New group iterative schemes in the numerical solution of the two-dimensional time fractional advection-diffusion equation

Transcription

1 Balasim & Hj. Mohd. Ali Cogent Mathematics 7 https//doi.org/./ COMPUTATIONAL SCIENCE RESEARCH ARTICLE New group iterative schemes in the numerical solution of the two-dimensional time fractional advection-diffusion equation Received April 7 Accepted 7 November 7 First Published 6 December 7 *Corresponding author Alla Tareq Balasim School of Mathematical Sciences Universiti Sains Malaysia Penang Malaysia; Department of Mathematics College of Basic Education University of Al-mustansiria Baghdad Iraq alkhazrejy@yahoo.com Reviewing editor Fawang Liu Queensland University of Technology Australia Additional information is available at the end of the article Alla Tareq Balasim * and Norhashidah Hj. Mohd. Ali Abstarct Numerical schemes based on small fixed-size grouping strategies have been successfully researched over the last few decades in solving various types of partial differential equations where they have been proven to possess the ability to increase the convergence rates of the iteration processes involved. The formulation of these strategies on fractional differential equations however is still at its infancy. Appropriate discretization formula will need to be derived and applied to the time and spatial fractional derivatives in order to reduce the computational complexity of the schemes. In this paper the design of new group iterative schemes applied to the solution of the D time fractional advection-diffusion equation are presented and discussed in detail. The Caputo fractional derivative is used in the discretization of the fractional group schemes in combination with the Crank Nicolson difference approximations on the standard grid. Numerical experiments are conducted to determine the effectiveness of the proposed group methods with regard to execution times number of iterations and computational complexity. The stability and convergence properties are also presented using a matrix method with mathematical induction.the numerical results will be proven to agree with the theoretical claims. ABOUT THE AUTHORS Alla Tareq Balasim is a lecture at the Department of Mathematics Almustansiria University Iraq. He received his Master in Mathematics from the Department of Mathematics Baghdad University Iraq. He is currently a PhD student and his research interests are numerical deferential equations and dynamical system. Norhashidah Hj. Mohd. Ali is a Professor at the School of Mathematical Sciences Universiti Sains Malaysia. She received her PhD in Industrial Computing from Universiti Kebangsaan Malaysia. Her areas of expertise include numerical differential equations and parallel numerical algorithms. She has over 5 years teaching and research experiences and has written more than articles in these areas. PUBLIC INTEREST STATEMENT Many phenomena in science and engineering can be modelled as two dimensional fractional advection-diffusion equations satisfying appropriate initial and boundary conditions. With the advent of computing technology effective numerical methods have been extensively formulated in solving these equations due to their simplicity and accuracy. In particular finite difference formulas are oftenly used to numerically discretize the differential equations to produce sparse systems of linear equations which may be suitably solved by iterative solvers. However iterative solvers have the disadvantage of being too slow to converge which may increase the computation timings. To overcome this problem suitable small fixed-size grouping strategies are applied to the mesh points of the solution domain following the discretization of the differential equation which results in schemes with faster convergence and lower computational complexity with comparable accuracies. This is very useful to engineers and scientists who are involved in time consuming simulation modeling processes. 7 The Authors. This open access article is distributed under a Creative Commons Attribution CC-BY. license. Page of 9

2 Balasim & Hj. Mohd. Ali Cogent Mathematics 7 https//doi.org/./ Subjects Analysis - Mathematics; Applied Mathematics; Computer Mathematics Keywords group methods; two-dimensional problem; time fractional advection-diffusion; Caputo fractional derivative; Crank Nicolson difference schemes; stability & convergence analysis. Introduction Fractional calculus which is the calculus of integrals and derivatives in random order dates back as far as the more popular integer calculus and has been gaining significant interest over the past few years with its history and development being explored in detail by Oldham and Spanier 97 Miller and Ross 993 Samko Kilbas and Marichev 993 and Podlubny 99. Fractional differential equations FDEs can be used to model many problems in a wide field of applications. They are defined as equations that utilize fractional derivatives and considered as powerful tools that can describe the memory and hereditary characteristics of various materials. Several researchers have explored the use of FDEs in the fields of chemistry Gorenflo Mainardi Moretti Pagnini & Paradisi ; Seki Wojcik & Tachiya 3 physics Henry & Wearne ; Metzler Barkai & Klafter 999; Metzler & Klafter ; Wyss 96 and other scientific and engineering spheres Baeumer Benson Meerschaert & Wheatcraft ; Benson Wheatcraft & Meerschaert ; Cushman & Ginn ; Mehdinejadiani Naseri Jafari Ghanbarzadeh & Baleanu 3. Since there are at most no exact solutions to the majority of fractional differential equations it is necessary to resort to approximation and numerical methods Abdelkawy Zaky Bhrawy & Baleanu 5; Balasim & Ali 5; Baleanu Agheli & Al Qurashi 6; Bhrawy & Baleanu 3. Over the past decade there has been an influx of numerical methods development for solving various types of FDEs Agrawal ; Ali Abdullah & Mohyud-Din 7; Chen Deng & Wu 3; Chen & Liu ; Chen Liu Anh & Turner ; Chen Liu & Burrage ; Leonenko Meerschaert & Sikorskii 3; Li Zeng & Liu ; Liu Zhuang Anh Turner & Burrage 7; Shen Liu & Anh ; Sousa & Li 5; Su Wang & Wang ; Uddin & Haq ; Zhang Huang Feng & Wei 3; Zheng Li & Zhao ; Zhuang Gu Liu Turner & Yarlagadda ; Zhuang Liu Anh & Turner 9. Chen et al. used implicit and explicit difference techniques to solve time fractional reaction-diffusion equations while Liu et al. 7 proposed for these techniques to be employed in solving the space-time fractional advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative and the first-order and second-order space derivatives by the Riemman Liouville fractional derivatives. The use of radial basis function RBF approximation method was discussed in Uddin and Haq to solve the time fractional advection-dispersion equation in a bounded domain. The application of finite element method was also seen in Zheng et al. in solving the space fractional advection-diffusion equation under non-homogeneous initial boundary conditions. In Chen et al. a numerical method with first-order temporal accuracy and second-order spatial accuracy was established for solving the variable order Galilei advection-diffusion equation with a nonlinear source term while Zhuang et al. 9 used the explicit and implicit Euler approximation to solve a variable order fractional advection-diffusion equation with a nonlinear source term. A new numerical solution for the D fractional advection-dispersion equation with variable coefficients in a finite field was also introduced by Chen and Liu. In Shen et al. proposed the explicit and implicit finite difference approximations for the space-time Riesz Caputo fractional advection-diffusion equation where the implicit scheme was proven to be unconditionally stable while the explicit scheme exhibits a conditionally stable property. The development of an implicit meshless method was also seen in Zhuang et al. in solving the time-dependent fractional advection-diffusion equation where a discretized system of equations was obtained using the moving least squares MLS meshless shape functions. In general finding numerical solutions to FDEs using iterative finite difference schemes is not a straightforward process and can be a challenging task due to several reasons. Firstly all the earlier solutions have to be saved if the current solution is to be computed making the calculations even more complex and costly in terms of CPU usage time in cases where traditional point implicit difference approaches are used. Secondly although the point implicit techniques are stable a significant Page of 9

3 Balasim & Hj. Mohd. Ali Cogent Mathematics 7 https//doi.org/./ amount of CPU usage time is required when many unknowns are involved. In recent decades grouping strategies have been proven to possess characteristics that are able to reduce the spectral radius of the generated matrix resulting from the finite difference discretization of the differential equation and therefore increase the convergence rates of the iterative algorithms. They have been shown to reduce the computational timings compared to their point wise counterparts in solving several types of partial differential equation Ali & Kew ; Evans & Yousif 96; Kew & Ali 5; Ng & Ali ; Othman & Abdullah ; Tan Ali & Lai ; Yousif & Evans 995. These group methods are also easy to implement and suitable to be implemented on parallel computers due to their explicit nature. However till date these strategies have not been tested on solving FDEs particularly for the D cases. Therefore in this study the formulation of new group iterative methods is presented in solving the following Dl time fractional advection diffusion equation α u t α a u x x + a y u y b u x x b u y + f x y t y where <α< a x a y b x b y are positive constants and fx y t is nonhomogeneous term subjected to the following initial and Dirichlet boundary conditions ux y gx y u y t g y t u y t g y t ux t g 3 x t ux t g x t This equation plays an important role in describing transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns Zhuang Gu Liu Turner & Yarlagadda. This paper is outlined as follows. Section presents the proposed group iterative methods obtained from the Crank Nicolson difference approximation followed by the stability and convergence analysis of the difference schemes in Sections 3 and respectively. Section 5 presents the discussion on the computational effort involved in solving Equation using the proposed methods with regard to the arithmetic operations for each iteration. Finally the results of the numerical experiments are presented and discussed in Section 6.. Standard approximation schemes for fractional advection-diffusion equations The Caputo fractional derivative D α of the order-α is expressed as follows Li Qian & Chen D α where Γ is the Euler Gamma function. Further details on the definitions and properties of fractional derivatives are available in Podlubny 99. We need to apply appropriate finite difference approximations to the time and space derivaties of let h > be the space step and k > be the time step. The domain is assumed to be uniform in both x and y directions. Define x i ih y j jh {i j... n} and a mesh size of h where n is n an arbitrary positive integer and t k kτ k... l. various approximations formulas could be obtained for at the point of x i y j t k. By taking the average of the central difference approximations to the left side of at the points i j k + and i j k the Caputo time fractional approximation can be transformed to the following form Karatay Kale & Bayramoglu 3 α ux i y j t k+ t α where σ Γm α x f m t dt α> m <α<m m N x > α m+ x t w u k + k s +Oτ α [ wk s+ w k s ] u s ij w k u i j τ α Γ α w s σs + α s α. uk+ ij u + σ k ij α Using 3 in combination with the second-order Crank Nicolson difference scheme for the right side of will result in the following fractional standard point FSP formula 3 Page 3 of 9

4 Balasim & Hj. Mohd. Ali Cogent Mathematics 7 https//doi.org/./ w u k + k [ ] s wk s+ w k s u s w ij k u i j a x u k+ + a y u k+ b y u k+ i j uk+ ij +u k+ i+j h ij uk+ ij +u k+ ij+ h ij+ uk+ ij h + u k i j uk ij +uk i+j h + u k ij uk ij +uk ij+ h b x u k+ uk+ ij u + σ k ij α i+j uk+ i j Equation can be simplified to become as follows + s x u k+ ij s x + c x uk+ i j +s x c x uk+ i+j +s y + c y uk+ ij c y uk+ + ij+ α w s s x y u k ij +s x + c x uk i j h + uk i+j uk i j h + uk ij+ uk ij + f k+ + Oτ α + x + y. h i j 5 + s x c x uk i+j +s y + c y uk ij +s y c y uk + ij+ α w k u ij + α k [ s w ] k s w k s+ u s + m f k+ ij ij Figure. Computational molecule of the standard point approximation. ij+k+ s y/-c y/ i-jk+ ijk+ i+jk+ s x/+c x/ i j-k+ +s x+sx y i+jk s y/-c y/ s x/-c x/ Time level k+ s y/+c y/ i-jk ijk i+jk s x/+c x/ ij-k -w -s x-s y s x/-c x/ Time level k s y/+c y/ ijk- w -w Time level k- ijk- w -w 3 Time level k-.. ij. w k--w k Time level ij w k Time level Page of 9

5 Balasim & Hj. Mohd. Ali Cogent Mathematics 7 https//doi.org/./ where m α t α Γ α w s s + α s α s x a x m h s y a y m Figure shows the computational molecule for Equation 5. c h x b m x h c y b m y. h.. Fractional explicit group method In formulating the fractional explicit group FEG method 5 is applied to any group of four points in the solution domain to generate a system of equation as follows D a b a D b b D a b a D sx where D + s x a c x R α w s x s y u ij u i+j u i+j+ u ij+ rhs ij rhs i+j rhs i+j+ rhs ij+ b sy c y 6 sx a + c sy x b + c y rhs ij a u k+ + i j uk +b i j uk+ + ij uk +a ij uk + b i+j uk ij+ k + α w k u + ij Ruk + ij α [w k s w k s+ ]us + m f k+ i.j ij s rhs i+j a u k+ + i+j uk +b i+j uk+ + i+j uk +b i+j uk + a i+j+ uk ij k + α w k u + i+j Ruk + i+j α [w k s w k s+ ]us + m f k+ i+j i+j s rhs i+j+ a u k+ + i+j+ uk +b i+j+ uk+ + i+j+ uk +b i+j+ uk + a i+j uk ij+ k + α w k u + i+j+ Ruk + i+j+ α [w k s w k s+ ]us + m f k+ i+j+ i+j+ rhs ij+ a u k+ + i j+ uk +b i j+ uk+ + ij+ uk +a ij+ uk + b i+j+ uk ij k + α w k u + ij+ Ruk + ij+ α [w k s w k s+ ]us + m f k+. ij+ ij+ s s Mathematical software can be used to easily invert 6 to obtain the FEG formula Figure. Grouping of the points for the FEG method n. Page 5 of 9

6 Balasim & Hj. Mohd. Ali Cogent Mathematics 7 https//doi.org/./ u ij u i+j u i+j+ u ij+ where r r r r 3 r r 5 r r r 6 r 7 r r r 5 r r 9 r r rhs ij rhs i+j rhs i+j+ rhs ij+ r 56 c + x c + y c y + 5s + s + x y 3s + s y x + 6[9s + x s3 x + + s y + 3s y + s x + s y + 39s y + 6s x + s y + s + y 3s3 y ] +c x c y + + 3s + s + x y 5s + s y x r + s 6 x c x + cy + + 3s + s + x y 3s + s y x r c 6 x s x c x c + y + 3s + s + x y 5s + s y x r 3 c x s x c y s y + s x r c 6 y s y c + x c + y + 5s + s + x y 3s + s y x r 5 c 6 x + s x c x c + y + 3s + s + x y 5s + s y x r 6 c x + s x c y s y + s x r 7 c x + s x + s x c y + s y r 6 r 9 c y + s y c x + c y + + 5s x + s y + 3s y + s x c x s x + s x c y + s y. 7 Figure shows the construction of blocks of four points in the solution domain for the case n. Note that if n is even there will be ungrouped points near the upper and right sides of the boundary. The FEG method proceeds with the iterative evaluation of solutions in these blocks of four points using Equation 7 throughout the whole solution domain until convergence is achieved. For the case of even n the solutions at the ungrouped points near the boundaries are computed using 5... The fractional modified explicit group method In this new method we consider the nodal points with grid size spacing h. The standard fractional formula is generated through the application specific finite difference approximations with n h-spaced points. Using the Caputo time fractional approximation 3 at the left-hand side of and the second order Crank Nicolson difference scheme with h-spaced points at the right hand side of the following approximation formula is obtained w u k + k [ ] s wk s+ w k s u s w ij k u i j a x u k+ i j uk+ ij +u k+ i+j h + a y u k+ ij uk+ ij +u k+ ij+ h b y u k+ ij+ uk+ ij h + u k i j uk ij +uk i+j h + u k ij uk ij +uk ij+ h b x u k+ uk+ ij u + σ k ij α i+j uk+ i j h + uk i+j uk i j h + uk ij+ uk ij + f k+ + Oτ α + x + y. h i j On simplification we obtain the following Page 6 of 9

7 Balasim & Hj. Mohd. Ali Cogent Mathematics 7 https//doi.org/./ s + s x y u k+ ij s x + c x uk+ +s x i j c x c y uk+ + ij+ α w s + s x y uk+ +s y i+j + c y uk+ ij u k +s x ij + c x uk i j + s x c x uk +s y i+j + c y uk +s y ij c y uk + ij+ α w k u ij k [ + α w ] k s w k s+ u s + m f k+ ij. ij s 9 Applying 9 to any group of four points i j i + j i + j + and i j + in the solution domain will result in the following system D a b a D b b D a b a D u ij u i+j u i+j+ u ij+ rhs ij+ where D + s +s x y a sx c sy x b c y sx a + c sy x b + c y Q α w s +s x y rhs ij rhs i+j rhs i+j+ rhs ij a u k+ + i j uk +b i j uk+ + ij uk +a ij uk + b i+j uk ij+ k + α w k u + ij Quk + ij α [w k s w k s+ ]us + m f k+ i.j ij s rhs i+j a u k+ + i+j uk +b i+j uk+ + i+j uk +b i+j uk + a i+j+ uk ij k + α w k u + i+j Quk + i+j α [w k s w k s+ ]us + m f k+ i+j i+j s rhs i+j+ a u k+ + i+j+ uk +b i+j+ uk+ + i+j+ uk +b i+j+ uk + a i+j uk ij+ k + α w k u + i+j+ quk + i+j+ α [w k s w k s+ ]us + m f k+ i+j+ i+j+ rhs ij+ a u k+ + i j+ uk +b i j+ uk+ + ij+ uk +a ij+ uk + b i+j+ uk ij k + α w k u + ij+ Quk + ij+ α [w k s w k s+ ]us + m f k+. ij+ ij+ s The four-point FMEG equation below is obtained by inverting as follows s u k+ i.j u k+ i+ j u k+ i+ j+ u k+ i j+ const r r r 33 r r 55 r r r 66 r 77 r r r 55 r r 99 r r rhs ij rhs i+j rhs i+j+ rhs ij+ Page 7 of 9

8 Balasim & Hj. Mohd. Ali Cogent Mathematics 7 https//doi.org/./ Figure 3. Grouping of the points for the FMEG method n cons c + x c + 96s + y x s + x 9s3 + x 9s + 96s x y + 37s x s y + 7s s + x y s3s + x y s + 7s y x s + y 7s x s + y 9s3 y + s x s 3 + y 9s + y c y 6 + 5s + 3s + x y 3s + s y x + c x 6 c + y 3s + 3s + x y 5s + s y x + s x 6 + c + x c + 3s + y x 3s + 3s + s s + x y x y 3s y r 56 r cx s 5 x 6 + c x c + 3s + y x 3s + 3s + s s + x y x y y 5s r 33 cx s 5 x 6 + c x c + 3s + y x 3s + 3s + s s + x y x y y 5s r c 5 y s y 6 c + x c + 3s + y x 5s + 3s + s s + x y x y 3s y r 55 5 r 66 cx + s x 6 + c x c y + 3s x + 3s x + 3s y + s x s y + 5s y cx + s x c y s y + s x r 77 cx + s x c y + s x r c 5 y 6 c + x c + 3s + y x 5s + 3s + s s + x y x y 3s y r 99 cx s x c y + s x. To use this method the grid points in the solution domain are divided into three types of points denoted by the symbols and in alternate ordering as shown in Figure 3. Note that the evaluation of require points of type only. Thus we can construct the FMEG method by generating the iterations on this type of points only followed by the evaluation of solutions directly once on points of type and. The FMEG algorithm can then be summarized as follows Divide the grid points into three types and in alternate order as depicted in Figure 3. Set the initial guess for the iterations. Page of 9

9 Balasim & Hj. Mohd. Ali Cogent Mathematics 7 https//doi.org/./ Evaluate the solutions at points using Equation iteratively at the time level k +. Step 5 is performed if the iteration converges. Otherwise Step 3 is repeated until a convergence is attained. 5 The steps below are performed directly once for points and in Figure 3 at the time level k + a For type points the rotated h spaced five-point approximation formula derived by rotating the x y axis clockwise at 5 o was used Tan Ali & Lai. This approximation formula was applied to the right side of Equation in combination with 3 being applied to the left side of to obtain the following rotated C-N formula + s + s x y u k+ ij s x + c c x y + c + c x y u k+ +s x i j+ c + c x y u k+ i+j+ +s y u k+ + i j α w s + s x y + c c y x u k +s x ij u k+ i+j + c c x y u k i j+ + s x c + c x y u k i+j+ +s y + c c y x u k i+j +s y + c + c x y u k i j + α w k u ij k [ + α w ] k s w k s+ u s + m f k+ ij. ij s b For type points the typical h spaced formula 5 is used. Note that formula 5 involve points of type only. 3. Stability analysis A scheme is considered to be stable if the errors cease to increase with the passing of time and gradually become inconsequential as the computation progresses. Even though the spacing is different Equation 5 gives rise to both the FEG and FMEG methods. Therefore the stability of both methods can be analyzed in similar ways. Here we will show the stability of the FMEG scheme using eigenvalues of the generated matrices with mathematical induction. Form we obtain AU BU k AU k+ BU k C U k + k s k s Us + C k U o + b k > where A R G G 3 G G G 3 G G G 3 G G R R R 3 R R R 3 R R R 3 R R G 5 G 5 B S S S 3 S S S 3 S S S 3 S G 5 G 5 R 3 G G b W W W W G G Page 9 of 9

10 Balasim & Hj. Mohd. Ali Cogent Mathematics 7 https//doi.org/./ S H H 3 H H H 3 H H H 3 H H S H 5 H 5 H 5 H 5 S 3 H H H H C k M k M k M k M k M k s C k s M k s M k s M k s s... k C M M M M W L L L L G D a b a D b b D a b a D G b b G 3 b b G a a G 5 a a H Q a b a Q b b Q a b a Q H b b H 3 b b H a a H 5 a a w w k s k s+ w M k s α w k s k s+ t α w w k s k s+ w w k s k s+ M α t α w w w w w k w M k α k t α w k w k L α Γ[ α] f ij f i+j f i+j+ f ij+ i j 6... n. Page of 9

11 Balasim & Hj. Mohd. Ali Cogent Mathematics 7 https//doi.org/./ The following lemma is important to prove the stability. [Karatay Kale & Bayramoglu 3] In the coefficients w s s... k satisfy the fol- Lemma lowing w > k s w k s+ s... k. k S [w k s w k s+ ]+w k w s... k. For simplicity we assume s x s y S Γ α α h Q tα α w S Theorem If t α S α w + t α C +S c c C Γ α α D x y h + S tα > then the FMEG scheme is stable. and Proof To prove the stability of we suppose that u k ij i j... n k... l are the approximate solutions to the exact solution U k ij of the error εk ij U k ij uk ij be the error at time level k. From the error satisfies AE BE k AE k+ BE k C E k + k s C k s Us + C k E o k > 3 where E k+ E k+ E k+ E k+ E k+ E k+ ε k+ ε k+ ε k+ m ε k+ m ε k+ i ε k+ ij ε k+ i+j ε k+ i+j+ ε k+ ij+ From the above equations the following are obtained. i 6... n j 6... n A G + G + G + G 3 + G 5 B H + H + H + H 3 + H 5 C M C k s M k s C k M k It is worthy to note that from the eigenvalues of the matrices A B C C k s and C k are a k b k c c k s and c k respectively where Table. Number of various types of mesh points for the point and explicit group methods Point types Number of points FSP FEG FMEG Iterative group points m m m Iterative ungrouped points m- Total iterative points m m m Direct h-spaced rotated points m+ Direct h-spaced standard points m Total direct points 3m +m Page of 9

12 Balasim & Hj. Mohd. Ali Cogent Mathematics 7 https//doi.org/./ Table. Computational complexity for the point and explicit group methods Per iteration After convergence Methods Additional Multiplication Additional Multiplication SFP + 3k m +k m FEG 5 + 3k m + + 3k m +k m + +k m FMEG 5+3k m +k m + 3k 3m +m +k 3m +m { a k t α + S t α + S { b k t α S t α S c α t α w c k s α t α w k s w k s+ c k α t α w k For k E A BE E ρa B E E E t α + S C + S t α S C + S Supposing that Es E s... k. S + t α + S + t α C +S Need to prove this inequality holds for s k +. C +S E t α + S + C + S } t α S + C + S } k E k+ A B C E k + A C k s E s + A C k E Using Lemma we get Ek+ s ρa B C E k + ρa C k s E + ρa C k E t α s Therefore under the conditions is stable. C +S S α w + t α + S + C +S t α E k+ E t α E + S α w + t α α t α w + S + t α C +S C +S E > the FMEG iterative scheme Page of 9

13 Balasim & Hj. Mohd. Ali Cogent Mathematics 7 https//doi.org/./ Table 3. Comparison of the number of iterations Execution times average and maximum error for different time step and mesh size at α.75 and α.95 Example α.75 α.95 h t Method Time Ite. Ave Max Time Ite. Ave Max FSP E-.3E E- 9.76E- /6 / FEG E-.9E E E- FMEG.3.63E E E-3 3.5E-3 FSP E-.9E E-.66E- / /35 FEG E- 3.7E E- 3.6E- FMEG E-.7E E-.E-3 FSP E-.353E E- 5.E- / /5 FEG E-.353E E-5.E- FMEG 7. 3.E E E- 6.37E- FSP E- 6.E E- 3.E- / /6 FEG E- 3.E E-5.7E- FMEG E-.5E E- 3.7E- Figure. A comparison of the numerical solution of the example at α.75 y /6 t between A FSP and FMEG B FEG and FMEG. a b Figure 5. A comparison of the numerical solution of the example at α.75 y /6 t betweena FSP and FMEG B FEG and FMEG. a b Page 3 of 9

14 Balasim & Hj. Mohd. Ali Cogent Mathematics 7 https//doi.org/./ Figure 6. Experimental results for the mesh size against the Time at α 75 A Example B Example. Time 5 a FSP FEG FMEG 5 Time b FSP FEG FMEG 5 5 meshsize 6 6 meshsize 6 6 Figure 7. Experimental results for the mesh size against the Ite. at α 75 A Example B Example. ite 6 a FSP FEG FMEG ite b FSP FEG FMEG mesh size 6 mesh size 6 Table. Comparison of the number of iterations Execution times average and maximum error for different time step and mesh size at α.75 and α.95 Example α.75 α.95 h t Method Time Ite. Ave Max Time Ite. Ave Max FSP E- 6.69E E- 6.7E- /6 / FEG E E E- 6.7E- FMEG.7.36E-3.76E E-3.3E-3 FSP E-.97E E-.35E- / /35 FEG E-.9E E-.3E- FMEG E- 9.75E E- 9.5E- FSP E-5.E E E-5 / /5 FEG E-5 9.5E E-5.E- FMEG E-.999E E- 5.7E- FSP E-5.6E- 3..7E-5.5E-5 / /6 FEG E-6.5E E-5 3.9E-5 FMEG E- 3.57E E- 3.7E- Page of 9

15 Balasim & Hj. Mohd. Ali Cogent Mathematics 7 https//doi.org/./ Table 5. Total computing operations involved for the point and grouping methods Total operations Example Total operations Example h t Method α.75 α.95 α.75 α.95 FSP /6 / FEG FMEG FSP / /35 FEG FMEG FSP / /5 FEG FMEG FSP / /6 FEG FMEG Convergence analysis Theorem The finite difference FMEG scheme is convergent and the following estimate hold ek+ C k τ α + x + y if S α w + C +S t α t α >. Let us denote the truncation error at x i y j t k+ by R k+. From we have Rk+ cτ α + x + y Define η k+ Ux i y j t k+ u k+ ij i j.. n k... l and e k+ e k+ e k+... e k+ T using e where e k+ into produces η k+ η k+ η k+ m η k+ m ε k+ i η k+ ij η k+ i+j η k+ i+j+ η k+ ij+ {i j} 6... n. Substitution Ae R Ae k+ Be k C e k + k C s k s + R k+ es. Using mathematical induction to prove the above theorem set C. For k Ae R e ρa R C τ α + x + y. Assume that e k C k Rk+. Need to prove it hold for s k + Page 5 of 9

16 Balasim & Hj. Mohd. Ali Cogent Mathematics 7 https//doi.org/./ Ae k+ B C e k k + C k s e s + R k+ s ek+ ρa B C e k + ek+ C k Rk t α k ρa C k s e s + ρa R k+ s S α w + C +S t α + + S + C +S t α α t α w + S + t α α t α w k τ α + x + y k α k α τ α α k + α k α τ α + x + y α α τ α + x + y C +S C k Rk+ Since lim k α k. k+ α k α α 5. Computational complexity This section presents an analysis of the computational complexity with regard to the three techniques described for solving which is the number of arithmetic operations per iteration. For simplicity we assume s x s y c x c y. Suppose m internal points exist within the solution with m n where n has an even mesh size then the ungrouped points will be found close to the right/upper boundaries. Internal mesh points have two main types namely the iteration points that participate in the iteration process and the direct points that are directly calculated once using the rotated and standard difference formulas following the attainment of the iteration convergence. Table lists the number of internal mesh points for the three earlier methods whereas Table provides a summary of the number of arithmetic operations that are needed for each iteration and the direct solution following the convergence not only for the explicit group methods but also for the fractional standard point FSP method. 6. Numerical experiments and discussion of results Several numerical examples are presented in this section to prove the effectiveness of the fractional explicit group methods in solving the -D TFADE with a Dirichlet boundary condition. A computer with Windows 7 Professional and Mathematica software having a Core i7 GHZ and GB of RAM was used for conducting the experiments. Example The time fractional initial boundary value problem below was considered Zhuang Gu Liu Turner & Yarlagadda α u u + u u u +.5 Γ 3 +α t e x+y where t α x y x y Ω { x y x y } is the solution domain with the exact solution being t +α e x+y. Example The following time fractional advection-diffusion equation was also considered Mohebbi & Abbaszadeh 3 α u u + u u u + t α sin x+sin y + tcos x + sin x + cos y + sin y t α x y x y Γ α with the initial and boundary conditions are given as u x y u y t t sin y u y t t sin + sin y u x t t sin x ux t t sin x+sin < t < < x y <. Different mesh sizes of 6 and and various time steps which satisfy the stability conditions with a fixed relaxation factor Gauss Seidel relaxation scheme of. were used to run the experiments. During all the experiments the norm for the convergence criteria was ι while error tolerance was set at ε 5. Numerical results were obtained using of the methods described in Page 6 of 9

17 Balasim & Hj. Mohd. Ali Cogent Mathematics 7 https//doi.org/./ Section for different values of α. The number of iterations the execution time and the error analysis are presented in Tables 3 and and Figures 7 for the fractional point method and the fractional group methods. Figure 6 shows the execution times for various mesh sizes for both Examples &. Its clear that when fractional group methods are used the results are just as accurate as the FSP method. From the results obtained it is observed that the execution timings were reduced by as much as and % of the FSP method when the FEG and FMEG methods were used respectively. In contrast to the other tested methods the fractional group methods specifically the FMEG method took a least times to compute the solutions. As shown in Tables 3 & and Figure 6 the time taken for FMEG to be executed was only approximately % of the FEG method and was.3 3.9% of the FSP method. To gauge the computational complexity obtaining an approximation of the amount of calculations involved for each method is necessary. The estimation for the amount of computational work involved was determined by arithmetic operations that were performed for a single iteration as outlined in Section 5. With the help of Tables & and based on the assumption that an approximately equal amount of time was needed for adding subtracting multiplying dividing and assigning a summary of the total number of operations involved in the iterative methods is presented in Table 5. This shows that the FMEG requires the least number of operations thus confirming the theoretical complexity analysis. 7. Conclusion This paper presented the development and formulation of new fractional explicit group iterative methods for solving the D-TFADE. The C-N difference schemes with h spacing and h spacing gave rise to the FEG and FMEG methods respectively. The stability and convergence of the proposed methods were analyzed using the matrix form with mathematical induction. Through the numerical experiments the FMEG method stood out amongst all the other tested methods when it comes to its execution time and number of iterations as it requires the least number of operation counts. It is noted that in terms of accuracy the FMEG method is just as good as the FSP method and the FEG method. This work confirms the suitability and feasibility of the grouping strategies in solving the D time fractional advection-diffusion equation. The implementation of similar group methods on solving other fractional differential equations will be reported soon. Funding This work was supported by the Research University Individual RUI under [grant number / PMATHS/6]. Author details Alla Tareq Balasim alkhazrejy@yahoo.com ORCID ID http//orcid.org/ Norhashidah Hj. Mohd. Ali shidah_ali@usm.my School of Mathematical Sciences Universiti Sains Malaysia Penang Malaysia. Department of Mathematics College of Basic Education University of Al-mustansiria Baghdad Iraq. Citation information Cite this article as New group iterative schemes in the numerical solution of the two-dimensional time fractional advection-diffusion equation Alla Tareq Balasim & Norhashidah Hj. Mohd. Ali Cogent Mathematics 7. References Abdelkawy M. Zaky M. Bhrawy A. & Baleanu D. 5. Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model. Romanian Reports in Physics Agrawal O. P.. A general finite element formulation for fractional variational problems. Journal of Mathematical Analysis and Applications 337. Ali N. H. M. & Kew L. M.. New explicit group iterative methods in the solution of two dimensional hyperbolic equations. Journal of Computational Physics Ali U. Abdullah F. A. & Mohyud-Din S. T. 7. Modified implicit fractional difference scheme for d modified nomalous fractional sub-diffusion equation. Advances in Difference Equations 7 5. Baeumer B. Benson D. A. Meerschaert M. M. & Wheatcraft S. W.. Subordinated advection-dispersion equation for contaminant transport. Water Resources Research Balasim A. T. & Ali N. H. M. 5. A rotated Crank--Nicolson iterative method for the solution of two-dimensional time-fractional diffusion equation. Indian Journal of Science and Technology 3. Baleanu D. Agheli B. & Al Qurashi M. M. 6. Fractional advection differential equation within caputo and caputofabrizio derivatives. Advances in Mechanical Engineering. Benson D. A. Wheatcraft S. W. & Meerschaert M. M.. Application of a fractional advection-dispersion equation. Water Resources Research Bhrawy A. & Baleanu D. 3. A spectral legendre-gausslobatto collocation method for a space-fractional advection diffusion equations with variable coefficients. Reports on Mathematical Physics Chen C.-M. Liu F. Anh V. & Turner I.. Numerical simulation for the variable-order galilei invariant advection diffusion equation with a nonlinear source term. Applied Mathematics and Computation Page 7 of 9

18 Balasim & Hj. Mohd. Ali Cogent Mathematics 7 https//doi.org/./ Chen C.-M. Liu F. & Burrage K.. Finite difference methods and a fourier analysis for the fractional reactionsubdiffusion equation. Applied Mathematics and Computation Chen M. Deng W. & Wu Y. 3. Superlinearly convergent algorithms for the two-dimensional space-time caputoriesz fractional diffusion equation. Applied Numerical Mathematics 7. Chen S. & Liu F.. Adi-euler and extrapolation methods for the two-dimensional fractional advection-dispersion equation. Journal of Applied Mathematics and Computing Cushman J. H. & Ginn T. R.. Fractional advectiondispersion equation A classical mass balance with convolution-fickian flux. Water Resources Research Evans D. & Yousif W. 96. Explicit group iterative methods for solving elliptic partial differential equations in 3-space dimensions. International Journal of Computer Mathematics Gorenflo R. Mainardi F. Moretti D. Pagnini G. & Paradisi P.. Discrete random walk models for space-time fractional diffusion. Chemical Physics 5 5. Henry B. I. & Wearne S. L.. Fractional reactiondiffusion. Physica A Statistical Mechanics and its Applications Karatay I. Kale N. & Bayramoglu S. R. 3. A new difference scheme for time fractional heat equations based on the crank-nicholson method. Fractional Calculus and Applied Analysis Kew L. M. & Ali N. H. M. 5. New explicit group iterative methods in the solution of three dimensional hyperbolic telegraph equations. Journal of Computational Physics 9 3. Leonenko N. N. Meerschaert M. M. & Sikorskii A. 3. Fractional pearson diffusions. Journal of Mathematical Analysis and Applications Li C. Qian D. & Chen Y... On Riemann-Liouville and caputo derivatives. Discrete Dynamics in Nature and Society. Li C. Zeng F. & Liu F.. Spectral approximations to the fractional integral and derivative. Fractional Calculus and Applied Analysis Liu F. Zhuang P. Anh V. Turner I. & Burrage K. 7. Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Applied Mathematics and Computation 9. Mehdinejadiani B. Naseri A. A. Jafari H. Ghanbarzadeh A. & Baleanu D. 3. A mathematical model for simulation of a water table profile between two parallel subsurface drains using fractional derivatives. Computers & Mathematics with Applications Metzler R. Barkai E. & Klafter J Anomalous diffusion and relaxation close to thermal equilibrium A fractional Fokker-Planck equation approach. Physical Review Letters Metzler R. & Klafter J.. Boundary value problems for fractional diffusion equations. Physica A Statistical Mechanics and its Applications Miller K. S. & Ross B An introduction to the fractional calculus and fractional differential equations. New York NY Wiley. Mohebbi A. & Abbaszadeh M. 3. Compact finite difference scheme for the solution of time fractional advection-dispersion equation. Numerical Algorithms Ng K. F. & Ali N. H. M.. Performance analysis of explicit group parallel algorithms for distributed memory multicomputer. Parallel Computing Oldham K. B. & Spanier J. 97. The fractional calculus. New York NY Academic Press. Othman M. & Abdullah A.. An efficient four points modified explicit group poisson solver. International Journal of Computer Mathematics Podlubny I. 99. Fractional differential equations An introduction to fractional derivatives fractional differential equations to methods of their solution and some of their applications Vol. 9. New York NY Academic Press. Samko S. G. Kilbas A. A. & Marichev O. I Fractional integrals and derivatives. London Taylor & Francis. Seki K. Wojcik M. & Tachiya M. 3. Fractional reactiondiffusion equation. The Journal of Chemical Physics Shen S. Liu F. & Anh V.. Numerical approximations and solution techniques for the space-time riesz-caputo fractional advection-diffusion equation. Numerical Algorithms Sousa E. & Li C. 5. A weighted finite difference method for the fractional diffusion equation based on the Riemann--Liouville derivative. Applied Numerical Mathematics Su L. Wang W. & Wang H.. A characteristic difference method for the transient fractional convection-diffusion equations. Applied Numerical Mathematics Tan K. B. Ali N. H. M. & Lai C.-H.. Parallel block interface domain decomposition methods for the d convection-diffusion equation. International Journal of Computer Mathematics Uddin M. & Haq S.. Rbfs approximation method for time fractional partial differential equations. Communications in Nonlinear Science and Numerical Simulation 6. Wyss W. 96. The fractional diffusion equation. Journal of Mathematical Physics Yousif W. & Evans D. J Explicit de-coupled group iterative methods and their parallel implementations. Parallel Algorithms and Applications Zhang X. Huang P. Feng X. & Wei L. 3. Finite element method for two-dimensional time-fractional tricomi-type equations. Numerical Methods for Partial Differential Equations Zheng Y. Li C. & Zhao Z.. A note on the finite element method for the space-fractional advection diffusion equation. Computers & Mathematics with Applications Zhuang P. Gu Y. Liu F. Turner I. & Yarlagadda P.. Time dependent fractional advection-diffusion equations by an implicit mls meshless method. International Journal for Numerical Methods in Engineering Zhuang P. Liu F. Anh V. & Turner I. 9. Numerical methods for the variable-order fractional advectiondiffusion equation with a nonlinear source term. SIAM Journal on Numerical Analysis Page of 9

19 Balasim & Hj. Mohd. Ali Cogent Mathematics 7 https//doi.org/./ The Authors. This open access article is distributed under a Creative Commons Attribution CC-BY. license. You are free to Share copy and redistribute the material in any medium or format Adapt remix transform and build upon the material for any purpose even commercially. The licensor cannot revoke these freedoms as long as you follow the license terms. Under the following terms Attribution You must give appropriate credit provide a link to the license and indicate if changes were made. You may do so in any reasonable manner but not in any way that suggests the licensor endorses you or your use. No additional restrictions You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits. Cogent Mathematics ISSN is published by Cogent OA part of Taylor & Francis Group. Publishing with Cogent OA ensures Immediate universal access to your article on publication High visibility and discoverability via the Cogent OA website as well as Taylor & Francis Online Download and citation statistics for your article Rapid online publication Input from and dialog with expert editors and editorial boards Retention of full copyright of your article Guaranteed legacy preservation of your article Discounts and waivers for authors in developing regions Submit your manuscript to a Cogent OA journal at Page 9 of 9