NUMERICAL SIMULATION OF TIME VARIABLE FRACTIONAL ORDER MOBILE-IMMOBILE ADVECTION-DISPERSION MODEL

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1 Romanian Reports in Physics, Vol. 67, No. 3, P , 2015 NUMERICAL SIMULATION OF TIME VARIABLE FRACTIONAL ORDER MOBILE-IMMOBILE ADVECTION-DISPERSION MODEL M.A. ABDELKAWY 1,a, M.A. ZAKY 2,b, A.H. BHRAWY 3,4,c, D. BALEANU 5,6,d 1 Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt 2 Department of Theoretical Physics, National Research Center, Cairo, Egypt 3 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia 4 Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt 5 Department of Mathematics and Computer Sciences, Canaya University, Esisehir Yolu 29.m, Anara, Turey 6 Institute of Space Sciences, P.O. BOX MG-23, RO , Magurele-Bucharest, Romania a melawy@yahoo.com; b ma.zay@yahoo.com c alibhrawy@yahoo.co.u; d dumitru@canaya.edu.tr Received October 27, 2014 Abstract. This paper reports a novel numerical technique for solving the time variable fractional order mobile-immobile advection-dispersion (TVFO-MIAD) model with the Coimbra variable time fractional derivative, which is preferable for modeling dynamical systems. The main advantage of the proposed method is that two different collocation schemes are investigated for both temporal and spatial discretizations of the TVFO-MIAD model. The problem with its boundary and initial conditions is then reduced to a system of algebraic equations that is far easier to be solved. Numerical results are consistent with the theoretical analysis and indicate the high accuracy and effectiveness of this algorithm. Key words: mobile-immobile advection-dispersion equation; collocation method; Jacobi-Gauss-Lobatto quadrature; Jacobi-Gauss-Radau quadrature; Coimbra fractional derivative. 1. INTRODUCTION Fractional differential equations model many phenomena in several fields such as fluid mechanics, chemistry [1, 2], biology [3], viscoelasticity [4], engineering, finance, and physics [5 7]. Therefore, we can see a several papers applying fractional calculus in control, signal processing, modeling, mechanics, electromagnetism, physics, chemistry, bioengineering, biology, medicine and in many other areas. Most fractional differential equations do not have exact analytic solutions, so approximation and numerical techniques must be used. Finite element methods have been introduced in [8 11] to obtain the numerical solutions of fractional differential equations. The numerical solutions based on finite difference methods for fractional differential

2 774 M.A. Abdelawy et al. 2 equations were reported in Refs. [12 14]. Moreover, several spectral algorithms are designed for fractional differential equations, see, for example, Refs. [15 19]. The fractional order derivative in these papers is constant fractional order, not variable fractional order. In this paper, we focus on presenting a novel spectral method for solving mobile-immobile advection-dispersion model with a Coimbra time variable fractional derivative. Advection-dispersion equation [20] is used to model many physical, chemical engineering, and earth sciences such as solute transport in surface and subsurface water flows, deeper river flows, ocean currents, streams and groundwater. Recently, researchers have also found that many dynamic processes exhibit fractional order behavior that may vary with time or space, which indicates that variable-order calculus is a natural candidate to provide an effective mathematical framewor for the description of complex dynamical problems, see, for instance Refs. [21 26]. Different definitions of variable fractional operators in various settings and various numerical techniques for solving variable fractional differential equations were discussed in Refs. [27 33]. In recent years there has been a high level of interest of employing spectral methods for numerically solving many types of integral and differential equations, due to their ease of applying them for finite and infinite domains [34 38]. The speed of convergence is one of the great advantages of spectral method. Besides that the spectral methods have exponential rates of convergence; they also have high level of accuracy [39 47]. The main idea of all versions of spectral methods is to express the spectral solution of the problem as a finite sum of certain basis functions (orthogonal polynomials or combination of orthogonal polynomials) and then to choose the coefficients in order to minimize the difference between the exact and numerical 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. The main aim of this article is to present a novel spectral scheme to obtain high accurate solution of time variable fractional order mobile-immobile advectiondispersion (TVFO-MIAD) model. Using shifted Jacobi-Gauss-Lobatto method in combination with shifted Jacobi-Gauss-Radau method, we introduce a fully collocation algorithm to numerically solve TVFO-MIAD model in Coimbra sense with different inds of non-local conditions. The widely applicable, efficiency, and high accuracy are the more advantages of the present method. The series expansion in shifted Jacobi polynomials N a i,j P (θ 1,ϑ 1 ) (x)p (θ 2,ϑ 2 ) (t), is the main step for solving the above mentioned problems. The expansion coefficients are then determined by reducing the TVFO-MIAD model equation with its boundary and initial

3 3 Numerical simulation of time variable fractional order advection-dispersion model 775 conditions to a system of algebraic equations for these coefficients. This system may be solved numerically in a step-by-step manner by using Newton s iterative method. Numerical results are consistent with the theoretical analysis and indicate the high accuracy and effectiveness of this algorithm. This paper is organized as follows. We present few relevant properties of shifted Jacobi polynomials in the next section. The mentioned scheme is implemented for the TVFO-MIAD model in Section 3. Section 4 is devoted to solve five test examples. Finally, some concluding remars are given in the last section. 2. PRELIMINARIES AND NOTATION We recall in this section some basic nowledge of orthogonal shifted Jacobi polynomials that are most relevant to spectral approximations. First, we introduce some properties about the standard Jacobi polynomials. A basic property of the Jacobi polynomials [48] is that they are the eigenfunctions to a singular Sturm- Liouville problem (1 x 2 )ψ (x) + [ϑ θ + (θ + ϑ + 2)x]ψ (x) + ( + θ + ϑ + 1)ψ(x) 0. (1) The Jacobi polynomials are generated from the three-term recurrence relation: where P (θ,ϑ) +1 (x) (a(θ,ϑ) x b (θ,ϑ) )P (θ,ϑ) (x) c (θ,ϑ) P (θ,ϑ) 1 P (θ,ϑ) 0 (x) 1, P (θ,ϑ) 1 (x) 1 2 (θ + ϑ + 2)x + 1 (θ ϑ), 2 a (θ,ϑ) (2 + θ + ϑ + 1)(2 + θ + ϑ + 2), 2( + 1)( + θ + ϑ + 1) (x), 1, b (θ,ϑ) (ϑ 2 θ 2 )(2 + θ + ϑ + 1) 2( + 1)( + θ + ϑ + 1)(2 + θ + ϑ), c (θ,ϑ) ( + θ)( + ϑ)(2 + θ + ϑ + 2) ( + 1)( + θ + ϑ + 1)(2 + θ + ϑ). The Jacobi polynomials satisfy the relations P (θ,ϑ) ( x) ( 1) P (θ,ϑ) (x), P (θ,ϑ) ( 1) ( 1) Γ( + ϑ + 1). (2)!Γ(ϑ + 1) Moreover, the qth derivative of P (θ,ϑ) (x), can be obtained from D (q) P (θ,ϑ) Γ(j + θ + ϑ + q + 1) (x) 2 q Γ(j + θ + ϑ + 1) P (θ+q,ϑ+q) q (x). (3)

4 776 M.A. Abdelawy et al. 4 Let us define the space L 2 with the weight function w (θ, ϑ) (x) (1 w (θ,ϑ) x) θ (1 + x) ϑ. The norm and the inner product of this weighted space are given by: u w (θ,ϑ) (u,u) 1 2 w (θ,ϑ), (u,v) w (θ,ϑ) The set of P (θ,ϑ) (x) is a complete L 2 -orthogonal system with w (θ,ϑ) 1 1 u(x) v(x) w (θ,ϑ) (x) dx. (4) P (θ,ϑ) 2 θ+ϑ+1 Γ( + θ + 1) Γ( + ϑ + 1) w (θ,ϑ) h (2 + θ + ϑ + 1) Γ( + 1) Γ( + θ + ϑ + 1). (5) If we define the shifted Jacobi polynomial of degree by P (θ,ϑ) (θ,ϑ) L, (x) P ( 2x L 1), L > 0, and in virtue of (2) and (3), then it can be easily shown that D q P (θ,ϑ) L, (0) ( 1) q Γ( + ϑ + 1)( + θ + ϑ + 1) q L q. (6) Γ( q + 1)Γ(q + ϑ + 1) D q P (θ,ϑ) L, (L) Γ( + θ + 1)( + θ + ϑ + 1) q L q Γ( q + 1)Γ(q + θ + 1). (7) D m P (θ,ϑ) Γ(m + + θ + ϑ + 1) L, (x) L m Γ( + θ + ϑ + 1) P (θ+m,ϑ+m) L, m (x). (8) Next, let w (θ,ϑ) L (x) (L x) θ x ϑ, then we define the weighted space L 2 [0,L] w (θ,ϑ) L in the usual way, with the following inner product and norm, (u,v) w (θ,ϑ) L L 0 u(x)v(x)w (θ,ϑ) L (x)dx, v w (θ,ϑ) L (v,v) 1 2 w (θ,ϑ) L. (9) The set of shifted Jacobi polynomials forms a complete L 2 [0, L]-orthogonal system. Moreover, due to (9), we w (θ,ϑ) L have P (θ,ϑ) L, 2 ( L w (θ,ϑ) L 2 )θ+ϑ+1 h (θ,ϑ) h (θ,ϑ) L,. (10) N,j We denote by x (θ,ϑ) N,j, 0 j N, the nodes of the standard Jacobi-Gauss interpolation on the interval [ 1, 1]. Their corresponding Christoffel numbers are ϖ (θ,ϑ), 0 j N. The nodes of the shifted Jacobi-Gauss interpolation on the interval [0,L] are the zeros of P (θ,ϑ) L,N+1 Clearly x (θ,ϑ) L,N,j L 2 (x(θ,ϑ) N,j ϖ (θ,ϑ) L,N,j ( L 2 )θ+ϑ+1 ϖ (θ,ϑ) N,j (x), which we denote by x(θ,ϑ) L,N,j, 0 j N. + 1), and their corresponding Christoffel numbers are, 0 j N. Let S N[0,L] be the set of polynomials of degree at most N. Thans to the property of the standard Jacobi-Gauss quadrature,

5 5 Numerical simulation of time variable fractional order advection-dispersion model 777 it follows that for any φ S 2N+1 [0,L], L 0 (L x) θ x ϑ φ(x)dx ( L 2 )θ+ϑ ( L 2 )θ+ϑ+1 N j0 j0 ϖ (θ,ϑ) L,N,j φ(x(θ,ϑ) L,N,j ). (1 x) θ (1 + x) ϑ φ( L (x + 1))dx 2 ϖ (θ,ϑ) N,j φ(l 2 (x(θ,ϑ) N,j + 1)) Consequently, the qth-order derivative of shifted Jacobi polynomial can be written in terms of the shifted Jacobi polynomials themselves as where D q P (θ,ϑ) L, q (x) i0 (11) C q (,i,θ,ϑ)p (θ,ϑ) (x), (12) C q (,i,θ,ϑ) ( + λ) q( + λ + q) i (i + θ + q + 1) i q Γ(i + λ) L q ( i q)! Γ(2i + λ) + i + q, + i + λ + q, i + θ F 2 i + θ + q + 1, 2i + λ + 1 ;1. (13) 3. SHIFTED JACOBI COLLOCATION METHOD In this section, we derive the shifted Jacobi collocation method and describe its implementation for solving the TVFO-MIAD model subject to different inds of boundary conditions. Here, we approximate the solution of TVFO-MIAD model for space and time variables by using a shifted Jacobi Gauss-Lobatto method in combination with shifted Jacobi Gauss-Radau collocation method. By using the collocation method, exponential convergence can be achieved for approximating the solution of the proposed problem. The computerized mathematical algorithm is the main ey to apply this method for solving the proposed problem INITIAL-BOUNDARY CONDITIONS The main objective of this subsection is to develop the shifted Jacobi collocation method to the TVFO-MIAD equation: a 1 D t u + a 2 D ν(x,t) t u a 3 D 2 x u + a 4 D x u + R(x,t), (x,t) [0,L] [0,T ], (14)

6 778 M.A. Abdelawy et al. 6 subject to the initial condition and the boundary conditions u(x,0) g 0 (x), x [0,L], (15) u(0,t) g 1 (t), u(l,t) g 2 (t), t [0,T ], (16) where, a 1, a 2, a 3, and a 4, are constants, while R(x,t), g 0 (x), g 1 (t), and g 2 (t) are given functions. Here, we use the set of shifted Jacobi Gauss-Lobatto points for the space approximation while, shifted Jacobi Gauss-Radau points will be used for the time variable. Now, we outline the main steps of the shifted Jacobi collocation method for solving initial-boundary TVFO-MIAD equation. The approximate solution has a series of the form, u(x,t) a i,j P (θ 1,ϑ 1 ) (x)p (θ 2,ϑ 2 ) (t) 0 (x,t), where f i,j 0 (x,t) P (θ 1,ϑ 1 ) (x)p (θ 2,ϑ 2 ) (t). Furthermore, the approximation of the spatial partial derivative D x u(x,t) can be computed as D x u(x,t) a i,j D x (P (θ 1,ϑ 1 ) (x))p (θ 2,ϑ 2 ) (t) 1 (x,t), where f i,j 1 (x,t) D x(p (θ 1,ϑ 1 ) (x))p (θ 2,ϑ 2 ) (t). The approximation of the time partial derivative D t u(x,t) can be computed as (17) (18) D t u(x,t) a i,j P (θ 1,ϑ 1 ) (x)d t (P (θ 2,ϑ 2 ) (t)) 2 (x,t), (19) where f i,j 2 (x,t) P (θ 1,ϑ 1 ) (x)d t (P (θ 2,ϑ 2 ) (t)).

7 7 Numerical simulation of time variable fractional order advection-dispersion model 779 Similar steps can be applied to the second spatial partial derivative to get D 2 xu(x,t) a i,j Dx(P 2 (θ 1,ϑ 1 ) (x))p (θ 2,ϑ 2 ) (t) 3 (x,t), (20) where f i,j 3 (x,t) D2 x(p (θ 1,ϑ 1 ) (x))p (θ 2,ϑ 2 ) (t). Moreover, the fractional derivative can be computed by D ν(x,t) t u(x,t) a i,j P (θ 1,ϑ 1 ) (x)d ν(x,t) 4 (x,t), t (P (θ 2,ϑ 2 ) (t)) (21) where f i,j 4 (x,t) P (θ 1,ϑ 1 ) (x)d ν(x,t) t (P (θ 2,ϑ 2 ) (t)). Here, we use fractional derivatives in the Coimbra sense. Definition 3.1 The Coimbra fractional derivatives of variable order ν(x, t) is defined as (see [49] for more details) 1 t f(t) Γ(1 ν(x,t)) D ν(x,t) t < ν(x,t) 1, t > 0, df(τ) dτ (t τ) ν(x,t) dτ + f(0+ ) f(0 ) Γ(1 ν(x,t))t ν(x,t), (22) if f(0 + ) f(0 ), the Coimbra fractional derivative related to the Caputo fractional derivative. Therefore, adopting (17)-(22), enable one to write (14)-(16) in the form: a 1 N 2 (x,t) + a 2 4 (x,t) a 3 + a 4 N 3 (x,t) + R(x,t) 1 (x,t), (23)

8 780 M.A. Abdelawy et al. 8 while the numerical treatments of initial and boundary conditions are u(x,0) u(0,t) u(l,t) 0 (x,0) g 0(x), 0 (0,t) g 1(t), 0 (L,t) g 2(t). (24) In the proposed shifted Jacobi collocation method the residual of (14) is set to zero at ((N 1)M) of collocation points. Moreover, the initial-boundary conditions in (24) will be collocated at collocation points. First, we have (M(N 1)) algebraic equations for (M + 1)(N + 1) unnown of a i,j, where, F i,j r,s a i,j R(x L N,r,t T M,s), r 1,,N 1; s 1,,M, (25) F i,j r,s a 1 f i,j 2 (xl N,r,t T M,s)+a 2 f i,j 4 (xl N,r,t T M,s) a 3 f i,j 3 (xl N,r,t T M,s) a 4 f i,j 1 (xl N,r,t T M,s), and due to initial condition, we have other (N 1) algebraic equations 0 (xl N,r,0) g 0 (x L N,r), r 1,,N 1, (26) using the boundary conditions, we have (2M + 2) algebraic equations 0 (0,tT M,s) g 1 (t T M,s), 0 (L,tT M,s) g 2 (t T M,s), s 0,,M, (27)

9 9 Numerical simulation of time variable fractional order advection-dispersion model 781 and this in turn, yields (M + 1)(N + 1) algebraic equations F i,j r,s a i,j R(x L N,r,t T M,s), r 1,,N 1; s 1,,M, 0 (xl N,r,0) g 0 (x L N,r), r 1,,N 1, 0 (0,tT M,s) g 1 (t T M,s), s 0,,M, 0 (L,tT M,s) g 2 (t T M,s), s 0,,M. (28) 3.2. NEUMANN BOUNDARY CONDITIONS In the present subsection, we develop the shifted Jacobi collocation method to numerically solve the TVFO-MIAD equation with Neumann boundary conditions a 1 D t u + a 2 D ν(x,t) t u a 3 D 2 x u + a 4 D x u + R(x,t), (x,t) [0,L] [0,T ], (29) subject to the initial condition and the Neumann boundary conditions a 1 N u(x,0) g 0 (x), x [0,L], (30) D x u(0,t) g 1 (t), D x u(l,t) g 2 (t), t [0,T ]. (31) Based on the information mentioned in the last subsection, we obtain 2 (x,t) + a 2 4 (x,t) a 3 + a 4 N 3 (x,t) + R(x,t) 1 (x,t), (32)

10 782 M.A. Abdelawy et al. 10 in addition to the previous equation, we obtain the following u(x,0) D x u(0,t) D x u(l,t) 0 (x,0) g 0(x), 1 (0,t) g 1(t), 1 (L,t) g 2(t). Then a similar analysis to that given in subsection (3.1), yields (M +1)(N +1) algebraic equations F i,j r,s a i,j R(x L N,r,t T M,s), r 1,,N 1; s 1,,M, 0 (xl N,r,0) g 0 (x L N,r), r 1,,N 1, 1 (0,tT M,s) g 1 (t T M,s), s 0,,M, 1 (L,tT M,s) g 2 (t T M,s), s 0,,M. (33) (34) 3.3. MIXED BOUNDARY CONDITIONS This subsection is devoted to develop an algorithm based on the shifted Jacobi collocation method to numerically solve the TVFO-MIAD equation subject to two non-local boundary conditions; namely a 1 D t u + a 2 D ν(x,t) t u a 3 D 2 x u + a 4 D x u + R(x,t), (x,t) [0,L] [0,T ], (35) subject to the initial condition and the mixed boundary conditions a2 u(x,0) g 0 (x), x [0,L], (36) D x u(0,t) g 1 (t), u(x,t)dx g 2 (t), 0 a 1 a 2 L, t [0,T ]. (37) a 1

11 11 Numerical simulation of time variable fractional order advection-dispersion model 783 Using the previous initial and mixed boundary conditions, we derive the following equations u(x,0) 0 (x,0) g 0(x) a2 D x u(0,t) a 1 u(x,t)dx 1 (0,t) g 1(t), a2 a i,j ( (P (θ 1,ϑ 1 ) a 1 (x))dx)p (θ 2,ϑ 2 ) (t) g 2 (t), accordingly, the following system of (M + 1)(N + 1) algebraic equations is obtained F i,j r,s a i,j R(x L N,r,t T M,s), r 1,,N 1; s 1,,M, 0 (xl N,r,0) g 0 (x L N,r), r 1,,N 1, 1 (0,tT M,s) g 1 (t T M,s), s 0,,M, a2 a i,j ( a 1 P (θ 1,ϑ 1 ) (x)dx)p (θ 2,ϑ 2 ) (t T M,s) g 2 (t T M,s), s 0,,M. Note: The systems (28), (34) and (39) of algebraic equations can be solved using Newton s iterative method. (38) (39) 4. NUMERICAL RESULTS In the present section, the applicability of our method will be tested by five examples. The programs used here have been coded in Mathematica version 9. The results obtained using our technique are also introduced for comparison. The highly accurate results produced in this paper are measured via the absolute error given by E(x,t) u(x,t) u N (x,t), (x,t) [0,L] [0,T ], (40) where u(x,t) and u N (x,t) are the exact solution and the numerical solution at the point (x,t), respectively, and the following maximum absolute errors (MAEs) MAEs Max{E(x,t) : (x,t) [0,L] [0,T ]}. (41)

12 784 M.A. Abdelawy et al. 12 Table 1 Comparing absolute errors of problem (42) for our method at N M 10 and the method in [50]. x t Our method Method in [50] where 4.1. TEST PROBLEM 1 Consider the following equation [50] D t u + D ν(x,t) t u D 2 x u D x u + R(x,t), (x,t) [0,1] [0,1], (42) ν(x,t) 1 0.5e t( x), ( (x 1) 2 x 2 t 1 ν(x,t) R(x,t) t((x 1)x(2x 7) 1)+ Γ(2 ν(x,t)) ) (x 1)x(x(x + 3) 14) 2, with the initial-boundary conditions u(0,t) 0, u(1,t) 0, u(x,0) 10x 2 (1 x) 2, (x,t) [0,1] [0,1]. (43) the exact solution may be given by u(x,t) 10(t + 1)x 2 (1 x) 2, (x,t) [0,1] [0,1]. (44) From the first loo in Tables 1 and 2, we see the highly accurate results based on absolute errors and MAEs in our paper compared with those obtained using implicit finite difference method [50]. In Fig. 1, we display the curves of numerical u N (x,t) and exact u(x,t) solutions of equation (42), with N M 10 and θ 1 θ 2 ϑ 1 ϑ 2 0 at three different values of t TEST PROBLEM 2 Consider the following TVFO-MIAD equation [50] D t u + D ν(x,t) t u Dx 2 u D x u + R(x,t), (x,t) [0,1] [0,1], (45)

13 13 Numerical simulation of time variable fractional order advection-dispersion model 785 Table 2 Comparison based on MAEs of problem (42). Our method θ 1 ϑ 1 1 2, θ 1 ϑ 1 1 2, θ 1 ϑ 1 1 2, N M θ 2 ϑ 2 0, θ 2 ϑ 2 1 2, θ 2 ϑ 2 1 2, Implicit finite difference method [50] h τ MAEs Fig. 1 x-directions curves of numerical u N (x,t) and exact u(x,t) solutions of equation (42).

14 786 M.A. Abdelawy et al. 14 Table 3 Comparison based on MAEs of the problem (45). Our method θ 1 θ 2 1 2, θ 1 ϑ 1 1 2, θ 1 ϑ 1 1 2, N M ϑ 1 ϑ 2 0, θ 2 ϑ 2 0, θ 2 ϑ 2 1 2, Implicit finite difference method [50] h τ MAEs where ν(x,t) sin(x)cos(tx), 5(x 1)xt1 ν(x,t) R(x,t) 5 ( t(2x 3) + x 2 + x 3 ) Γ(2 ν(x,t)) with the initial-boundary conditions u(0,t) 0, u(1,t) 0, u(x,0) 5x(1 x), (x,t) [0,1] [0,1]. (46) The exact solution may be given by u(x,t) 5(t + 1)x(1 x), (x,t) [0,1] [0,1]. (47) MAEs of u(x,t) related to (45) are introduced in Table 3 using shifted Jacobi collocation method with two choices of N and M, and are compared by the results in Ref. [50]. We plot in Fig. 2, the x-direction curve of the absolute error of problem (45), where N M 8 and θ 1 θ 2 ϑ 1 ϑ where 4.3. TEST PROBLEM 3 In this example, we consider the problem [51] D ν(x,t) t u D 2 x u + R(x,t), (x,t) [0,1] [0,1], (48) ν(x,t) 1 (sin(tx) + 2), 4 R(x,t) 20(t + 1) 2 (3x 1) 20(x 1)x2 t 1 ν(x,t) ( ν(x,t) + t + 2), Γ(3 ν(x,t)) with the initial-boundary conditions u(0,t) 0, u(1,t) 0, u(x,0) 10x 2 (1 x), (x,t) [0,1] [0,1]. (49)

15 15 Numerical simulation of time variable fractional order advection-dispersion model 787 Fig. 2 x-direction curve of the absolute error of problem (45). Table 4 Comparison based on MAEs of problem (48). Our method θ 1 ϑ 1 0, θ 1 ϑ 1 1 2, θ 1 ϑ 1 1 2, N M θ 2 ϑ 2 1 2, θ 2 ϑ 2 0, θ 2 ϑ 2 1 2, [51] (h,τ) ( 1 4, 1 16 ) ( 1 8, 1 64 ) ( 1 16, ) MAEs The exact solution may be given by u(x,t) 10(t + 1) 2 x 2 (1 x), (x,t) [0,1] [0,1]. (50) A comparison based on the MAEs obtained by our method and those in [51] of the problem (48) is given in Table 4. We setch in Fig. 3, the space-graph absolute error of equation (48), where N M 8, θ 1 ϑ and θ 2 ϑ 2 1. where 4.4. TEST PROBLEM 4 Consider the following TVFO-MIAD with Neumann boundary conditions D ν(x,t) t u D t u D 2 x u u + R(x,t), (x,t) [0,L] [0,T ], (51) ν(x,t) 0.005sin(x)cos(tx) + 0.8,

16 788 M.A. Abdelawy et al. 16 Fig. 3 The space-graph of absolute error of problem (48).! 2t1 ν(x,t) 2 R(x, t) t sin(2πx) + 4π t + t 2, Γ(3 ν(x, t)) with the initial and Neumann boundary conditions u(x, 0) 0, Dx u(0, t) 2πt2, Dx u(1, t) 2πt2, (x, t) [0, 1] [0, 1]. (52) The exact solution may be given by u(x, t) t2 sin(2πx), (x, t) [0, 1] [0, 1]. (53) In Table 5, we exhibit the MAEs of problem (51) for various choices of N and Table 5 MAEs of problem (51). N M θ1 ϑ1 0, θ2 ϑ2 0, Our method θ1 ϑ1 0, θ2 ϑ2 12, θ1 ϑ1 12, θ2 ϑ2 0, θ1 ϑ1 12, θ1 ϑ1 12,

17 17 Numerical simulation of time variable fractional order advection-dispersion model 789 Fig. 4 x-directions curves of exact and numerical solutions of problem (51). M. We plot in Fig. 4, the x-directions curves of of exact and numerical solutions of problem (51), where N M 20, θ 1 ϑ 1 0, and θ 2 ϑ where 4.5. TEST PROBLEM 5 Finally, we discuss the following equation with mixed boundary conditions D ν(x,t) t u D 2 x u + v 2 u + R(x,t), (x,t) [0,L] [0,T ], (54) ν(x,t) 1 (sin(tx) + 2), ( 4 ) R(x,t) e t Γ(1 ν(x,t),t) sin(πx) Γ(1 ν(x,t)) et sin(πx) + π 2 + 2, where Γ(a,t) is the incomplete gamma function of t, with the initial and non-local conditions u(x,0) sin(πx), 1 The exact solution may be given by M. 0 u(x,t) 2et π, D xu(0,t) πe t, (x,t) [0,1] [0,1]. (55) u(x,t) e t sin(πx), (x,t) [0,1] [0,1]. (56) In Table 6, we exhibit the MAEs of problem (54) for various choices of N and

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