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 [1β 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 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 [14β17]. 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: ππ ππ 1 4 ππΌ π 1 2 π2 π σ΅¨ σ΅¨2 + π + π + πσ΅¨σ΅¨σ΅¨πσ΅¨σ΅¨σ΅¨ π β π π = 0, 2 ππ¦πΌ 2 ππ₯2 ππ‘ ππ₯ σ΅¨ σ΅¨2 πΌ π (σ΅¨σ΅¨σ΅¨πσ΅¨σ΅¨σ΅¨ ) π2 π 2π π βπ β 2π = 0, ππ₯2 ππ¦πΌ ππ₯ 1 < πΌ β€ 2,
π β R,
(1)
π β C.
The special case {πΌ = 2, π = 1} is called the classical DS-I equation, while {πΌ = 2, π = Β±ββ1} is the classical DS-II equation. The parameter π 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
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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, 22β24]. Definition 1. A real function π’(π‘), π‘ > 0, is said to be in the space πΆπΌ , πΌ β R, if there exists a real number π (>πΌ), such that π’(π‘) = π‘π π’1 (π‘), where π’1 (π‘) β πΆ[0, β), and it is said to be in the space πΆπΌπ , π β N β {0}, if and only if π’(π) (π‘) β πΆπΌ . Definition 2. The (left sided) Riemann-Liouville fractional integral of order πΌ > 0 of a function π’(π‘) β πΆπΌ , πΌ β₯ β1, is defined as π‘ π’ (π) 1 { { ππ, πΌ > 0, π‘ > 0, β« πΌ πΌπ‘ π’ (π‘) = { Ξ (πΌ) 0 (π‘ β π)1βπΌ { π’ , { (π‘)
πΌπ‘πΌ π’ (π₯, π‘) =
π’ (π₯, π ) 1 ππ , β« Ξ (πΌ) 0 (π‘ β π )1βπΌ
Theorem 5. Assume that the continuous function π’(π‘) has a fractional derivative of order πΌ; then one has π½βπΌ
2. Preliminaries and Notations
π‘
2.1. The Relation between Fractional Derivative and Fractional Integral
π½ π·π‘πΌ πΌπ‘ π’ (π‘)
πΌ π’ (π‘) { {π‘ = {π’ (π‘) { βπ½+πΌ {π·π‘ π’ (π‘)
πΌ < π½, πΌ = π½, πΌ > π½,
πβ1
πΌ πΌπ‘πΌ π·βπ‘ π’ (π‘) = π’ (π‘) β β π’(π) (0+ ) π=0
π β 1 < πΌ β€ π, πΌ πΌ π·βπ‘ πΌπ‘ π’ (π‘) = {
π‘π , π!
π β N,
π’ (π‘) , π β 1 < πΌ β€ π, π β N, πΌ πΌπ‘πΌ π·βπ‘ π’ (π‘) + π’ (0) , 0 < πΌ < 1. (6)
3. Homotopy Analysis Method Let us consider the following system of differential equations: (2)
πΌ > 0, π‘ > 0,
Nπ [π’1 (π₯, π¦, π‘) , . . . , π’π (π₯, π¦, π‘) , π₯, π¦, π‘] = 0,
π = 1, 2, . . . , π, (7)
where Nπ are nonlinear operators and π’π (π₯, π¦, π‘) 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 derivative of π’(π‘), π’(π‘) β πΆβ1 , π β N β{0}, of order πΌ is defined as
(1 β π) Lπ [ππ (π₯, π¦, π‘; π) β π’π0 (π₯, π¦, π‘)] = πβπ Nπ [π1 (π₯, π¦, π‘; π) , . . . , ππ (π₯, π¦, π‘; π) , π₯, π¦, π‘] , (8) π = 1, 2, . . . , π,
π·π‘πΌ π’ (π‘) =
π
π πβπΌ πΌ π’ (π‘) , ππ‘π π‘
π β 1 < πΌ β€ π, π β N.
(3)
Definition 4. The (left sided) Caputo fractional derivative of π , π β N β{0}, is defined as π’(π‘), π’(π‘) β πΆβ1 πβπΌ (π) {[πΌπ‘ π’ (π‘)] π β 1 < πΌ < π, π β N, πΌ π·βπ‘ π’ (π‘) = { ππ π’ (π‘) πΌ = π, { ππ‘π πΌ π·βπ‘ π’ (π₯, π‘) = πΌπ‘πβπΌ πΌ π·π‘π π’ (π‘) π·βπ‘
=
ππ π’ (π₯, π‘) , ππ‘π
πΌ+π π·βπ‘ π’ (π‘) ,
where π β [0, 1] is the embedding parameter, βπ =ΜΈ 0 are nonzero auxiliary parameters, and Lπ are auxiliary linear operators with the following property: Lπ [π] = 0,
π = 1, 2, . . . , π,
where π is constant. π’π0 (π₯, π¦, π‘) are initial guesses of π’π (π₯, π¦, π‘), ππ (π₯, π¦, π‘; π) are unknown functions, respectively. It is important that one has great freedom to choose auxiliary things in HAM. Obviously, when π = 0 and π = 1, it holds ππ (π₯, π¦, π‘; 0) = π’π0 (π₯, π¦, π‘) , ππ (π₯, π¦, π‘; 1) = π’π (π₯, π¦, π‘) ,
π β 1 < πΌ < π,
(10)
π = 1, 2, . . . , π,
π = 0, 1, . . . , π β 1 < πΌ < π. (4)
Property. Similar to integer-order differentiation, fractional differentiation is a linear operation:
respectively. Thus, as π increases from 0 to 1, the solution ππ (π₯, π¦, π‘; π) varies from the initial guesses π’π0 (π₯, π¦, π‘) to the solution π’π (π₯, π¦, π‘). Expanding ππ (π₯, π¦, π‘; π) in Taylor series with respect to π, we have +β
ππ (π₯, π¦, π‘; π) = π’π0 (π₯, π¦, π‘) + β π’ππ (π₯, π¦, π‘) ππ , π=1
πΌ πΌ πΌ π·βπ‘ π’ (π‘) + ππ·βπ‘ V (π‘) . (πΎπ’ (π‘) + πV (π‘)) = πΎπ·βπ‘
(9)
(5)
π = 1, 2, . . . , π,
(11)
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3 In this way, it is easy to obtain π’ππ (π₯, π¦, π‘) for π β©Ύ 1, at πthorder; we have
where π’ππ (π₯, π¦, π‘) =
σ΅¨ π 1 π ππ (π₯, π¦, π‘; π) σ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ π! πππ σ΅¨π=0,
π
If the auxiliary linear operators, the initial guesses, the auxiliary parameters βπ are so properly chosen, the series (11) converges at π = 1; then we have +β
π=1
(13)
π = 1, 2, . . . , π. Define the vector π’βππ (π₯, π¦, π‘) = {π’π0 (π₯, π¦, π‘) , π’π1 (π₯, π¦, π‘) , . . . , π’ππ (π₯, π¦, π‘)} , π = 1, 2, . . . , π,
π = 1, 2, . . . , π.
(18)
π=0
π = 1, 2, . . . , π.
π’ππ (π₯, π¦, π‘) = π’π0 (π₯, π¦, π‘) + β π’ππ (π₯, π¦, π‘) ,
π’π (π₯, π¦, π‘) = β π’ππ (π₯, π¦, π‘) ,
(12)
When π β 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 into real part and imaginary part; that is, π = π’ + πV, (π2 = β1). Then we rewrite the fractional DaveyStewartson equations in the following form: ππΌ π’ 1 π2 π’ 2 πV + β 4 πΌ 2 2 ππ¦ π ππ₯ π ππ‘
(14)
π = 0, 1, 2, . . . .
+
Differentiating (8) π times with respect to the embedding parameter π, then setting π = 0, and finally dividing them by π!, we obtain the πth-order deformation equations:
2π 3 2 ππ (π’ + V2 π’) β 4 ( π’) = 0, 4 π π ππ₯
1 π2 V 2 ππ’ ππΌ V + + ππ¦πΌ π2 ππ₯2 π4 ππ‘
Lπ [π’ππ (π₯, π¦, π‘) β ππ π’π πβ1 (π₯, π¦, π‘)]
+
= βπ π
π,π (π’β1 πβ1 (π₯, π¦, π‘) , . . . , π’βπ πβ1 (π₯, π¦, π‘) , π₯, π¦, π‘) ,
(19)
2π 3 2 ππ (V + π’2 V) β 4 ( V) = 0, π4 π ππ₯
2 2 ππΌ π 1 π2 π 2π π (π’ + V ) β + = 0. ππ¦πΌ π2 ππ₯2 π2 ππ₯
π = 1, 2, . . . , (15)
To solve (19) by means of homotopy analysis method, we choose the linear operators
where π
π,π (π’β1 πβ1 , . . . , π’βπ πβ1 , π₯, π¦, π‘)
L1 = L2 = L3 =
1 = (π β 1)! ππβ1 Nπ [π1 (π₯, π¦, π‘; π) , . . . , ππ (π₯, π¦, π‘; π) , π₯, π¦, π‘] σ΅¨σ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨ Γ σ΅¨σ΅¨ πππβ1 σ΅¨π=0,
ππΌ , ππ¦πΌ
(20)
with the property Lπ [π] = 0, where π is constant. We now define nonlinear operators as N1 =
ππΌ π’ 1 π2 π’ 2 πV 2π 3 2 ππ + 2 2β 4 (π’ + V2 π’) β 4 ( π’) , + πΌ ππ¦ π ππ₯ π ππ‘ π4 π ππ₯
(16)
N2 =
ππΌ V 1 π2 V 2 ππ’ 2π 3 2 ππ + + (V + π’2 V) β 4 ( V) , + ππ¦πΌ π2 ππ₯2 π4 ππ‘ π4 π ππ₯
The solution of the πth-order deformation equation (15) is readily found to be
N3 =
2 2 ππΌ π 1 π2 π 2π π (π’ + V ) β + . ππ¦πΌ π2 ππ₯2 π2 ππ₯
ππ = {
0, 1,
π β©½ 1, π > 1.
(21)
π’ππ (π₯, π¦, π‘)
The initial guesses are considered as follows:
= ππ π’π πβ1 (π₯, π¦, π‘) + βπ Lβ1 π Γ [π
π,π (π’β1 πβ1 (π₯, π¦, π‘) , . . . , π’βπ πβ1 (π₯, π¦, π‘) , π₯, π¦, π‘)] , π = 1, 2, . . . . (17)
π’0 (π₯, π¦, π‘) = π’ (π₯, 0, π‘) , V0 (π₯, π¦, π‘) = V (π₯, 0, π‘) , π0 (π₯, π¦, π‘) = π (π₯, 0, π‘) .
(22)
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β = β1.1 1.2787 Γ 10β8 1.918 Γ 10β8 9.9302 Γ 10β7 2.1807 Γ 10β5
20 17 14 11
0.1 β = β1 1.2831 Γ 10β8 1.9662 Γ 10β8 1.0008 Γ 10β6 2.1951 Γ 10β5
π‘ β = β1.1 8.3105 Γ 10β9 2.602 Γ 10β8 1.4136 Γ 10β6 0.2546 Γ 10β5
VIM 1.284 Γ 10β8 1.9759 Γ 10β8 9.9505 Γ 10β7 2.1948 Γ 10β5
0.5 β = β1 8.355 Γ 10β9 2.6289 Γ 10β8 1.4301 Γ 10β6 0.2551 Γ 10β5
VIM 8.3639 Γ 10β9 2.6326 Γ 10β8 1.4247 Γ 10β6 2.5473 Γ 10β5
0.5 β = β1 4.9292 Γ 10β8 5.1921 Γ 10β7 4.8858 Γ 10β6 4.2922 Γ 10β5
VIM 4.7834 Γ 10β8 5.0445 Γ 10β7 4.8504 Γ 10β6 4.1826 Γ 10β5
0.5 β = β1 6.661 Γ 10β16 5.306 Γ 10β14 5.655 Γ 10β12 5.952 Γ 10β10
VIM 9.992 Γ 10β16 1.122 Γ 10β13 1.219 Γ 10β11 1.329 Γ 10β9
Table 2: Absolute errors of V(π₯, π¦, π‘). π₯
β = β1.1 4.7113 Γ 10β8 5.1811 Γ 10β7 5.1731 Γ 10β6 4.6771 Γ 10β5
20 17 14 11
0.1 β = β1 4.8836 Γ 10β8 5.2365 Γ 10β8 5.2219 Γ 10β6 4.8037 Γ 10β5
π‘ β = β1.1 4.7378 Γ 10β8 5.0412 Γ 10β7 4.8430 Γ 10β6 4.1509 Γ 10β5
VIM 4.7367 Γ 10β8 5.1823 Γ 10β7 5.1745 Γ 10β6 4.6775 Γ 10β5
Table 3: Absolute errors of π(π₯, π¦, π‘). π₯
β = β1.1 4.440 Γ 10β16 5.584 Γ 10β14 5.956 Γ 10β12 6.288 Γ 10β10
20 17 14 11
0.1 β = β1 8.881 Γ 10β16 8.815 Γ 10β14 9.577 Γ 10β12 1.397 Γ 10β9
π‘
In view of the discussion in Section 3, we get the following recursive relations: π’1 (π₯, π¦, π‘) = βπΌπ¦πΌ [R11 (π’β0 , Vβ0 , π0β , π₯, π¦, π‘)] , V1 (π₯, π¦, π‘) =
βπΌπ¦πΌ
[R21 (π’β0 , Vβ0 , π0β , π₯, π¦, π‘)] ,
(23)
π1 (π₯, π¦, π‘) = βπΌπ¦πΌ [R31 (π’β0 , Vβ0 , π0β , π₯, π¦, π‘)] , π’π+1 (π₯, π¦, π‘)
Vπ+1 (π₯, π¦, π‘)
β , π₯, π¦, π‘)] , = ππ (π₯, π¦, π‘) + βπΌπ¦πΌ [R3π+1 (π’βπ , Vβπ , ππ π = 1, 2, . . . ,
ππΌ π’π 1 π2 π’π 2 πV + β 4 π ππ¦πΌ π2 ππ₯2 π ππ‘
(24)
ππΌ Vπ 1 π2 Vπ 2 ππ’ + + 4 π πΌ 2 2 ππ¦ π ππ₯ π ππ‘ πβπ πβπ π } 2π { π V V V ) + ( V ( β β β π π πβπβπ π β π’π π’πβπβπ )} { 4 π π=0 π=0 { π=0 π=0 }
} 2 { π πππ Vπβπ } , β { 4 π π=0 ππ₯ { } β , π₯, π¦, π‘) R3π+1 (π’βπ , Vβπ , ππ β
ππ+1 (π₯, π¦, π‘)
=
} 2 { π πππ π’πβπ } , β { 4 π π=0 ππ₯ { } β R2π+1 (π’βπ , Vβπ , ππ , π₯, π¦, π‘) β
+
β , π₯, π¦, π‘)] , = Vπ (π₯, π¦, π‘) + βπΌπ¦πΌ [R2π+1 (π’βπ , Vβπ , ππ
β , π₯, π¦, π‘) R1π+1 (π’βπ , Vβπ , ππ
πβπ πβπ π } 2π { π + 4 {( βπ’π β π’π π’πβπβπ ) + ( β π’π β Vπ Vπβπβπ )} π π=0 π=0 { π=0 π=0 }
=
β , π₯, π¦, π‘)] , = π’π (π₯, π¦, π‘) + βπΌπ¦πΌ [R1π+1 (π’βπ , Vβπ , ππ
where
β = β1.1 4.737 Γ 10β16 8.459 Γ 10β14 9.205 Γ 10β12 1.007 Γ 10β9
VIM 9.992 Γ 10β16 1.169 Γ 10β13 1.269 Γ 10β11 1.383 Γ 10β9
=
+
ππΌ ππ 1 π2 ππ β ππ¦πΌ π2 ππ₯2 π π } 2π { π π’ π’ ) + ( Vπ Vπβπ ))} . (( β β π πβπ { 2 π ππ₯ π=0 π=0 { }
(25)
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5
0.4
4
0.2 0 β10
3
0 β10
2 0
4
0.2
3 β5
0.4
β5
1
5
2 0
1
5
10 0
(a)
10 0
(b) 0
u[x, 0.3, 10]
β0.1 β0.2 β0.3 β0.4 0
1
2
3
4
5
x
(c)
Figure 1: (a) and (b) The surface shows the solution π’(π₯, π¦, π‘) for (8): (a) approximate solution for πΌ = 1.98, β = β1.1; (b) exact solution. (c) Four profiles of approximate solutions π’(π₯, π¦, π‘) for some values of πΌ: blue line (πΌ = 2), mauve line (πΌ = 1.9), green line (πΌ = 1.8), and red line (πΌ = 1.5), when π1 = 0.1, π2 = 0.03, π3 = β0.3, π = πΌ, and π = 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: π’ (π₯, 0, π‘) = π sech [π (π₯ β ππ‘)] cos [(π1 π₯ + π3 π‘)] , V (π₯, 0, π‘) = π sech [π (π₯ β ππ‘)] sin [(π1 π₯ + π3 π‘)] ,
π’1 = β [β
2πππ π¦πΌ sech[π (βππ‘ + π₯)]3 cos [π₯π1 + π‘π3 ] π4 Ξ [πΌ + 1]
β
ππ 2 π¦πΌ sech[π (βππ‘ + π₯)]3 cos [π₯π1 + π‘π3 ] π2 Ξ [πΌ + 1]
+
2π3 π¦πΌ π cos [π₯π1 + π‘π3 ] sech[π (βππ‘ + π₯)]3 π4 Ξ [πΌ + 1]
+
2π3 π¦πΌ πsech[π (βππ‘ + π₯)]3 sin [π₯π1 + π‘π3 ] cos [π₯π1 + π‘π3 ] π4 Ξ [πΌ + 1]
β
ππ¦πΌ sech [π (βππ‘ + π₯)] cos [π₯π1 + π‘π3 ] π12 π2 Ξ [πΌ + 1]
β
2ππ¦πΌ sech [π (βππ‘ + π₯)] cos [π₯π1 + π‘π3 ] π3 π4 Ξ [πΌ + 1]
β
2πππ π¦πΌ sin [π₯π1 + π‘π3 ] sech [π (βππ‘ + π₯)] tanh [π (βππ‘ + π₯)] π4 Ξ [πΌ + 1]
(26)
π (π₯, 0, π‘) = π tanh [π (π₯ β ππ‘)] , where
3
2
π = π2 + π π1 ,
2
(2π3 + π12 π2 + π22 ) β π= β , π π = π=
2 2 2 β (2π3 + π1 π + π2 )
π2
(2πββπ) (1 β π2 )
(27) ,
,
and ππ (π = 1, 2, 3) are arbitrary constants.
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0 β0.1 β0.2 β0.3 β10
0 β0.1 β0.2 β0.3 β10
4 3 2
β5
0
4 3
0
1 5
2
β5
1 5
10 0
(a)
10 0
(b)
0
[x, 0.3, 10]
β0.02 β0.04 β0.06 β0.08 0
1
2
3
4
5
x
(c)
Figure 2: (a) and (b) The surface shows the solution V(π₯, π¦, π‘) for (8): (a) approximate solution for πΌ = 1.98, β = β1.1; (b) exact solution. (c) Four profiles of approximate solutions V(π₯, π¦, π‘) for some values of πΌ: blue line (πΌ = 2), mauve line (πΌ = 1.9), green line (πΌ = 1.8), and red line (πΌ = 1.5), when π1 = 0.1, π2 = 0.03, π3 = β0.3, π = πΌ, and π = 1.
+
2ππ π¦πΌ sin [π₯π1 + π‘π3 ] sech [π (βππ‘ + π₯)] π1 tanh [π (βππ‘ + π₯)] π2 Ξ [πΌ + 1]
+
2πππ π¦πΌ cos [π₯π1 + π‘π3 ] sech [π (βππ‘ + π₯)] tanh [π (βππ‘ + π₯)] π4 Ξ [πΌ + 1]
+
ππ 2 π¦πΌ sech [π (βππ‘ + π₯)] cos [π₯π1 + π‘π3 ] tanh [π (βππ‘ + π₯)]2 ], Ξ [πΌ + 1] π2
β
2ππ π¦πΌ cos [π₯π1 + π‘π3 ] sech [π (βππ‘ + π₯)] π1 tanh [π (βππ‘ + π₯)] Ξ [πΌ + 1] π2
+
ππ 2 π¦πΌ sech [π (βππ‘ + π₯)] sin [π₯π1 + π‘π3 ] tanh [π (βππ‘ + π₯)]2 ], Ξ [πΌ + 1] π2 (28)
V1 = β [β
2πππ π¦πΌ sech[π (βππ‘ + π₯)]3 sin [π₯π1 + π‘π3 ] Ξ [πΌ + 1] π4
ππ 2 π¦πΌ sech[π (βππ‘ + π₯)]3 sin [π₯π1 + π‘π3 ] β π2 Ξ [πΌ + 1]
π1 =
2
+
2π3 π¦πΌ π cos [π₯π1 + π‘π3 ] sech[π (βππ‘ + π₯)]3 sin [π₯π1 + π‘π3 ] π4 Ξ [πΌ + 1]
+
2π3 π¦πΌ πsech[π (βππ‘ + π₯)]3 sin [π₯π1 + π‘π3 ] π4 Ξ [πΌ + 1]
β
ππ¦πΌ sech [π (βππ‘ + π₯)] sin [π₯π1 + π‘π3 ] π12 π2 Ξ [πΌ + 1]
3
ππ¦πΌ sech [π (βππ‘ + π₯)] sin [π₯π1 + π‘π3 ] π3 β π4 Ξ [πΌ + 1]
2βπ π¦πΌ (ππ β 2π2 π) sech[π (βππ‘ + π₯)]2 tanh [π (βππ‘ + π₯)] Ξ [πΌ+1] π2
.
(29)
In the same manner, using recurrence relations in (24) the other components V2 (π₯, π¦, π‘), V3 (π₯, π¦, π‘), . . . , π’2 (π₯, π¦, π‘), π’3 (π₯, π¦, π‘), . . ., and π2 (π₯, π¦, π‘), π3 (π₯, π¦, π‘), . . . 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.
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1 0.5 0 β0.5 β1 β10
7
4 3 2 β5
1 0.5 0 β0.5 β1 β10
4 3
1
0 5
2
β5
1
0 5
10 0
10 0
(a)
(b)
π [x, 0.3, 10]
β0.85 β0.9 β0.95 β1 0
1
2
3
4
5
x (c)
Figure 3: (a) and (b) The surface shows the solution π(π₯, π¦, π‘) for (8): (a) approximate solution for πΌ = 1.98, β = β1.1; (b) exact solution. (c) Four profiles of approximate solutions π(π₯, π¦, π‘) for some values of πΌ: blue line (πΌ = 2), mauve line (πΌ = 1.9), green line (πΌ = 1.8), and red line (πΌ = 1.5), when π1 = 0.1, π2 = 0.03, π3 = β0.3, π = πΌ, and π = 1.
Theorem 6. If the series π’π (π₯, π¦, π‘) = ββ π=0 π’ππ (π₯, π¦, π‘), π = 1, 2, . . . , π, converges, where π’ππ (π₯, π¦, π‘) 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 π’(π₯, π¦, π‘), V(π₯, π¦, π‘), and π(π₯, π¦, π‘) (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 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 πΌ = 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 π¦ = 0.2, π1 = 0.1, π2 = 0.03, π3 = β0.3, π = π, and π = 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 πΌ. We pointed out
8 that the corresponding analytical and numerical solutions are obtained using Mathematica.
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