-GR-C have been used for the space (x, y) and time t approximations, respectively.

Now, we outline the main steps of the collocation method for solving the two-dimensional VO-NGIADE. Let

[mathematical expression not reproducible], (4.2)

where [mathematical expression not reproducible].

Then the first spatial and temporal partial derivatives [partial derivative]u(x, y, t)/[partial derivative]x, [partial derivative]u(x, y, t)/[partial derivative]y and [partial derivative]u(x, y, t)/[partial derivative]t can be computed as

[mathematical expression not reproducible], (4.3)

where

[mathematical expression not reproducible].

While, the second spatial partial derivatives [[partial derivative].sup.2](x, y, t)/[partial derivative][x.sup.2] and [[partial derivative].sup.2]u(x, y, t)/[partial derivative][y.sup.2] are given by

[mathematical expression not reproducible],

where

[mathematical expression not reproducible].

Moreover, the variable order Riemann-Liouville fractional derivatives [D.sup.1-[gamma](x, y, t).sub.t] [[partial derivative].sup.2]u(x, y, t)[partial derivative][x.sup.2] and [D.sup.1-[gamma](x, y, t).sub.t] [[partial derivative].sup.2]u(x, y, t)[partial derivative][y.sup.2]

[mathematical expression not reproducible],

where

[mathematical expression not reproducible].

Therefore, adopting (4.2)-(4.3), enable one to write (4.1) in the form:

[mathematical expression not reproducible],

where

[mathematical expression not reproducible].

Moreover, the collocation treatments of the initial-boundary conditions immediately give

[mathematical expression not reproducible].

In the proposed method, the residual of (4.1) is set to be zero at (N - 1) x (M - 1) x K of collocation points

[mathematical expression not reproducible],

r = 1, ..., N - 1, s = 1, ..., M - 1, [zeta] = 1, ..., K, where,

[mathematical expression not reproducible],

and from the initial conditions, we have, namely (1+N+2KN+M(1+2K+N)) algebraic equations

[mathematical expression not reproducible],

and this in turn, yields (M+1) x (N +1) x (K +1) nonlinear algebraic equations

[mathematical expression not reproducible],

r = 1, ..., N - 1, [??] = 0, ..., K.

The previous system of nonlinear algebraic equations can be easily solved. After the coefficients [a.sub.i,j,k] are determined, we compute the approximate solution [u.sub.N,M,K](x, y, t) at any value of (x, y, t) in the given domain.

5 Numerical results and comparisons

This section listed several numerical examples to demonstrate the accuracy of the proposed method. Also, we compare our numerical results with the existing numerical results [7]. The obtained results of these examples show that the proposed method, by selecting a few number nodes, has high level of accuracy.

The difference between the measured value of approximate solution and exact solution is defined absolute error (AE), given by

E(x, t) = [absolute value of (u(x, t) - [u.sub.N,M(x, t)])],

where u(x, t) and [u.sub.N,M](x, t) are the exact and the approximate solutions at the point (x, t), respectively.

Moreover, the maximum absolute error (MAE) is given by

[M.sub.E] = max{E(x,t) : [for all](x, t) [member of] [0, L] x [0, T]}.

5.1 One-dimensional space of VO-NGIADE

Example 1. We start with the VO-NGIADE in the following form [7]:

[mathematical expression not reproducible],

where (x, t) [member of] [0, 1] x [0, 1], and [gamma](x, t) = 1/300(10 - tx), knowing that the exact solution is given by u(x, t) = [t.sup.2][e.sup.x].

In Table 1, we display a comparison based on the MAEs between our results (with various choices of N, M, [[alpha].sub.1], [[beta].sub.1], [[alpha].sub.2], and [[beta].sub.2]) and the finite difference method (FDM) [7].

Figure 1 display the space-time graph of the AEs with N = M = 14, and [[alpha].sub.1] = [[beta].sub.1] = [[alpha].sub.2], = [[beta].sub.2] = 0. While, Figure 2 compare graphically the curves of numerical and exact solutions of Example 1 for the different values of x at N = M = 14, [[alpha].sub.1] = [[beta].sub.1] = 1/2 and [[alpha].sub.2], = [[beta].sub.2] = 0.

The t-direction curve of AEs of Example 1 for N = M = 14, and [[alpha].sub.1] = [[beta].sub.1] = [[alpha].sub.2] = [[beta].sub.2] = 0, is displayed in Figure 3. Moreover, we plot in Figure 4 the logarithmic graphs of MAEs (i.e., [log.sub.10][M.sub.E]) obtained by the present method with different values of (N = M = 2,4, 6, ... , 14) at three choices of [[alpha].sub.1], [[beta].sub.1], [[alpha].sub.2], and [[beta].sub.2]. This demonstrates that the proposed method leads to an accurate approximation and yields exponential convergence rates.

Example 2. Consider the VO-NGIADE in the following form [7]:

[mathematical expression not reproducible],

where (x, t) [member of] [0,1] x [0,1], and

[gamma](x, t) = 1/500 ([(tx).sup.2] - [sin.sup.3](tx) + [cos.sup.4](tx) + 16),

is the exact solution is given by u(x, t) = [t.sup.2][e.sup.x].

Table 2, displays the MAEs using the present method together with the results obtained in [7] for different choices of N, M, [[alpha].sub.1], [[beta].sub.1], [[alpha].sub.2], and [[beta].sub.2]. From the results of this example, we observe that the approximate solution obtained by our method is more better than those obtained in [7].

Example 3. Here, we test the VO-NGIADE in the following form:

[mathematical expression not reproducible],

where (x, t) [member of] [0,1] x [0,1], and f(x,t) is a given function such that [gamma](x, t) = 1/600 (20 - [e.sup.tx]), and the exact solution is u(x, t) = [t.sup.2.8][e.sup.x].

The MAEs for Example 3 are listed in Table 3 at different choice of N, M, [[alpha].sub.1], [[beta].sub.1], [[alpha].sub.2], and [[beta].sub.2]. This table confirm the high accuracy of the present scheme.

Example 4. Here, we test the VO-NGIADE in the following form:

[mathematical expression not reproducible],

where (x, t) [member of] [0,1] x [0,10], [gamma](x, t) = 1/500 ([(tx).sup.2] - [sin.sup.3](tx) + [cos.sup.4](tx) + 266), knowing that the exact solution is given by u(x, t) = [t.sup.2][e.sup.-x].

The MAEs for Example 4 are listed in Table 4 at different choice of N, M, [[alpha].sub.1], [[beta].sub.1], [[alpha].sub.2], and [[beta].sub.2].

The space-time graph of AEs of Example 4, for N = 10, M = 20, and [[alpha].sub.1] = [[beta].sub.1] = [[alpha].sub.2], = [[beta].sub.2] = 0, is sketched in Figure 5.

5.2 Two-dimensional space of VO-NGIADE

Example 5. We consider the following two-dimensional space VO-NGIADE

[mathematical expression not reproducible],

(x, y, t) [member of] [0, 1] x [0, 1] x [0, 1], [gamma](x, y, t) = 1/400(t(-x)y + cos(txy) + 13) and [mathematical expression not reproducible], knowing that the exact solution is given by u(x, y, t) = [t.sup.2] sin(x + y).

The MAEs obtained by our method are summarized in Table 5, for several choices of N, M, K, [[alpha].sub.1], [[beta].sub.1], [[alpha].sub.2], [[beta].sub.2], [[alpha].sub.3], and [[beta].sub.3]. This results confirm the high accuracy of the present scheme.

The three-dimensional graph of the AEs of Example 5 at t = 1/2, N = M = 8, K = 4, [[alpha].sub.1] = 1/2, [[beta].sub.1] = 0, [[alpha].sub.2] = 1/2, [[beta].sub.2] = 0, and [[alpha].sub.3] = [[beta].sub.3] = 1/2 is displayed in Figure 6. In addition, the curve of the AEs of Example 5, at y = t = 1/2, is displayed in Figure 7 with the choice N = M = 8, K = 4 and [[alpha].sub.1] = 1/2, [[beta].sub.1] = 0, [[alpha].sub.2] = 1/2, [[beta].sub.2] = 0, [[alpha].sub.3] = [[beta].sub.3] = 1/2.

Example 6. Finally, we introduce the following two-dimensional VO-NGIADE

[mathematical expression not reproducible],

where (x, y, t) [member of] [0,1] x [0,1] x [0,1], y(x, y, t) = 1/310(150 - txy) and f(x, t) = [te.sup.x+y] (t + 2 - [4t.sup.[gamma](x,y,t)]/[GAMMA]([gamma](x,y,t)+2), knowing that the exact solution is given by u(x, y, t) = [t.sup.2][e.sup.x+y].

Table 6 lists the MAEs obtained by using the proposed method for various choices of N, M, K, [[alpha].sub.1], [[beta].sub.1], [[alpha].sub.2], [[beta].sub.2], and [[beta].sub.3]. The numerical results presented in this table show that the results are vary accurate for small value of N, M and K.

6 Conclusions

By means of SJ-GL-C and SJ-GR-C schemes, we have introduced a space-time spectral algorithm for solving VO-NGIADs . According to the numerical results obtained above, we can concluded the high accuracy of our technique.

Comparisons between our approximate solutions of the problems, with their exact solutions or with the approximate solutions achieved by other methods, were also included to confirm the validity and accuracy of the new scheme.

https://doi.org/10.3846/13926292.2017.1258014

References

[1] D. Baleanu and K. Sayevand. Performance evaluation of matched asymptotic expansions for fractional differential equations with multi-order. B. Math. Soc. Sci. Math., 59(1):3-12, 2016.

[2] A.H. Bhrawy. An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system. Appl. Math. Comput., 247:30-46, 2014. https://doi.org/10.1016/j.amc.2014.08.062.

[3] A.H. Bhrawy and M.A. Zaky. Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation. Nonlinear Dyn., 80(1):101-116, 2015. https://doi.org/10.1007/s11071-014-1854-7.

[4] A.H. Bhrawy, M.A. Zaky and J.A. Tenreiro Machado. Efficient Legendre spectral tau algorithm for solving the two-sided space-time Caputo fractional advection-dispersion equation. J. Vib. Control., 22(8):2053-2068, 2015. https://doi.org/10.1177/1077546314566835.

[5] C-M. Chen, F. Liu, V. Anh and I. Turner. Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation. Math. Math. Comput., 81(277):345-366, 2012. https://doi.org/10.1090/S0025-5718-2011-024476.

[6] C.M. Chen. Numerical methods for solving a two-dimensional variable-order modified diffusion equation. Appl. Math. Comput., 225:62-78, 2013. https://doi.org/10.1016/j.amc.2013.08.064.

[7] C.M. Chen, F. Liu, V. Anh and I. Turner. Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term. Appl. Math. Comput., 217(12):5729-5742, 2011. https://doi.org/10.1016/j.amc.2010.12.049.

[8] C.M. Chen, F. Liu, I. Turner and V. Anh. Numerical methods with fourth-order spatial accuracy for variable-order nonlinear Stokes first problem for a heated generalized second grade fluid. Comput. Math. Appl., 62(3):971-986, 2011. https://doi.org/10.1016/j.camwa.2011.03.065.

[9] C.F.M. Coimbra. Mechanics with variable-order differential operators. Ann. Phys., 12(11-12):692-703, 2003. https://doi.org/10.1002/andp.200310032.

[10] E. H Doha, A H Bhrawy and M A. Abdelkawy. An accurate Jacobi pseudo-spectral algorithm for parabolic partial differential equations with nonlocal boundary conditions. J. Comput. Nonlinear Dyn., 10(2):021016, 2015. https://doi.org/10.1115/1.4026930.

[11] V.J. Ervin and J.P.Roop. Variational solution of fractional advection-dispersion equations on bounded domains in Rd. Numer. Methods Partial Differ. Equ., 23(2):256-281, 2007. https://doi.org/10.1002/num.20169.

[12] M. Giona and H.E. Roman. Fractional diffusion equation for transport phenomena in random media. Phys. A, 185(1-4):87-97, 1992. https://doi.org/10.1016/0378-4371(92)90441-R.

[13] H. Sun H. Sheng, C. Coopmans, Y.Q. Chen and G.W. Bohannan. Physical experimental study of variable-order fractional integrator and differentiator. In Fractional Differentiation and its Applications, 2010.

[14] R. Hilfer. Applications of Fractional Calculus in Physics. Word Scientific, Singapore, 2000. https://doi.org/10.1142/3779.

[15] J.W. Kirchner, X. Feng and C. Neal. Fractal stream chemistry and its implications for containant transport in catchments. Nature, 403:524-526, 2000. https://doi.org/10.1038/35000537.

[16] R. Lin, F. Liu, V. Anh and I. Turner. Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Appl. Math. Comput., 212(2):435-445, 2009. https://doi.org/10.1016/j.amc.2009.02.047.

[17] C.F. Lorenzo. Initialization, conceptualization, and application in the generalized (fractional) calculus CRC Crit. Rev. Biomed. Eng., 35(6):447-553, 2007. https://doi.org/10.1615/CritRevBiomedEng.v35.i6.10.

[18] C.F. Lorenzo and T.T. Hartley. Variable order and distributed order fractional operators. Nonlinear Dyn., 29(1):57-98, 2002. https://doi.org/10.1023/A:1016586905654.

[19] R.L. Magin. Fractional Calculus in Bioengineering. Begell House Publishers, 2006.

[20] I. Podlubny. Fractional Differential Equations, in: Mathematics in Science and Engineering. Academic Press Inc., San Diego, CA, 1999.

[21] A. Quarteroni, C. Canuto, M.Y. Hussaini and T.A. Zang. Spectral Methods: Fundamentals in Single Domains. Springer-Verlag, New York, 2006.

[22] S.G. Samko and B. Ross. Integration and differentiation to a variable fractional order. Integral Transforms Spec. Funct., 1(4):277-300, 1993. https://doi.org/10.1080/10652469308819027.

[23] S. Shen, F. Liu and V. Anh. Numerical approximations and solution techniques for the space-time Riesz-Caputo fractional advection-diffusion equation. Numer. Algor., 56(3):383-403, 2011. https://doi.org/10.1007/s11075-010-9393-x.

[24] S. Shen, F. Liu, V. Anh, I. Turner and J. Chen. A characteristic difference method for the variable-order fractional advection-diffusion equation. J. Appl. Math. Comput., 42(1):371-386, 2013. https://doi.org/10.1007/s12190-012-0642-0.

[25] H. Sheng, H. Sun, Y-Q. Chen and T. Qiu. Synthesis of multifractional Gaussian noises based on variable-order fractional operators. Signal Processing, 91(7):1645-1650, 2011. https://doi.org/10.1016/j.sigpro.2011.01.010.

[26] H. Sun, W. Chen and Y-Q. Chen. Variable-order fractional differential operators in anomalous diffusion modeling. Phys. A, 388(21):4586-4592, 2009. https://doi.org/10.1016/j.physa.2009.07.024.

[27] H.G. Sun, W. Chen, C. Li and Y-Q. Chen. Finite difference schemes for variable-order time fractional diffusion equation. Int. J. Bifurcat. Chaos., 22(4), 2012. https://doi.org/10.1142/S021812741250085X.

[28] H.G. Suna, Yong Zhangb and D.M. Reevesd W. Chenc. Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media. J. Contam. Hydrol., 157:47-58, 2014. https://doi.org/10.1016/j.jconhyd.2013.11.002.

[29] C.C. Tseng. Design of variable and adaptive fractional order FIR differentiators. Signal Processing, 86(10):2554-2566, 2006. https://doi.Org/10.1016/j.sigpro.2006.02.004.

[30] Y. Xu and V.S. Erturk. A finite difference technique for solving variable-order fractional integro-differential equations. Bull. Iranian Math. Soc., 40:699-712, 2014.

[31] H. Zhang, F. Liu, M.S. Phanikumar and M.M. Meerschaert. A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model. Comput. Math. Appl., 66(5):693-701, 2013. https://doi.org/10.1016/jxamwa.2013.01.031.

[32] H. Zhang, F. Liu, P. Zhuang, I. Turner and V. Anh. Numerical analysis of a new space-time variable fractional order advection-dispersion equation. Appl. Math. Comput., 242:541-550, 2014. https://doi.org/10.1016/j.amc.2014.06.003.

[33] X. Zhao, Z.Z. Sun and G.E. Karniadakis. Second-order approximations for variable order fractional derivatives: Algorithms and applications. J. Comput. Phys., 293:184-200, 2015. https://doi.org/10.1016/j.jcp.2014.08.015.

[34] P. Zhuang, F. Liu, V. Anh and I. Turner. Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM J. Numer. Anal., 47(3):1760-1781, 2009. https://doi.org/10.1137/080730597.

Mohamed A. Abd-Elkawy (a, b) and Rubayyi T. Alqahtani (a)

(a) Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University (IMSIU) Riyadh, Saudi Arabia

(b) Department of Mathematics, Faculty of Science, Beni-Suef University Beni-Suef, Egypt

E-mail(corresp.): melkawy@yahoo.com

E-mail: rr-gahtani@hotmail.com

Received March 23, 2016; revised October 26, 2016; published online January 5, 2017

Caption: Figure 1. Space-time graph of the AEs of Example 1.

Caption: Figure 2. t-direction curves of exact and numerical solutions of Example 1.

Caption: Figure 3. t-direction curve of the AEs of Example 1.

Caption: Figure 4. [M.sub.E] convergence of Example 1.

Caption: Figure 5. Space-time graph of the AEs of Example 4.

Caption: Figure 6. Space graph of the AEs for Example 5.

Caption: Figure 7. The AEs versus x of Example 5.

Table 1. The MAEs of Example 1. Our method with several choices of N, M [[alpha].sub.1], 2 6 [[beta].sub.1], [[alpha].sub.2], [[beta].sub.2] (0,0,0,0) 1:36284 x [10.sup.-3] 5:42912 x [10.sup.-8] (1/2, 1/2, 0, 0) 1:36284 x [10.sup.-3] 4:08291 x [10.sup.-8] (- 1/2, 0, 1/2, 1/2) 1:26147 x [10.sup.-2] 7:07546 x [10.sup.-8] Our method with several choices of N, M [[alpha].sub.1], 10 14 [[beta].sub.1], [[alpha].sub.2], [[beta].sub.2] (0,0,0,0) 2:66454 x [10.sup.-14] 6:66134 x [10.sup.-16] (1/2, 1/2, 0, 0) 2:62013 x [10.sup.-14] 4:44089 x [10.sup.-16] (- 1/2, 0, 1/2, 1/2) 3:59712 x [10.sup.-14] 6:66134 x [10.sup.-16] FDM [7] with several choices of [h.sub.t], [h.sub.x] [h.sub.t] = [h.sup.2.sub.x] = 1/16 -- 5.5308 x [10.sup.-4] [h.sub.t] = [h.sub.t] = [h.sup.2.sub.x] = 1/64 [h.sup.2.sub.x] = 1/256 1.4567 x [10-.sup.4] 6.1896 x [10.sup.-5] Table 2. The MAEs of Example 2. Our method with several choices of N, M [[alpha].sub.1], 4 6 [[beta].sub.1], [[alpha].sub.2], [[beta].sub.2] (0,0,0,0) 3.0160 x [10.sup.5] 5.4289 x [10.sup.8] (0, 1/2, 1, 1/2) 3.5922 x [10.sup.5] 5.7451 x [10.sup.8] (1, 1/2, -1/2, 0) 4.8971 x [10.sup.5] 7.9121 x [10.sup.8] Our method with several choices of N, M [[alpha].sub.1], 8 10 [[beta].sub.1], [[alpha].sub.2], [[beta].sub.2] (0,0,0,0) 4.6928 x [10.sup.11] 2.5757 x [10.sup.14] (0, 1/2, 1, 1/2) 7.9964 x [10.sup.11] 4.4631 x [10.sup.14] (1, 1/2, -1/2, 0) 1.1243 x [10.sup.10] 7.7716 x [10.sup.14] FDM [7] with several choices of [h.sub.t], [h.sub.x] -- -- -- -- [h.sub.t] = [h.sub.t] = [h.sup.2.sub.x] [h.sup.2.sub.x] 0.0625 = 1/64 5.6574 x [10.sup.-4] 1.4910 x [10.sup.-4] Table 3. The MAEs of Example 3. Our method with several choices of (N, M) [[alpha].sub.1], [[beta].sub.1], (8,4) (8,8) [[alpha].sub.2], [[beta].sub.2] (0,0,0,0) 9.92325e-4 1.4186e-5 (1/2, 1/2, 1/2, 1/2) 1.46369e-3 2.4485e-5 (- 1/2, - 1/2, 1/2, 1/2) 1.46369e-3 2.4485e-5 Our method with several choices of (N, M) [[alpha].sub.1], [[beta].sub.1], (8,16) (8,20) (8, 24) [[alpha].sub.2], [[beta].sub.2] (0,0,0,0) 1.9261e-7 4.7128e-8 1.4803e-8 (1/2, 1/2, 1/2, 1/2) 3.7493e-7 9.4460e-8 3.0290e-8 (- 1/2, - 1/2, 1/2, 1/2) 3.7493e-7 9.4460e-8 3.0290e-8 Table 4. The MAEs of Example 4. Our method with several choices of [[alpha].sub.1], [[beta].sub.1], [[alpha].sub.2], [[beta].sub.2], (N,M) (1/2, 1/2, 1/2, 1/2,) (- 1/2, - 1/2, 1/2, 1/2,) (5,5) 3.6056e-5 7.9412 x [10.sup.-5] (10,20) 7.3896e-13 1.4495 x [10.sup.-12] Our method with several choices of [[alpha].sub.1], [[beta].sub.1], [[alpha].sub.2], [[beta].sub.2], (N,M) (1/2, 1/2, (- 1/2, - 1/2, 1, 0) (0, 0, 0, 0) 0, 0) (5,5) 3.6056e-5 7.9405 x [10.sup.-5] 5.5766 x [10.sup.-5] (10,20) 7.9581e-13 1.4353 x [10.sup.-12] 1.0090 x [10.sup.-12 Table 5. The MAEs of Example 5. Our method with several choices of (N, M, K) [[alpha].sub.1], [[beta].sub.1], (2,2,2) (4,4,4) [[alpha].sub.2], [[beta].sub.2], [[alpha].sub.3], [[beta].sub.3] (0,0,0,0,0,0) 2.8713e-3 2.50895 x [10.sup.-5] (1/2, 0, 1/2, 0, 1/2, 1/2) 2.3442e-3 2.49389 x [10.sup.-5] (1/2, - 1/2, 0, 0, 1/2, 1/2) 1.0031e-2 3.636267 x [10.sup.-5] Our method with several choices of (N, M, K) [[alpha].sub.1], [[beta].sub.1], (6, 6, 6) (8, 8, 4) [[alpha].sub.2], [[beta].sub.2], [[alpha].sub.3], [[beta].sub.3] (0,0,0,0,0,0) 5.23130 x [10.sup.-8] 4.2599e-11 (1/2, 0, 1/2, 0, 1/2, 1/2) 3.82665 x [10.sup.-8] 3.4212e-11 (1/2, - 1/2, 0, 0, 1/2, 1/2) 4.15992 x [10.sup.-8] 4.4601e-11 Table 6. The MAEs of Example 6. Our method with several choices of (N, M, K) [[alpha].sub.1], [[beta].sub.1], (2,2,2) [[alpha].sub.2], [[beta].sub.2], [[alpha].sub.3], [[beta].sub.3] (0,0,0,0,0,0) 1.0934 x [10.sup.-3] (1/2, 0, 1/2, 0, 1/2, 1/2) 2.1502 x [10.sup.-2] (1/2, - 1/2, 0, 0, 1/2, 1/2) 4.7764 x [10.sup.-2] Our method with several choices of (N, M, K) [[alpha].sub.1], [[beta].sub.1], (4,4,4) [[alpha].sub.2], [[beta].sub.2], [[alpha].sub.3], [[beta].sub.3] (0,0,0,0,0,0) 1.3013 x [10.sup.-4] (1/2, 0, 1/2, 0, 1/2, 1/2) 7.1109 x [10.sup.-5] (1/2, - 1/2, 0, 0, 1/2, 1/2) 9.9759 x [10.sup.-5] Our method with several choices of (N, M, K) [[alpha].sub.1], [[beta].sub.1], (6,6,6) (8,8,4) [[alpha].sub.2], [[beta].sub.2], [[alpha].sub.3], [[beta].sub.3] (0,0,0,0,0,0) 2.4714 x [10.sup.-7] 1.96E-10 (1/2, 0, 1/2, 0, 1/2, 1/2) 1.5411 x [10.sup.-7] 2.6118e-10 (1/2, - 1/2, 0, 0, 1/2, 1/2) 1.9505 x [10.sup.-7] 3.3079e-10

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Author: | Abd-Elkawy, Mohamed A.; Alqahtani, Rubayyi T. |
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Publication: | Mathematical Modeling and Analysis |

Article Type: | Report |

Date: | Jan 1, 2017 |

Words: | 5534 |

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