Numerical solution for space and time fractional order

Alexandria Engineering Journal (2017) xxx, xxx–xxx

H O S T E D BY

Alexandria University

Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com

ORIGINAL ARTICLE

Numerical solution for space and time fractional order Burger type equation Asıf Yokus Firat University, Faculty of Science, Department of Actuary, Elazig, Turkey Received 8 February 2017; revised 16 May 2017; accepted 22 May 2017

KEYWORDS Finite difference method; Generalized Taylor series method; Space and time order Burger type equation; Gru¨nwald formula; Caputo formula

Abstract In this paper, we study the fractional differential operators thereby considering the space and time order Burger type equation with initial condition. The extended finite difference method which is based on shifted Gru¨nwald, Caputo and Riemann formulas are used. The fractional terms are approximated using two fractional difference schemes. Time Order Burger type Equation, Space Order Burger type Equation and Space-Time Order Burger type Equation are discussed with an example and error estimates obtained for the Finite difference method (FDM) and Generalized Taylor Series Methods (GTSM). The numerical methods have been applied to solve a numerical example, and results are compared with the exact solutions. By these methods, the numerical solutions of space and time order Burger type equation are obtained with the help of Wolfram Mathematica 11 software package. We present the tables of the obtained numerical results. Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction In recent years, a rapid development on scientific theory of the fractional calculus has been seen in technic and applied sciences [1–4]. The exact solutions of the fractional differential equations may not be easily obtained, so we need numerical methods for fractional differential equations. One of them is finite difference method and it is one of the most popular methods regarding to numerical solutions of partial differential equations. Classical partial differential equations have been extended to the fractional partial differential equations. There are many applications of such equations in the literature. The fractional partial differential equations have been used in several fields such as fluid dynamics, heat flow, finance, E-mail address: [email protected] Peer review under responsibility of Faculty of Engineering, Alexandria University.

hydrology and others [5–13]. In this paper we investigate finite difference numerical methods to solve the space and time order Burger type equation of the form @ b uðx; tÞ @ a uðx; tÞ @ 2 uðx; tÞ þ uðx; tÞ  ¼ 0; @tb @xa @x2

ð1Þ

uðx; 0Þ ¼ u0 ; a 6 x 6 b and uða; tÞ ¼ uðb; tÞ ¼ 0; 0 < t 6 T; where 0 < a 6 1, 0 < b 6 1. Eq. (1) uses a Riemann fractional derivative of ordera. In most of the related literature, it is findable to solve fractional partial differential equations. Various methods have been used to obtain these solutions such as Fourier transform methods, Laplace transform methods and Mellin transform methods. On the other hand, numerical solutions of fractional partial differential equations are in the science literature such as standard or shifted Gru¨nwald-Letnikov formulae, convolution formulae, homotopy perturbation method and

http://dx.doi.org/10.1016/j.aej.2017.05.028 1110-0168 Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: A. Yokus, Numerical solution for space and time fractional order Burger type equation, Alexandria Eng. J. (2017), http://dx.doi. org/10.1016/j.aej.2017.05.028

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A. Yokus

so on [15–17]. In this paper, based on shifted Gru¨nwaldLetnikov formula, we consider using the finite difference method.

uiþ1;j ¼ ui1;j þ c½ui;j þ ui;jþ1 þ

j X rk ðui;jk þ ui;1þjk Þ k¼1

! iþ1 X 2 þ ðDtÞ2a gk;a uiþ1k;j ;

þ ui;j 2. Analysis of the finite difference method

ð9Þ

k¼1 b

We need same notations to describe the finite forward difference method and this includes Dx which is the spatial step, Dt which is the time step, xi ¼ a þ iDx; i ¼ 0; 1; 2; . . . ; N points are the coordinates of T ; tj ¼ jDt; j ¼ 0; 1; 2; . . . ; M; M ¼ Dt . The mesh and N ¼ ba Dx function uðx; tÞ which represents the values of the solution at these grid points is given by uðxi ; tj Þ ffi ui;j , where we denote by ui;j the numerical estimate of the exact value of uðx; tÞ at the point ðxi ; tj Þ. The forward time difference scheme is @u ui;jþ1  ui;j ffi ; @t Dt

3. Analysis of the generalized Taylor series method There are many studies in the literature on fractional derivatives Taylor Series. One of these studies has been awarded the generalized Taylor series by Odibat [19]. uðx; t þ DtÞ and uðx þ Dx; tÞ can be expanded in generalized Taylor series of fractional order as following

ð2Þ uðx; t þ DtÞ ¼

and central space differences are @u uiþ1;j  ui1;j ; ffi 2Dx @x

ð3Þ

@ 2 u uiþ1;j  2ui;j þ ui1;j ffi ; @x2 ðDxÞ2

ð4Þ

uðx þ Dx; tÞ ¼

according to the shifted Gru¨nwald-Letnikov definition [6], we have the following; iþ1 @ a uðx; tÞ 1 X ¼ a ga uikþ1;j ; a @x Dx k¼0 k

ð5Þ

  a a here, gak ¼ ð1Þk ð Þ, ga0 ¼ 1 and gak ¼ 1  aþ1 gk1 . According k k to the shifted Caputo definition [18], we have the following; 8 b h > ðu  ui;j Þ > > Cð2bÞ i;jþ1 > > i < b X @ uðx; tÞ hb ðui;jþ1k  ui;jk Þððk þ 1Þ1b  k1b Þ; j P 1  þ Cð2bÞ > @tb > k¼1 > > > : hb ðu  u Þ: j¼0 i;1 i;0 Cð2bÞ

ð6Þ

We consider space and time order Burger type equation of the form (1) with the coefficients b ¼ 1, 0 < a 6 1 which is called the space order Burger type equation. Considering the Gru¨nwald-Letnikov definition (2), (4) and (5) the space order Burger type equation has the following indexed form; ! iþ1 X a uiþ1;j ¼ ui1;j þ w ui;jþ1 þ ui;j 2  w þ / gk uikþ1;j ; ð7Þ

2

ðDxÞ , rk ¼ ðk1b þ ð1 þ kÞ1b Þ and the initial where c ¼ ðDtÞCð2bÞ values ui;0 ¼ u0 ðxi Þ.

1 X

ðDtÞbk @ ðbkÞ uðx; tÞ ; 0 < b 6 1; Cðbk þ 1Þ @xðbkÞ k¼0

ð10Þ

1 X ðDxÞak @ ðakÞ uðx; tÞ ; 0 < a 6 1: Cðak þ 1Þ @xðakÞ k¼0

ð11Þ

(10) and (11) supplied with the generalized Taylor series, and the second term is the next terms are neglected and made the necessary arrangements, we get the following difference scheme of fractional derivatives; @ ðbÞ uðx; tÞ ffi Cðb þ 1ÞðDtÞb ½ui;jþ1  ui;j ; @tðbÞ

ð12Þ

@ ðaÞ uðx; tÞ ffi Cða þ 1ÞðDxÞa ½uiþ1;j  ui;j : @xðaÞ

ð13Þ

When Eqs. (3), (4) and (12) are substituted into the time order Burger type equation and supplied with the necessary arrangements are made and then we would have the following equation as; uiþ1;j ¼

4ui;j þ ui1;j ð2 þ Dxui;j Þ þ dðui;j  ui;jþ1 Þ ; 2 þ Dxui;j

ð14Þ

where d ¼ 2ðDtÞa ðDxÞ2 Cð1 þ bÞ. Also, Eqs. (2), (4) and (13) are written into the space order Burger type equation and after the necessary arrangements are made the following equation can be found

k¼0 2

; / ¼ ha ðDxÞ2 . Eq. (1) with the coeffiwhere w ¼ ðDxÞ Dt cientsa ¼ 1, 0 < b 6 1 is time order Burger type equation. Considering Eqs. (3), (4) and (6) the time order Burger type equation has the following indexed form; uiþ1;j ¼

ui1;j þ ui;j ð2 þ Dxui;j Þ  cðui;j þ ui;jþ1 þ 1 þ Dxui;j

Pj

k¼1 rk ðui;jk

þ ui;1þjk Þ

:

uiþ1;j ¼

ð15Þ

where e ¼ 2ðDxÞ2 Cð1 þ bÞ. But only, Eqs. (4), (12) and (13) can be written into the time and space order Burger type Eq. (1) and after the necessary arrangements are made the following equation can be found;

ð8Þ

Eq. (1) with the coefficients 0 < a 6 1, 0 < b 6 1 is time and space order Burger type equation. Considering Eqs. (4), (5) and (6) the space and time order Burger type equation can be written in the following indexed form;

e ui;j þ ðDxÞa ð4ui;j þ ui1;j ð2  Dxui;j Þ ; e þ ðDxÞa ð2 þ Dxui;j Þ

uiþ1;j ¼

ðDxÞa ðui1;j  2ui;j Þ  wu2i;j þ qðui;j  ui;jþ1 Þ ; ðDxÞa  wui;j

ð16Þ

where w ¼ ðDxÞ2 Cð1 þ aÞ, q ¼ ðDtÞb ðDxÞaþ2 Cð1 þ bÞ.

Please cite this article in press as: A. Yokus, Numerical solution for space and time fractional order Burger type equation, Alexandria Eng. J. (2017), http://dx.doi. org/10.1016/j.aej.2017.05.028

Numerical solution for space and time fractional order

3

4. Consistency analysis In this section, we will investigate the consistency of the Eq. (1) by finite difference method. At first, the Taylor series expansions can be given in the form as follows, @u @2u þ ðDxÞ2 2 þ OðDx3 Þ; @x @x

ð17aÞ

@u @2u þ ðDtÞ2 2 þ OðDt3 Þ; @t @t

ð17bÞ

@u @2u þ ðDxÞ2 2  OðDx3 Þ; @x @x

ð17cÞ

uiþ1;j ¼ ui;j þ Dx ui;jþ1 ¼ ui;j þ Dt

ui1;j ¼ ui;j  Dx

let us define an operator L, L¼

@ @ @2 þu  2; @t @x @x

the indexed form of operator L can be written as Li;j ¼

ui;jþ1  ui;j uiþ1;j  ui;j @u uiþ1;j  2ui;j þ ui1;j  þu : @x Dt Dx ðDxÞ2

Fig. 1

ð18Þ

If we substitute the indexed forms (17a), (17b) and (17c) into Eq. (18) and do some necessary manipulations, then the approach will be Dt ! 0 and Dx ! 0 , and then Eq. (18) will be the same with the left hand side of the Eq. (1). This conclusion shows us that the Eq. (1) is consistent with finite difference method. 5. Truncation error and stability analysis In this section, we examine the stability and error analysis of the finite forward difference method. With the stability, we discuss whether there is a perturbation in the initial condition, and then the small change would not cause the large error in the numerical solution. Simply stability means that the scheme does not amplify errors and the error caused by a small perturbation in the numerical solution remains bound. Lemma 1. The truncation error of the finite difference method to nonlinear space and time fractional order Burger type Eq. (1) is OððDtÞ3 þ ðDtÞ3 Þ.

Numerical and exact solution of Eq. (1) for Finite Difference Method (a) and Generalized Taylor Series Method (b).

Fig. 2

Numerical and exact solution of Eq. (1) for Finite Difference Method and Generalized Taylor Series Method.

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Proof. Substituting Eqs. (17a), (17b), (17c) and (7) into the indexed Eq. (18) and doing some necessary manipulations, ^ and the then we obtain the following numerical solution U transaction error E

jEj < e:

! iþ1 X a ^ U ¼ ui;j 1 þ u gk uikþ1 ;

6. Examples ð19Þ

k¼1

@u w @2u @u 1 @2u þ ðDtÞ2 2 þ Dx  ðDxÞ2 2 @t 2 @t @x 2 @x þ OððDtÞ3 þ ðDtÞ3 Þ:

E ¼ wDt

ð20Þ

It is obviously seen that, in Eq. (20) the truncation error is OððDtÞ3 þ ðDtÞ3 Þ, which proves the Lemma 1. h In addition to this, a transaction error E can be written as ^ E ¼ jU  Uj; where U is the exact solution. Here, considering Lemma 1, if the values of Dt and Dx are chosen as small as necessary, transaction error will be obviously very small. What is more, limit of E can be written as lim E ¼ 0; Dx ! 0 Dt ! 0

ð21Þ

In here, if Dt and Dx are configured for a value close to zero e > 0, we may give the following inequality;

Table 1

This also shows that finite difference method is unconditionally stable.

6.1. Example 1 We consider space and time order Burger type equation of the form (1) as in [14] with the coefficient a ¼ 0, b ¼ 1, T ¼ 1, b ¼ 1, a ¼ 0:8 and 0 < t 6 1 the initial condition u0 ðxÞ ¼ x. This equation is space order Burger type equation. For a ¼ b ¼ 1, the exact solution of Eq. (1) turns to ([14]) x : ð22Þ uðx; tÞ ¼ 1þt The numerical solutions of space order Burger type equation are obtained from the difference schemes (7) and (15) discussed above. But only the exact solution is calculated according to Eq. (22). We know that errors depend on the choice of the Dx and Dt. These resulting behavior of the numerical and exact solutions can be seen in the following graph using values of Dx ¼ Dt ¼ 0:02 and i = 0, 1 ,2, . . . , 50. In this Fig. 1(a)-(b), the exact solution by dots and squares represents the numerical solution. In this example, the results are obtained more sensitive with finite difference method. This resulting behavior of the numerical and exact solutions can be

Numerical and exact solution of Eq. (1) and error when Dx ¼ 0:02.

xi

t

Finite Difference Method

Generalized Taylor Series Method

Exact solution

Error of Finite Difference Method

Error of Generalized Taylor Series Method

0.00 0.02 0.04 0.06 0.08 0.10 0.12

0.02 0.02 0.02 0.02 0.02 0.02 0.02

0.00 0.0197805 0.0395171 0.0592273 0.0789183 0.0985937 0.1182563

0.00 0.0198296 0.0396593 0.0594889 0.0793185 0.0991481 0.1189778

0.00 0.0196078 0.0392157 0.0588235 0.0784314 0.0980392 0.1176470

0.00 1.72650  104 3.01400  104 4.03808  104 4.86900  104 5.54538  104 6.09196  104

0.00 2.21786  104 4.43573  104 6.65359  104 8.87145  104 1.10893  103 1.33072  103

Fig. 3

Numerical and exact solution of Eq. (1) for Finite Difference Method (a) and Generalized Taylor Series Method (b).

Please cite this article in press as: A. Yokus, Numerical solution for space and time fractional order Burger type equation, Alexandria Eng. J. (2017), http://dx.doi. org/10.1016/j.aej.2017.05.028

Numerical solution for space and time fractional order

5

Fig. 4 Numerical and exact solution of Eq. (1) for Finite Difference Method and Generalized Taylor Series Method.

Table 2

Fig. 6 Numerical and exact solution of Eq. (1) for Finite Difference Method and Generalized Taylor Series Method.

Numerical and exact solution of Eq. (1) and error when Dx ¼ 0; 0.

xi

tj

Finite Difference Method

Generalized Taylor Series Method

Exact solution

Error of Finite Difference Method

Error of Generalized Taylor Series Method

0.02 0.04 0.06 0.08 0.10 0.12

0.02 0.02 0.02 0.02 0.02 0.02

0.0191968 0.0383938 0.0575907 0.0767876 0.0959844 0.1151813

0.0190608 0.0381217 0.0571826 0.0762434 0.0953044 0.1143652

0.0196078 0.0392157 0.0588235 0.0784314 0.0980392 0.1176470

4.10956  104 8.21912  104 1.23287  103 1.64382  103 2.05478  103 2.46574  103

5.46972  104 1.09394  10 -3 1.64092  103 2.18789  103 2.73486  103 3.28183  103

Table 3

Numerical and exact solution of Eq. (1) and error when Dx ¼ 0:02.

xi

tj

Finite Difference Method

Generalized Taylor Series Method

Exact solution

Error of Finite Difference Method

Error of Generalized Taylor Series Method

0.00 0.02 0.04 0.06 0.08 0.10 0.12

0.02 0.02 0.02 0.02 0.02 0.02 0.02

0.00 0.0195593 0.0390304 0.0584487 0.0778281 0.0971766 0.1164989

0.00 0.0196 0.0392 0.0588 0.0784 0.0979 0.1176

0.00 0.0196078 0.0392157 0.0588235 0.0784314 0.0980392 0.1176470

0.00 4.85641  105 1.85272  104 3.74867  104 6.03246  104 8.62651  104 1.14812  103

0.00 7.84314  106 1.56863  10 -5 2.35294  105 3.13725  105 3.92157  105 4.70588  105

Fig. 5

Numerical and exact solution of Eq. (1) for Finite Difference Method (a) and Generalized Taylor Series Method (b).

Please cite this article in press as: A. Yokus, Numerical solution for space and time fractional order Burger type equation, Alexandria Eng. J. (2017), http://dx.doi. org/10.1016/j.aej.2017.05.028

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A. Yokus

Fig. 7

Numerical solution of Eq. (1) for Finite Difference Method.

seen in the following graph using values of Dx ¼ Dt ¼ 0:1 and i = 0, 1, . . . , 10. In Fig. 2, precise numerical results are given using the finite difference method. We can see this situation in the Table 1 as numerical. 6.2. Example 2 We consider space and time order Burger type equation of the form (1) with the coefficient a ¼ 0, b ¼ 1, T ¼ 1, b ¼ 0; 8. a ¼ 1. 0 < t 6 1 the initial condition u0 ðxÞ ¼ x. This equation is time order Burger type equation. This fractional partial differential equation together with the above initial condition is constructed such that the exact solution is Eq. (22). The numerical solutions are obtained from the difference schemes (8) and (14) discussed above. We know that error depends on the choice of the Dx and Dt, these resulting behavior of the numerical and exact solutions can be seen in the following graph using values of Dx ¼ Dt ¼ 0:02 and i = 0, 1, 2, . . . , 50. In this Fig. 3(a)-(b), the exact solution by dots and squares represents the numerical solution. In this example, the results are obtained more sensitive with finite difference method. This resulting behavior of the numerical and exact solutions can be seen in the following graph using values of Dx ¼ Dt ¼ 0; 1and i = 0, 1, . . . , 10. In the Fig. 4, numerical and exact solution of Eq. (1) for finite difference method and generalized Taylor series method are given more precisely. We can see this situation in the Table 2 as numerical.

the numerical and exact solutions can be seen in the following graph using values of Dx ¼ Dt ¼ 0:1 and i = 0, 1, 2, . . . , 10. In Fig. 6 generalized Taylor series method gives more precise results. We can see this situation in Table 3 as numerical (see Fig. 5). Considering the Eq. (9) on the brink of 0 < a 6 1; 0 < b 6 1 which is obtained using finite difference method, we can say that when a decreases the term of Gru¨nwald-Letnikov definition (5) and Caputo definition (6) will decrease. It means potential u is decreasing. We can see this situation in the figures as follow (see Fig. 7). 7. Conclusions In this study, with the aid of Wolfram Mathematica 11 software, we utilized the Finite difference and generalized Taylor Series method in studying the numerical behavior of the time and space order Burger type equation. These methods can be applied to many other nonlinear equations of various forms. In addition, these methods are also computerizable, which allows us to perform complicated and tedious algebraic calculation on a computer. Fractional finite difference method is useful in solving the fractional differential equations. In some way, this numerical method has similar form as the case for classical equations, some of which can be seen as the generalizations of the Finite Difference Method and generalized Taylor Series method for the typical differential equations. Numerical method for solving the space and time order Burger type equation has been described and demonstrated. References

6.3. Example 3 We consider space and time order Burger type equation of the form (1) with the coefficient L ¼ 0, R ¼ 1, T ¼ 1, b ¼ 0:8, a ¼ 0:8, 0 < t 6 1. This equation is space and time order Burger type equation. This fractional partial differential equation together with the above initial condition is constructed such that the exact solution is Eq. (22). The numerical solutions are obtained from the difference schemes (9) and (16) discussed above. We know that error due to the choice of the Dx and Dt, these resulting behavior of the numerical and exact solutions can be seen in the following graph using values of Dx ¼ Dt ¼ 0:02 and i = 0, 1, 2, . . . , 50. In this example, the results are obtained more sensitive with generalized Taylor Series method. This resulting behavior of

[1] K. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. [2] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [3] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 2006. [4] G.M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep. 371 (2002) 461–580. [5] E. Sousa, Finite difference approximates for a fractional advection diffusion problem, J. Comput. Phys. 228 (2009) 4038–4054. [6] M.M. Meerschaert, C. Tadjeran, Finite difference approximations for fractional advection–dispersion flow equations, J. Comput. Appl. Math. 172 (2004) 65–77. [7] J. Singh, D. Kumar, R. Swroop, Numerical solution of timeand space-fractional coupled Burgers’ equations via homotopy algorithm, Alex. Eng. J. 55 (2016) 1753–1763.

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Numerical solution for space and time fractional order [8] F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage, Stability and convergence of the difference methods for the space-time fractional advection–diffusion equation, Appl. Math. Comput. 191 (2007) 2–20. [9] L. Su, W. Wang, Z. Yang, Finite difference approximations for the fractional advection–diffusion equation, Phys. Lett. A 373 (2009) 4405–4408. [10] D. Benson, S. Wheatcraft, M. Meerschaert, Application of a fractional advection–dispersion equation, Water Resour. Res. 36 (2000) 1403–1412. [11] D. Kumar, J. Singh, D. Baleanu, A hybrid computational approach for Klein-Gordon equations on Cantor sets, Nonlinear Dyn. 87 (2017) 511–517. [12] J. Singha, D. Kumar, J.J. Nieto, Analysis of an El NinoSouthern Oscillation model with a new fractional derivative, Chaos Solit. Fract. 99 (2017) 109–115. [13] E. Scalas, R. Gorenflo, F. Mainardi, Fractional calculus and continuous-time finance, Physica A 284 (2000) 376–384.

7 [14] A. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Higher Education Press, New York, 2009. [15] D. Kumar, J. Singh, D. Baleanu, Numerical computation of a fractional model of differential-difference equation, J. Comput. Nonlinear Dyn. 11 (6) (2016) 061004. [16] H.M. Srivastava, D. Kumar, J. Singh, An efficient analytical technique for fractional model of vibration equation, Appl. Math. Model. 45 (2017) 192–204. [17] S.B. Yuste, Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys. 216 (2006) 264– 274. [18] W. Chen, L. Ye, H. Sun, Fractional diffusion equations by the Kansa method, Comput. Math. Appl. 59 (2010) 1614– 1620. [19] Z.M. Odibat, N.T. Shawagfeh, Generalized Taylor’s formula, Appl. Math. Comput. 186 (2007) 286–293.

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