Davey-Stewartson Equation with Fractional Coordinate Derivatives

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Hindawi Publishing Corporation The Scientific World Journal Volume 2013, Article ID 941645, 8 pages http://dx.doi.org/10.1155/2013/941645

Research Article Davey-Stewartson Equation with Fractional Coordinate Derivatives H. Jafari,1,2 K. Sayevand,3 Yasir Khan,4 and M. Nazari1 1

Department of Mathematics and Computer Science, University of Mazandaran, P.O. Box 47416-1467, Babolsar, Iran International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa 3 Faculty of Mathematical Sciences, Malayer University, P.O. Box 65719-95863, Malayer, Iran 4 Department of Mathematics, Zhejiang University, Hangzhou 310027, China 2

Correspondence should be addressed to H. Jafari; [email protected] Received 4 August 2013; Accepted 19 September 2013 Academic Editors: B. Lamichhane, T. Li, and X. Song Copyright Β© 2013 H. Jafari et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We have used the homotopy analysis method (HAM) to obtain solution of Davey-Stewartson equations of fractional order. The fractional derivative is described in the Caputo sense. The results obtained by this method have been compared with the exact solutions. Stability and convergence of the proposed approach is investigated. The effects of fractional derivatives for the systems under consideration are discussed. Furthermore, comparisons indicate that there is a very good agreement between the solutions of homotopy analysis method and the exact solutions in terms of accuracy.

1. Introduction In recent years, fractional differential equations (FDEs) have been the focus of many studies due to their appearance in various fields such as physics, chemistry, and engineering [1– 7]. On the other hand, much attention has been paid to the solutions of fractional differential equations. Several techniques including Adomian decomposition method (ADM) [8, 9], Laplace decomposition method [10], homotopy perturbation method (HPM) [11], variational iteration method (VIM) [11], and differential transform method [12] have been used for solving a wide range of problems. Another powerful analytical method, called the homotopy analysis method (HAM), was first proposed by Liao in his Ph.D. thesis [13]. The HAM contains a certain auxiliary parameter β„Ž which provides us with a simple way to adjust and control the convergence region and rate of convergence of the series solution. This method has been successfully applied to solve many types of nonlinear problems [14–17]. For instance, Jafari and Seifi have solved diffusion-wave equations and system of nonlinear fractional partial differential equations using homotopy analysis method [18, 19].

In this paper, the homotopy analysis method [13, 20] is applied to solve fractional Davey-Stewartson equations: πœ•π‘ž πœ•πœ™ 1 4 πœ•π›Ό π‘ž 1 2 πœ•2 π‘ž 󡄨 󡄨2 + 𝜎 + 𝑖 + πœ†σ΅„¨σ΅„¨σ΅„¨π‘žσ΅„¨σ΅„¨σ΅„¨ π‘ž βˆ’ 𝜎 π‘ž = 0, 2 πœ•π‘¦π›Ό 2 πœ•π‘₯2 πœ•π‘‘ πœ•π‘₯ 󡄨 󡄨2 𝛼 πœ• (σ΅„¨σ΅„¨σ΅„¨π‘žσ΅„¨σ΅„¨σ΅„¨ ) πœ•2 πœ™ 2πœ• πœ™ βˆ’πœŽ βˆ’ 2πœ† = 0, πœ•π‘₯2 πœ•π‘¦π›Ό πœ•π‘₯ 1 < 𝛼 ≀ 2,

πœ† ∈ R,

(1)

𝜎 ∈ C.

The special case {𝛼 = 2, 𝜎 = 1} is called the classical DS-I equation, while {𝛼 = 2, 𝜎 = Β±βˆšβˆ’1} is the classical DS-II equation. The parameter πœ† characterizes the focusing or defocusing case. The classical Davey-Stewartson I and II are two well-known examples of integrable equations in two space dimensions, which arise as higher dimensional generalizations of the nonlinear SchrΒ¨odinger equation [21]. Although there are a lot of studies for the classical DaveyStewartson equation and some profound results have been established, it seems that detailed studies of the fractional Davey-Stewartson equation are only beginning. We intend to

2

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apply the homotopy analysis method to solve the fractional Davey-Stewartson equations. We will also present numerical results to show the nature of the curves/surfaces as the fractional derivative parameter changed.

This section deals with some preliminaries and notations regarding fractional calculus. For more details see [6, 22–24]. Definition 1. A real function 𝑒(𝑑), 𝑑 > 0, is said to be in the space 𝐢𝛼 , 𝛼 ∈ R, if there exists a real number 𝑝 (>𝛼), such that 𝑒(𝑑) = 𝑑𝑝 𝑒1 (𝑑), where 𝑒1 (𝑑) ∈ 𝐢[0, ∞), and it is said to be in the space πΆπ›Όπ‘š , π‘š ∈ N ⋃ {0}, if and only if 𝑒(π‘š) (𝑑) ∈ 𝐢𝛼 . Definition 2. The (left sided) Riemann-Liouville fractional integral of order 𝛼 > 0 of a function 𝑒(𝑑) ∈ 𝐢𝛼 , 𝛼 β‰₯ βˆ’1, is defined as 𝑑 𝑒 (𝜏) 1 { { π‘‘πœ, 𝛼 > 0, 𝑑 > 0, ∫ 𝛼 𝐼𝑑 𝑒 (𝑑) = { Ξ“ (𝛼) 0 (𝑑 βˆ’ 𝜏)1βˆ’π›Ό { 𝑒 , { (𝑑)

𝐼𝑑𝛼 𝑒 (π‘₯, 𝑑) =

𝑒 (π‘₯, 𝑠) 1 𝑑𝑠, ∫ Ξ“ (𝛼) 0 (𝑑 βˆ’ 𝑠)1βˆ’π›Ό

Theorem 5. Assume that the continuous function 𝑒(𝑑) has a fractional derivative of order 𝛼; then one has π›½βˆ’π›Ό

2. Preliminaries and Notations

𝑑

2.1. The Relation between Fractional Derivative and Fractional Integral

𝛽 𝐷𝑑𝛼 𝐼𝑑 𝑒 (𝑑)

𝐼 𝑒 (𝑑) { {𝑑 = {𝑒 (𝑑) { βˆ’π›½+𝛼 {𝐷𝑑 𝑒 (𝑑)

𝛼 < 𝛽, 𝛼 = 𝛽, 𝛼 > 𝛽,

π‘šβˆ’1

𝛼 𝐼𝑑𝛼 π·βˆ—π‘‘ 𝑒 (𝑑) = 𝑒 (𝑑) βˆ’ βˆ‘ 𝑒(π‘˜) (0+ ) π‘˜=0

π‘š βˆ’ 1 < 𝛼 ≀ π‘š, 𝛼 𝛼 π·βˆ—π‘‘ 𝐼𝑑 𝑒 (𝑑) = {

π‘‘π‘˜ , π‘˜!

π‘š ∈ N,

𝑒 (𝑑) , π‘š βˆ’ 1 < 𝛼 ≀ π‘š, π‘š ∈ N, 𝛼 𝐼𝑑𝛼 π·βˆ—π‘‘ 𝑒 (𝑑) + 𝑒 (0) , 0 < 𝛼 < 1. (6)

3. Homotopy Analysis Method Let us consider the following system of differential equations: (2)

𝛼 > 0, 𝑑 > 0,

N𝑖 [𝑒1 (π‘₯, 𝑦, 𝑑) , . . . , 𝑒𝑛 (π‘₯, 𝑦, 𝑑) , π‘₯, 𝑦, 𝑑] = 0,

𝑖 = 1, 2, . . . , 𝑛, (7)

where N𝑖 are nonlinear operators and 𝑒𝑖 (π‘₯, 𝑦, 𝑑) are unknown functions. By means of generalizing the traditional homotopy method, Liao [15] constructed the so-called zero-order deformation equations:

where Ξ“(β‹…) is the well-known Gamma function. Definition 3. The (left sided) Riemann-Liouville fractional π‘š derivative of 𝑒(𝑑), 𝑒(𝑑) ∈ πΆβˆ’1 , π‘š ∈ N ⋃{0}, of order 𝛼 is defined as

(1 βˆ’ 𝑝) L𝑖 [πœ™π‘– (π‘₯, 𝑦, 𝑑; 𝑝) βˆ’ 𝑒𝑖0 (π‘₯, 𝑦, 𝑑)] = π‘β„Žπ‘– N𝑖 [πœ™1 (π‘₯, 𝑦, 𝑑; 𝑝) , . . . , πœ™π‘› (π‘₯, 𝑦, 𝑑; 𝑝) , π‘₯, 𝑦, 𝑑] , (8) 𝑖 = 1, 2, . . . , 𝑛,

𝐷𝑑𝛼 𝑒 (𝑑) =

π‘š

𝑑 π‘šβˆ’π›Ό 𝐼 𝑒 (𝑑) , π‘‘π‘‘π‘š 𝑑

π‘š βˆ’ 1 < 𝛼 ≀ π‘š, π‘š ∈ N.

(3)

Definition 4. The (left sided) Caputo fractional derivative of π‘š , π‘š ∈ N ⋃{0}, is defined as 𝑒(𝑑), 𝑒(𝑑) ∈ πΆβˆ’1 π‘šβˆ’π›Ό (π‘š) {[𝐼𝑑 𝑒 (𝑑)] π‘š βˆ’ 1 < 𝛼 < π‘š, π‘š ∈ N, 𝛼 π·βˆ—π‘‘ 𝑒 (𝑑) = { π‘‘π‘š 𝑒 (𝑑) 𝛼 = π‘š, { π‘‘π‘‘π‘š 𝛼 π·βˆ—π‘‘ 𝑒 (π‘₯, 𝑑) = πΌπ‘‘π‘šβˆ’π›Ό 𝛼 π·π‘‘π‘š 𝑒 (𝑑) π·βˆ—π‘‘

=

πœ•π‘š 𝑒 (π‘₯, 𝑑) , πœ•π‘‘π‘š

𝛼+π‘š π·βˆ—π‘‘ 𝑒 (𝑑) ,

where 𝑝 ∈ [0, 1] is the embedding parameter, β„Žπ‘– =ΜΈ 0 are nonzero auxiliary parameters, and L𝑖 are auxiliary linear operators with the following property: L𝑖 [𝑐] = 0,

𝑖 = 1, 2, . . . , 𝑛,

where 𝑐 is constant. 𝑒𝑖0 (π‘₯, 𝑦, 𝑑) are initial guesses of 𝑒𝑖 (π‘₯, 𝑦, 𝑑), πœ™π‘– (π‘₯, 𝑦, 𝑑; 𝑝) are unknown functions, respectively. It is important that one has great freedom to choose auxiliary things in HAM. Obviously, when 𝑝 = 0 and 𝑝 = 1, it holds πœ™π‘– (π‘₯, 𝑦, 𝑑; 0) = 𝑒𝑖0 (π‘₯, 𝑦, 𝑑) , πœ™π‘– (π‘₯, 𝑦, 𝑑; 1) = 𝑒𝑖 (π‘₯, 𝑦, 𝑑) ,

π‘š βˆ’ 1 < 𝛼 < π‘š,

(10)

𝑖 = 1, 2, . . . , 𝑛,

π‘š = 0, 1, . . . , 𝑛 βˆ’ 1 < 𝛼 < 𝑛. (4)

Property. Similar to integer-order differentiation, fractional differentiation is a linear operation:

respectively. Thus, as 𝑝 increases from 0 to 1, the solution πœ™π‘– (π‘₯, 𝑦, 𝑑; 𝑝) varies from the initial guesses 𝑒𝑖0 (π‘₯, 𝑦, 𝑑) to the solution 𝑒𝑖 (π‘₯, 𝑦, 𝑑). Expanding πœ™π‘– (π‘₯, 𝑦, 𝑑; 𝑝) in Taylor series with respect to 𝑝, we have +∞

πœ™π‘– (π‘₯, 𝑦, 𝑑; 𝑝) = 𝑒𝑖0 (π‘₯, 𝑦, 𝑑) + βˆ‘ π‘’π‘–π‘š (π‘₯, 𝑦, 𝑑) π‘π‘š , π‘š=1

𝛼 𝛼 𝛼 π·βˆ—π‘‘ 𝑒 (𝑑) + πœ‚π·βˆ—π‘‘ V (𝑑) . (𝛾𝑒 (𝑑) + πœ‚V (𝑑)) = π›Ύπ·βˆ—π‘‘

(9)

(5)

𝑖 = 1, 2, . . . , 𝑛,

(11)

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3 In this way, it is easy to obtain π‘’π‘–π‘š (π‘₯, 𝑦, 𝑑) for π‘š β©Ύ 1, at π‘šthorder; we have

where π‘’π‘–π‘š (π‘₯, 𝑦, 𝑑) =

󡄨 π‘š 1 πœ• πœ™π‘– (π‘₯, 𝑦, 𝑑; 𝑝) 󡄨󡄨󡄨 󡄨󡄨 󡄨󡄨 π‘š! πœ•π‘π‘š 󡄨𝑝=0,

𝑀

If the auxiliary linear operators, the initial guesses, the auxiliary parameters β„Žπ‘– are so properly chosen, the series (11) converges at 𝑝 = 1; then we have +∞

π‘š=1

(13)

𝑖 = 1, 2, . . . , 𝑛. Define the vector π‘’βƒ—π‘–π‘š (π‘₯, 𝑦, 𝑑) = {𝑒𝑖0 (π‘₯, 𝑦, 𝑑) , 𝑒𝑖1 (π‘₯, 𝑦, 𝑑) , . . . , π‘’π‘–π‘š (π‘₯, 𝑦, 𝑑)} , 𝑖 = 1, 2, . . . , 𝑛,

𝑖 = 1, 2, . . . , 𝑛.

(18)

π‘š=0

𝑖 = 1, 2, . . . , 𝑛.

π‘’π‘–π‘š (π‘₯, 𝑦, 𝑑) = 𝑒𝑖0 (π‘₯, 𝑦, 𝑑) + βˆ‘ π‘’π‘–π‘š (π‘₯, 𝑦, 𝑑) ,

𝑒𝑖 (π‘₯, 𝑦, 𝑑) = βˆ‘ π‘’π‘–π‘š (π‘₯, 𝑦, 𝑑) ,

(12)

When 𝑀 β†’ ∞, we get an accurate approximation of the original equation (7).

4. Analysis of Fractional Davey-Stewartson Equations with the HAM In this section we apply the proposed approach for solving the fractional Davey-Stewartson equations. Without loss of generality, first we separate the amplitude of a surface wave packet π‘ž into real part and imaginary part; that is, π‘ž = 𝑒 + 𝑖V, (𝑖2 = βˆ’1). Then we rewrite the fractional DaveyStewartson equations in the following form: πœ•π›Ό 𝑒 1 πœ•2 𝑒 2 πœ•V + βˆ’ 4 𝛼 2 2 πœ•π‘¦ 𝜎 πœ•π‘₯ 𝜎 πœ•π‘‘

(14)

π‘š = 0, 1, 2, . . . .

+

Differentiating (8) π‘š times with respect to the embedding parameter 𝑝, then setting 𝑝 = 0, and finally dividing them by π‘š!, we obtain the π‘šth-order deformation equations:

2πœ† 3 2 πœ•πœ™ (𝑒 + V2 𝑒) βˆ’ 4 ( 𝑒) = 0, 4 𝜎 𝜎 πœ•π‘₯

1 πœ•2 V 2 πœ•π‘’ πœ•π›Ό V + + πœ•π‘¦π›Ό 𝜎2 πœ•π‘₯2 𝜎4 πœ•π‘‘

L𝑖 [π‘’π‘–π‘š (π‘₯, 𝑦, 𝑑) βˆ’ πœ’π‘š 𝑒𝑖 π‘šβˆ’1 (π‘₯, 𝑦, 𝑑)]

+

= β„Žπ‘– 𝑅𝑖,π‘š (𝑒⃗1 π‘šβˆ’1 (π‘₯, 𝑦, 𝑑) , . . . , 𝑒⃗𝑛 π‘šβˆ’1 (π‘₯, 𝑦, 𝑑) , π‘₯, 𝑦, 𝑑) ,

(19)

2πœ† 3 2 πœ•πœ™ (V + 𝑒2 V) βˆ’ 4 ( V) = 0, 𝜎4 𝜎 πœ•π‘₯

2 2 πœ•π›Ό πœ™ 1 πœ•2 πœ™ 2πœ† πœ• (𝑒 + V ) βˆ’ + = 0. πœ•π‘¦π›Ό 𝜎2 πœ•π‘₯2 𝜎2 πœ•π‘₯

π‘š = 1, 2, . . . , (15)

To solve (19) by means of homotopy analysis method, we choose the linear operators

where 𝑅𝑖,π‘š (𝑒⃗1 π‘šβˆ’1 , . . . , 𝑒⃗𝑛 π‘šβˆ’1 , π‘₯, 𝑦, 𝑑)

L1 = L2 = L3 =

1 = (π‘š βˆ’ 1)! πœ•π‘šβˆ’1 N𝑖 [πœ™1 (π‘₯, 𝑦, 𝑑; 𝑝) , . . . , πœ™π‘› (π‘₯, 𝑦, 𝑑; 𝑝) , π‘₯, 𝑦, 𝑑] 󡄨󡄨󡄨󡄨 󡄨󡄨 Γ— 󡄨󡄨 πœ•π‘π‘šβˆ’1 󡄨𝑝=0,

πœ•π›Ό , πœ•π‘¦π›Ό

(20)

with the property L𝑖 [𝑐] = 0, where 𝑐 is constant. We now define nonlinear operators as N1 =

πœ•π›Ό 𝑒 1 πœ•2 𝑒 2 πœ•V 2πœ† 3 2 πœ•πœ™ + 2 2βˆ’ 4 (𝑒 + V2 𝑒) βˆ’ 4 ( 𝑒) , + 𝛼 πœ•π‘¦ 𝜎 πœ•π‘₯ 𝜎 πœ•π‘‘ 𝜎4 𝜎 πœ•π‘₯

(16)

N2 =

πœ•π›Ό V 1 πœ•2 V 2 πœ•π‘’ 2πœ† 3 2 πœ•πœ™ + + (V + 𝑒2 V) βˆ’ 4 ( V) , + πœ•π‘¦π›Ό 𝜎2 πœ•π‘₯2 𝜎4 πœ•π‘‘ 𝜎4 𝜎 πœ•π‘₯

The solution of the π‘šth-order deformation equation (15) is readily found to be

N3 =

2 2 πœ•π›Ό πœ™ 1 πœ•2 πœ™ 2πœ† πœ• (𝑒 + V ) βˆ’ + . πœ•π‘¦π›Ό 𝜎2 πœ•π‘₯2 𝜎2 πœ•π‘₯

πœ’π‘š = {

0, 1,

π‘š β©½ 1, π‘š > 1.

(21)

π‘’π‘–π‘š (π‘₯, 𝑦, 𝑑)

The initial guesses are considered as follows:

= πœ’π‘š 𝑒𝑖 π‘šβˆ’1 (π‘₯, 𝑦, 𝑑) + β„Žπ‘– Lβˆ’1 𝑖 Γ— [𝑅𝑖,π‘š (𝑒⃗1 π‘šβˆ’1 (π‘₯, 𝑦, 𝑑) , . . . , 𝑒⃗𝑛 π‘šβˆ’1 (π‘₯, 𝑦, 𝑑) , π‘₯, 𝑦, 𝑑)] , π‘š = 1, 2, . . . . (17)

𝑒0 (π‘₯, 𝑦, 𝑑) = 𝑒 (π‘₯, 0, 𝑑) , V0 (π‘₯, 𝑦, 𝑑) = V (π‘₯, 0, 𝑑) , πœ™0 (π‘₯, 𝑦, 𝑑) = πœ™ (π‘₯, 0, 𝑑) .

(22)

4

The Scientific World Journal Table 1: Absolute errors of 𝑒(π‘₯, 𝑦, 𝑑). π‘₯

β„Ž = βˆ’1.1 1.2787 Γ— 10βˆ’8 1.918 Γ— 10βˆ’8 9.9302 Γ— 10βˆ’7 2.1807 Γ— 10βˆ’5

20 17 14 11

0.1 β„Ž = βˆ’1 1.2831 Γ— 10βˆ’8 1.9662 Γ— 10βˆ’8 1.0008 Γ— 10βˆ’6 2.1951 Γ— 10βˆ’5

𝑑 β„Ž = βˆ’1.1 8.3105 Γ— 10βˆ’9 2.602 Γ— 10βˆ’8 1.4136 Γ— 10βˆ’6 0.2546 Γ— 10βˆ’5

VIM 1.284 Γ— 10βˆ’8 1.9759 Γ— 10βˆ’8 9.9505 Γ— 10βˆ’7 2.1948 Γ— 10βˆ’5

0.5 β„Ž = βˆ’1 8.355 Γ— 10βˆ’9 2.6289 Γ— 10βˆ’8 1.4301 Γ— 10βˆ’6 0.2551 Γ— 10βˆ’5

VIM 8.3639 Γ— 10βˆ’9 2.6326 Γ— 10βˆ’8 1.4247 Γ— 10βˆ’6 2.5473 Γ— 10βˆ’5

0.5 β„Ž = βˆ’1 4.9292 Γ— 10βˆ’8 5.1921 Γ— 10βˆ’7 4.8858 Γ— 10βˆ’6 4.2922 Γ— 10βˆ’5

VIM 4.7834 Γ— 10βˆ’8 5.0445 Γ— 10βˆ’7 4.8504 Γ— 10βˆ’6 4.1826 Γ— 10βˆ’5

0.5 β„Ž = βˆ’1 6.661 Γ— 10βˆ’16 5.306 Γ— 10βˆ’14 5.655 Γ— 10βˆ’12 5.952 Γ— 10βˆ’10

VIM 9.992 Γ— 10βˆ’16 1.122 Γ— 10βˆ’13 1.219 Γ— 10βˆ’11 1.329 Γ— 10βˆ’9

Table 2: Absolute errors of V(π‘₯, 𝑦, 𝑑). π‘₯

β„Ž = βˆ’1.1 4.7113 Γ— 10βˆ’8 5.1811 Γ— 10βˆ’7 5.1731 Γ— 10βˆ’6 4.6771 Γ— 10βˆ’5

20 17 14 11

0.1 β„Ž = βˆ’1 4.8836 Γ— 10βˆ’8 5.2365 Γ— 10βˆ’8 5.2219 Γ— 10βˆ’6 4.8037 Γ— 10βˆ’5

𝑑 β„Ž = βˆ’1.1 4.7378 Γ— 10βˆ’8 5.0412 Γ— 10βˆ’7 4.8430 Γ— 10βˆ’6 4.1509 Γ— 10βˆ’5

VIM 4.7367 Γ— 10βˆ’8 5.1823 Γ— 10βˆ’7 5.1745 Γ— 10βˆ’6 4.6775 Γ— 10βˆ’5

Table 3: Absolute errors of πœ™(π‘₯, 𝑦, 𝑑). π‘₯

β„Ž = βˆ’1.1 4.440 Γ— 10βˆ’16 5.584 Γ— 10βˆ’14 5.956 Γ— 10βˆ’12 6.288 Γ— 10βˆ’10

20 17 14 11

0.1 β„Ž = βˆ’1 8.881 Γ— 10βˆ’16 8.815 Γ— 10βˆ’14 9.577 Γ— 10βˆ’12 1.397 Γ— 10βˆ’9

𝑑

In view of the discussion in Section 3, we get the following recursive relations: 𝑒1 (π‘₯, 𝑦, 𝑑) = β„ŽπΌπ‘¦π›Ό [R11 (𝑒⃗0 , Vβƒ—0 , πœ™0βƒ— , π‘₯, 𝑦, 𝑑)] , V1 (π‘₯, 𝑦, 𝑑) =

β„ŽπΌπ‘¦π›Ό

[R21 (𝑒⃗0 , Vβƒ—0 , πœ™0βƒ— , π‘₯, 𝑦, 𝑑)] ,

(23)

πœ™1 (π‘₯, 𝑦, 𝑑) = β„ŽπΌπ‘¦π›Ό [R31 (𝑒⃗0 , Vβƒ—0 , πœ™0βƒ— , π‘₯, 𝑦, 𝑑)] , π‘’π‘š+1 (π‘₯, 𝑦, 𝑑)

Vπ‘š+1 (π‘₯, 𝑦, 𝑑)

βƒ— , π‘₯, 𝑦, 𝑑)] , = πœ™π‘š (π‘₯, 𝑦, 𝑑) + β„ŽπΌπ‘¦π›Ό [R3π‘š+1 (π‘’βƒ—π‘š , Vβƒ—π‘š , πœ™π‘š π‘š = 1, 2, . . . ,

πœ•π›Ό π‘’π‘š 1 πœ•2 π‘’π‘š 2 πœ•V + βˆ’ 4 π‘š πœ•π‘¦π›Ό 𝜎2 πœ•π‘₯2 𝜎 πœ•π‘‘

(24)

πœ•π›Ό Vπ‘š 1 πœ•2 Vπ‘š 2 πœ•π‘’ + + 4 π‘š 𝛼 2 2 πœ•π‘¦ 𝜎 πœ•π‘₯ 𝜎 πœ•π‘‘ π‘šβˆ’π‘— π‘šβˆ’π‘— π‘š } 2πœ† { π‘š V V V ) + ( V ( βˆ‘ βˆ‘ βˆ‘ 𝑗 𝑖 π‘šβˆ’π‘—βˆ’π‘– 𝑗 βˆ‘ 𝑒𝑖 π‘’π‘šβˆ’π‘—βˆ’π‘– )} { 4 𝜎 𝑗=0 𝑖=0 { 𝑗=0 𝑖=0 }

} 2 { π‘š πœ•πœ™π‘— Vπ‘šβˆ’π‘— } , βˆ‘ { 4 𝜎 𝑗=0 πœ•π‘₯ { } βƒ— , π‘₯, 𝑦, 𝑑) R3π‘š+1 (π‘’βƒ—π‘š , Vβƒ—π‘š , πœ™π‘š βˆ’

πœ™π‘š+1 (π‘₯, 𝑦, 𝑑)

=

} 2 { π‘š πœ•πœ™π‘— π‘’π‘šβˆ’π‘— } , βˆ‘ { 4 𝜎 𝑗=0 πœ•π‘₯ { } βƒ— R2π‘š+1 (π‘’βƒ—π‘š , Vβƒ—π‘š , πœ™π‘š , π‘₯, 𝑦, 𝑑) βˆ’

+

βƒ— , π‘₯, 𝑦, 𝑑)] , = Vπ‘š (π‘₯, 𝑦, 𝑑) + β„ŽπΌπ‘¦π›Ό [R2π‘š+1 (π‘’βƒ—π‘š , Vβƒ—π‘š , πœ™π‘š

βƒ— , π‘₯, 𝑦, 𝑑) R1π‘š+1 (π‘’βƒ—π‘š , Vβƒ—π‘š , πœ™π‘š

π‘šβˆ’π‘— π‘šβˆ’π‘— π‘š } 2πœ† { π‘š + 4 {( βˆ‘π‘’π‘— βˆ‘ 𝑒𝑖 π‘’π‘šβˆ’π‘—βˆ’π‘– ) + ( βˆ‘ 𝑒𝑗 βˆ‘ V𝑖 Vπ‘šβˆ’π‘—βˆ’π‘– )} 𝜎 𝑗=0 𝑖=0 { 𝑗=0 𝑖=0 }

=

βƒ— , π‘₯, 𝑦, 𝑑)] , = π‘’π‘š (π‘₯, 𝑦, 𝑑) + β„ŽπΌπ‘¦π›Ό [R1π‘š+1 (π‘’βƒ—π‘š , Vβƒ—π‘š , πœ™π‘š

where

β„Ž = βˆ’1.1 4.737 Γ— 10βˆ’16 8.459 Γ— 10βˆ’14 9.205 Γ— 10βˆ’12 1.007 Γ— 10βˆ’9

VIM 9.992 Γ— 10βˆ’16 1.169 Γ— 10βˆ’13 1.269 Γ— 10βˆ’11 1.383 Γ— 10βˆ’9

=

+

πœ•π›Ό πœ™π‘š 1 πœ•2 πœ™π‘š βˆ’ πœ•π‘¦π›Ό 𝜎2 πœ•π‘₯2 π‘š π‘š } 2πœ† { πœ• 𝑒 𝑒 ) + ( V𝑗 Vπ‘šβˆ’π‘— ))} . (( βˆ‘ βˆ‘ 𝑗 π‘šβˆ’π‘— { 2 𝜎 πœ•π‘₯ 𝑗=0 𝑗=0 { }

(25)

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5

0.4

4

0.2 0 βˆ’10

3

0 βˆ’10

2 0

4

0.2

3 βˆ’5

0.4

βˆ’5

1

5

2 0

1

5

10 0

(a)

10 0

(b) 0

u[x, 0.3, 10]

βˆ’0.1 βˆ’0.2 βˆ’0.3 βˆ’0.4 0

1

2

3

4

5

x

(c)

Figure 1: (a) and (b) The surface shows the solution 𝑒(π‘₯, 𝑦, 𝑑) for (8): (a) approximate solution for 𝛼 = 1.98, β„Ž = βˆ’1.1; (b) exact solution. (c) Four profiles of approximate solutions 𝑒(π‘₯, 𝑦, 𝑑) for some values of 𝛼: blue line (𝛼 = 2), mauve line (𝛼 = 1.9), green line (𝛼 = 1.8), and red line (𝛼 = 1.5), when π‘˜1 = 0.1, π‘˜2 = 0.03, π‘˜3 = βˆ’0.3, 𝜎 = 𝐼, and πœ† = 1.

5. Results Analysis

By the same manipulation as in Section 4, we will have

In this section, some numerical results are presented to support our theatrical analysis. We consider the following initial conditions: 𝑒 (π‘₯, 0, 𝑑) = π‘Ÿ sech [𝑠 (π‘₯ βˆ’ 𝑐𝑑)] cos [(π‘˜1 π‘₯ + π‘˜3 𝑑)] , V (π‘₯, 0, 𝑑) = π‘Ÿ sech [𝑠 (π‘₯ βˆ’ 𝑐𝑑)] sin [(π‘˜1 π‘₯ + π‘˜3 𝑑)] ,

𝑒1 = β„Ž [βˆ’

2π‘“π‘Ÿπ‘ π‘¦π›Ό sech[𝑠 (βˆ’π‘π‘‘ + π‘₯)]3 cos [π‘₯π‘˜1 + π‘‘π‘˜3 ] 𝜎4 Ξ“ [𝛼 + 1]

βˆ’

π‘Ÿπ‘ 2 𝑦𝛼 sech[𝑠 (βˆ’π‘π‘‘ + π‘₯)]3 cos [π‘₯π‘˜1 + π‘‘π‘˜3 ] 𝜎2 Ξ“ [𝛼 + 1]

+

2π‘Ÿ3 𝑦𝛼 πœ† cos [π‘₯π‘˜1 + π‘‘π‘˜3 ] sech[𝑠 (βˆ’π‘π‘‘ + π‘₯)]3 𝜎4 Ξ“ [𝛼 + 1]

+

2π‘Ÿ3 𝑦𝛼 πœ†sech[𝑠 (βˆ’π‘π‘‘ + π‘₯)]3 sin [π‘₯π‘˜1 + π‘‘π‘˜3 ] cos [π‘₯π‘˜1 + π‘‘π‘˜3 ] 𝜎4 Ξ“ [𝛼 + 1]

βˆ’

π‘Ÿπ‘¦π›Ό sech [𝑠 (βˆ’π‘π‘‘ + π‘₯)] cos [π‘₯π‘˜1 + π‘‘π‘˜3 ] π‘˜12 𝜎2 Ξ“ [𝛼 + 1]

βˆ’

2π‘Ÿπ‘¦π›Ό sech [𝑠 (βˆ’π‘π‘‘ + π‘₯)] cos [π‘₯π‘˜1 + π‘‘π‘˜3 ] π‘˜3 𝜎4 Ξ“ [𝛼 + 1]

βˆ’

2π‘π‘Ÿπ‘ π‘¦π›Ό sin [π‘₯π‘˜1 + π‘‘π‘˜3 ] sech [𝑠 (βˆ’π‘π‘‘ + π‘₯)] tanh [𝑠 (βˆ’π‘π‘‘ + π‘₯)] 𝜎4 Ξ“ [𝛼 + 1]

(26)

πœ™ (π‘₯, 0, 𝑑) = 𝑓 tanh [𝑠 (π‘₯ βˆ’ 𝑐𝑑)] , where

3

2

𝑐 = π‘˜2 + 𝜎 π‘˜1 ,

2

(2π‘˜3 + π‘˜12 𝜎2 + π‘˜22 ) √ π‘Ÿ= βˆ’ , πœ† 𝑠= 𝑓=

2 2 2 √ (2π‘˜3 + π‘˜1 𝜎 + π‘˜2 )

𝜎2

(2πœŽβˆšβˆ’πœ†) (1 βˆ’ 𝜎2 )

(27) ,

,

and π‘˜π‘– (𝑖 = 1, 2, 3) are arbitrary constants.

6

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0 βˆ’0.1 βˆ’0.2 βˆ’0.3 βˆ’10

0 βˆ’0.1 βˆ’0.2 βˆ’0.3 βˆ’10

4 3 2

βˆ’5

0

4 3

0

1 5

2

βˆ’5

1 5

10 0

(a)

10 0

(b)

0

 [x, 0.3, 10]

βˆ’0.02 βˆ’0.04 βˆ’0.06 βˆ’0.08 0

1

2

3

4

5

x

(c)

Figure 2: (a) and (b) The surface shows the solution V(π‘₯, 𝑦, 𝑑) for (8): (a) approximate solution for 𝛼 = 1.98, β„Ž = βˆ’1.1; (b) exact solution. (c) Four profiles of approximate solutions V(π‘₯, 𝑦, 𝑑) for some values of 𝛼: blue line (𝛼 = 2), mauve line (𝛼 = 1.9), green line (𝛼 = 1.8), and red line (𝛼 = 1.5), when π‘˜1 = 0.1, π‘˜2 = 0.03, π‘˜3 = βˆ’0.3, 𝜎 = 𝐼, and πœ† = 1.

+

2π‘Ÿπ‘ π‘¦π›Ό sin [π‘₯π‘˜1 + π‘‘π‘˜3 ] sech [𝑠 (βˆ’π‘π‘‘ + π‘₯)] π‘˜1 tanh [𝑠 (βˆ’π‘π‘‘ + π‘₯)] 𝜎2 Ξ“ [𝛼 + 1]

+

2π‘π‘Ÿπ‘ π‘¦π›Ό cos [π‘₯π‘˜1 + π‘‘π‘˜3 ] sech [𝑠 (βˆ’π‘π‘‘ + π‘₯)] tanh [𝑠 (βˆ’π‘π‘‘ + π‘₯)] 𝜎4 Ξ“ [𝛼 + 1]

+

π‘Ÿπ‘ 2 𝑦𝛼 sech [𝑠 (βˆ’π‘π‘‘ + π‘₯)] cos [π‘₯π‘˜1 + π‘‘π‘˜3 ] tanh [𝑠 (βˆ’π‘π‘‘ + π‘₯)]2 ], Ξ“ [𝛼 + 1] 𝜎2

βˆ’

2π‘Ÿπ‘ π‘¦π›Ό cos [π‘₯π‘˜1 + π‘‘π‘˜3 ] sech [𝑠 (βˆ’π‘π‘‘ + π‘₯)] π‘˜1 tanh [𝑠 (βˆ’π‘π‘‘ + π‘₯)] Ξ“ [𝛼 + 1] 𝜎2

+

π‘Ÿπ‘ 2 𝑦𝛼 sech [𝑠 (βˆ’π‘π‘‘ + π‘₯)] sin [π‘₯π‘˜1 + π‘‘π‘˜3 ] tanh [𝑠 (βˆ’π‘π‘‘ + π‘₯)]2 ], Ξ“ [𝛼 + 1] 𝜎2 (28)

V1 = β„Ž [βˆ’

2π‘“π‘Ÿπ‘ π‘¦π›Ό sech[𝑠 (βˆ’π‘π‘‘ + π‘₯)]3 sin [π‘₯π‘˜1 + π‘‘π‘˜3 ] Ξ“ [𝛼 + 1] 𝜎4

π‘Ÿπ‘ 2 𝑦𝛼 sech[𝑠 (βˆ’π‘π‘‘ + π‘₯)]3 sin [π‘₯π‘˜1 + π‘‘π‘˜3 ] βˆ’ 𝜎2 Ξ“ [𝛼 + 1]

πœ™1 =

2

+

2π‘Ÿ3 𝑦𝛼 πœ† cos [π‘₯π‘˜1 + π‘‘π‘˜3 ] sech[𝑠 (βˆ’π‘π‘‘ + π‘₯)]3 sin [π‘₯π‘˜1 + π‘‘π‘˜3 ] 𝜎4 Ξ“ [𝛼 + 1]

+

2π‘Ÿ3 𝑦𝛼 πœ†sech[𝑠 (βˆ’π‘π‘‘ + π‘₯)]3 sin [π‘₯π‘˜1 + π‘‘π‘˜3 ] 𝜎4 Ξ“ [𝛼 + 1]

βˆ’

π‘Ÿπ‘¦π›Ό sech [𝑠 (βˆ’π‘π‘‘ + π‘₯)] sin [π‘₯π‘˜1 + π‘‘π‘˜3 ] π‘˜12 𝜎2 Ξ“ [𝛼 + 1]

3

π‘Ÿπ‘¦π›Ό sech [𝑠 (βˆ’π‘π‘‘ + π‘₯)] sin [π‘₯π‘˜1 + π‘‘π‘˜3 ] π‘˜3 βˆ’ 𝜎4 Ξ“ [𝛼 + 1]

2β„Žπ‘ π‘¦π›Ό (𝑓𝑠 βˆ’ 2π‘Ÿ2 πœ†) sech[𝑠 (βˆ’π‘π‘‘ + π‘₯)]2 tanh [𝑠 (βˆ’π‘π‘‘ + π‘₯)] Ξ“ [𝛼+1] 𝜎2

.

(29)

In the same manner, using recurrence relations in (24) the other components V2 (π‘₯, 𝑦, 𝑑), V3 (π‘₯, 𝑦, 𝑑), . . . , 𝑒2 (π‘₯, 𝑦, 𝑑), 𝑒3 (π‘₯, 𝑦, 𝑑), . . ., and πœ™2 (π‘₯, 𝑦, 𝑑), πœ™3 (π‘₯, 𝑦, 𝑑), . . . can be obtained.

6. Convergence and Stability Analysis This section is devoted to prove the convergence and stability of solutions for fractional initial value problems on a finite interval of the complex axis in spaces of continuous functions.

The Scientific World Journal

1 0.5 0 βˆ’0.5 βˆ’1 βˆ’10

7

4 3 2 βˆ’5

1 0.5 0 βˆ’0.5 βˆ’1 βˆ’10

4 3

1

0 5

2

βˆ’5

1

0 5

10 0

10 0

(a)

(b)

πœ™ [x, 0.3, 10]

βˆ’0.85 βˆ’0.9 βˆ’0.95 βˆ’1 0

1

2

3

4

5

x (c)

Figure 3: (a) and (b) The surface shows the solution πœ™(π‘₯, 𝑦, 𝑑) for (8): (a) approximate solution for 𝛼 = 1.98, β„Ž = βˆ’1.1; (b) exact solution. (c) Four profiles of approximate solutions πœ™(π‘₯, 𝑦, 𝑑) for some values of 𝛼: blue line (𝛼 = 2), mauve line (𝛼 = 1.9), green line (𝛼 = 1.8), and red line (𝛼 = 1.5), when π‘˜1 = 0.1, π‘˜2 = 0.03, π‘˜3 = βˆ’0.3, 𝜎 = 𝐼, and πœ† = 1.

Theorem 6. If the series 𝑒𝑖 (π‘₯, 𝑦, 𝑑) = βˆ‘βˆž π‘š=0 π‘’π‘–π‘š (π‘₯, 𝑦, 𝑑), 𝑖 = 1, 2, . . . , 𝑛, converges, where π‘’π‘–π‘š (π‘₯, 𝑦, 𝑑) is governed by (15) under the definitions (16), it must be the solution of (7).

evaluating the approximate solutions for Tables 1, 2, and 3. Both the exact solutions and the approximate solutions of 𝑒(π‘₯, 𝑦, 𝑑), V(π‘₯, 𝑦, 𝑑), and πœ™(π‘₯, 𝑦, 𝑑) (for the same parameters as mentioned before) are plotted in Figures 1, 2, and 3.

Proof. Proof is similar to Theorem 3.1 in [17]. Clear conclusion can be drawn from the numerical results and Theorem 6. Our approach provides highly accurate numerical solutions without spatial discretization for the problems. Overall, results show that the proposed approach is unconditionally stable and convergent. In other words, we can always find a proper value of the convergence control parameter β„Ž to ensure the convergent series solution, and our approximate results agree well with numerical ones. It should be pointed out that the response and stability of this type of problems in general can also be studied in a similar way. For further information see [25]. Tables 1, 2, and 3 show the absolute errors between the approximate solutions obtained for value of 𝛼 = 1.98 by the homotopy analysis method and the exact solutions. It is to be noted that only the two-order term of the homotopy analysis method solutions for the special case 𝑦 = 0.2, π‘˜1 = 0.1, π‘˜2 = 0.03, π‘˜3 = βˆ’0.3, 𝜎 = 𝑖, and πœ† = 1 is used in

7. Concluding Remarks In this paper, the homotopy analysis method has been successfully applied to find the solution of fractional order Davey-Stewartson equations. The convergence and stability of the HAM solution was examined. Results reveal that the solution obtained by the homotopy analysis method is an infinite power series for appropriate initial condition, which can, in turn, be expressed in a closed form, the exact solution. Moreover, in the comparison of HAM with VIM method we will find better approximations. The results show that the homotopy analysis method is a powerful mathematical tool for solving Davey-Stewartson equations of fractional order. In other words, the proposed approach is also a promising method to solve other nonlinear equations. Finally, HAM yields convergent solutions for all values of the relevant parameters whereas a previous study only provided convergent approximate solutions for small 𝛼. We pointed out

8 that the corresponding analytical and numerical solutions are obtained using Mathematica.

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