Davey-Stewartson Equation with Fractional Coordinate Derivatives

Abstract

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 ℏ 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: 12𝜎⁴𝜕^𝛼𝑞𝜕𝑦^𝛼+12𝜎²𝜕²𝑞𝜕𝑥²+𝑖𝜕𝑞𝑞𝜕𝑡+𝜆²𝑞−𝜕𝜙𝜕𝜕𝑥𝑞=0,₂𝜙𝜕𝑥²−𝜎²𝜕^𝛼𝜙𝜕𝑦^𝛼𝜕𝑞−2𝜆²𝜕𝑥=0,1<𝛼≤2,𝜆∈ℜ,𝜎∈𝒞. (1) The special case {𝛼=2,𝜎=1} is called the classical DS-I equation, while {𝛼=2,𝜎=±√−1} is the classical DS-II equation. The parameter 𝜆 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 Schrödinger equation [21]. Although there are a lot of studies for the classical Davey-Stewartson 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 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.

2. Preliminaries and Notations

This section deals with some preliminaries and notations regarding fractional calculus. For more details see [6, 22–24].

Definition 1. A real function 𝑢(𝑡),𝑡>0, is said to be in the space 𝐶_𝛼,𝛼∈ℜ, if there exists a real number 𝑝(>α), such that 𝑢(𝑡)=𝑡^𝑝𝑢₁(𝑡), where 𝑢₁(𝑡)∈𝐶[0,∞), and it is said to be in the space 𝐶^𝑚_𝛼,𝑚∈𝒩⋃{0}, if and only if 𝑢^(𝑚)(𝑡)∈𝐶_𝛼.

Definition 2. The (left sided) Riemann-Liouville fractional integral of order 𝛼>0 of a function 𝑢(𝑡)∈𝐶_𝛼, 𝛼≥−1, is defined as 𝐼^𝛼_𝑡𝑢(𝑡)=1Γ(𝛼)∫^𝑡₀𝑢(𝜏)(𝑡−𝜏)^1−𝛼𝑢(𝑡),𝐼𝑑𝜏,𝛼>0,𝑡>0,_𝛼𝑡𝑢()=1𝑥,𝑡Γ(𝛼)^𝑡₀𝑢(𝑥,𝑠)(𝑡−𝑠)^1−𝛼𝑑𝑠,𝛼>0,𝑡>0, (2) where Γ(⋅) is the well-known Gamma function.

Definition 3. The (left sided) Riemann-Liouville fractional derivative of 𝑢(𝑡),𝑢(𝑡)∈𝐶^𝑚₋₁,𝑚∈𝒩⋃{0}, of order 𝛼 is defined as 𝐷^𝛼_𝑡𝑢(𝑡)=𝑑^𝑚𝑑𝑡^𝑚𝐼_𝑡^𝑚−𝛼𝑢(𝑡),𝑚−1<𝛼≤𝑚,𝑚∈𝒩. (3)

Definition 4. The (left sided) Caputo fractional derivative of 𝑢(𝑡),𝑢(𝑡)∈𝐶^𝑚₋₁,𝑚∈𝒩⋃{0}, is defined as 𝐷^𝛼_∗𝑡𝑢(𝑡)=𝐼_𝑡^𝑚−𝛼𝑢^(𝑚)(𝑡𝑑)𝑚−1<𝛼<𝑚,𝑚∈𝒩,_𝑚𝑑𝑡^𝑚𝑢(𝑡)𝐷𝛼=𝑚,_𝛼∗𝑡𝑢(𝑥,𝑡)=𝐼_𝑡^𝑚−𝛼𝜕^𝑚𝑢(𝑥,𝑡)𝜕𝑡^𝑚𝐷,𝑚−1<𝛼<𝑚,_𝛼∗𝑡𝐷^𝑚_𝑡𝑢(𝑡)=𝐷^𝛼+𝑚_∗𝑡𝑢(𝑡),𝑚=0,1,…,𝑛−1<𝛼<𝑛. (4)Property. Similar to integer-order differentiation, fractional differentiation is a linear operation: 𝐷^𝛼_∗𝑡𝛾𝑢(𝑡)+𝜂𝑣(𝑡)=𝛾𝐷^𝛼_∗𝑡𝑢(𝑡)+𝜂𝐷^𝛼_∗𝑡𝑣(𝑡). (5)

2.1. The Relation between Fractional Derivative and Fractional Integral

Theorem 5. Assume that the continuous function 𝑢(𝑡) has a fractional derivative of order 𝛼; then one has 𝐷^𝛼_𝑡𝐼^𝛽_𝑡𝑢(𝑡)=𝐼_𝑡^𝛽−𝛼𝑢(𝑡)𝑢(𝑡)𝐷𝛼<𝛽,𝛼=𝛽,_{𝑡−𝛽+𝛼}𝑢(𝑡)𝐼𝛼>𝛽,_𝛼𝑡𝐷^𝛼_∗𝑡𝑢(𝑡)=𝑢(𝑡)−^𝑚−1_𝑘=0𝑢^(𝑘)0⁺𝑡^𝑘,𝐷𝑘!𝑚−1<𝛼≤𝑚,𝑚∈𝒩,_𝛼∗𝑡𝐼^𝛼_𝑡𝑢(𝑡)=𝑢(𝑡)𝐼,𝑚−1<𝛼≤𝑚,𝑚∈𝒩,_𝛼𝑡𝐷^𝛼_∗𝑡𝑢(𝑡)+𝑢(0),0<𝛼<1. (6)

3. Homotopy Analysis Method

Let us consider the following system of differential equations: 𝒩_𝑖𝑢₁𝑥,𝑦,𝑡,…,𝑢_𝑛𝑥,𝑦,𝑡,𝑥,𝑦,𝑡=0,𝑖=1,2,…,𝑛, (7) where 𝒩_𝑖 are nonlinear operators and 𝑢_𝑖(𝑥,𝑦,𝑡) are unknown functions. By means of generalizing the traditional homotopy method, Liao [15] constructed the so-called zero-order deformation equations: 1−𝑝ℒ_𝑖𝜙_𝑖𝑥,𝑦,𝑡;𝑝−𝑢_𝑖0𝑥,𝑦,𝑡=𝑝ℏ_𝑖𝒩_𝑖𝜙₁𝑥,𝑦,𝑡;𝑝,…,𝜙_𝑛𝑥,𝑦,𝑡;𝑝,𝑥,𝑦,𝑡,𝑖=1,2,…,𝑛, (8) where 𝑝∈[0,1] is the embedding parameter, ℏ_𝑖≠0 are nonzero auxiliary parameters, and ℒ_𝑖 are auxiliary linear operators with the following property: ℒ_𝑖[𝑐]=0,𝑖=1,2,…,𝑛, (9) where 𝑐 is constant. 𝑢_𝑖0(𝑥,𝑦,𝑡) are initial guesses of 𝑢_𝑖(𝑥,𝑦,𝑡), 𝜙_𝑖(𝑥,𝑦,𝑡;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 𝜙_𝑖𝑥,𝑦,𝑡;0=𝑢_𝑖0,𝜙𝑥,𝑦,𝑡_𝑖𝑥,𝑦,𝑡;1=𝑢_𝑖𝑥,𝑦,𝑡,𝑖=1,2,…,𝑛, (10) respectively. Thus, as 𝑝 increases from 0 to 1, the solution 𝜙_𝑖(𝑥,𝑦,𝑡;𝑝) varies from the initial guesses 𝑢_𝑖0(𝑥,𝑦,𝑡) to the solution 𝑢_𝑖(𝑥,𝑦,𝑡). Expanding 𝜙_𝑖(𝑥,𝑦,𝑡;𝑝) in Taylor series with respect to 𝑝, we have 𝜙_𝑖𝑥,𝑦,𝑡;𝑝=𝑢_𝑖0𝑥,𝑦,𝑡+^+∞_𝑚=1𝑢_𝑖𝑚𝑥,𝑦,𝑡𝑝^𝑚,𝑖=1,2,…,𝑛, (11) where 𝑢_𝑖𝑚=1𝑥,𝑦,𝑡𝜕𝑚!^𝑚𝜙_𝑖𝑥,𝑦,𝑡;𝑝𝜕𝑝^𝑚_𝑝=0,𝑖=1,2,…,𝑛. (12) If the auxiliary linear operators, the initial guesses, the auxiliary parameters ℏ_𝑖 are so properly chosen, the series (11) converges at 𝑝=1; then we have 𝑢_𝑖𝑚𝑥,𝑦,𝑡=𝑢_𝑖0𝑥,𝑦,𝑡+^+∞_𝑚=1𝑢_𝑖𝑚𝑥,𝑦,𝑡,𝑖=1,2,…,𝑛. (13) Define the vector 𝑢_𝑖𝑚=𝑢𝑥,𝑦,𝑡_𝑖0𝑥,𝑦,𝑡,𝑢_𝑖1𝑥,𝑦,𝑡,…,𝑢_𝑖𝑚𝑥,𝑦,𝑡,𝑖=1,2,…,𝑛,𝑚=0,1,2,…. (14) Differentiating (8) 𝑚 times with respect to the embedding parameter 𝑝, then setting 𝑝=0, and finally dividing them by 𝑚!, we obtain the 𝑚th-order deformation equations: ℒ_𝑖𝑢_𝑖𝑚𝑥,𝑦,𝑡−𝜒_𝑚𝑢_𝑖𝑚−1𝑥,𝑦,𝑡=ℏ_𝑖𝑅_𝑖,𝑚𝑢_1𝑚−1𝑥,𝑦,𝑡,…,𝑢_𝑛𝑚−1𝑥,𝑦,𝑡,𝑥,𝑦,𝑡,𝑚=1,2,…, (15) where 𝑅_𝑖,𝑚𝑢_1𝑚−1,…,𝑢_𝑛𝑚−1=1,𝑥,𝑦,𝑡()!×𝜕𝑚−1_𝑚−1𝒩_𝑖𝜙₁𝑥,𝑦,𝑡;𝑝,…,𝜙_𝑛𝑥,𝑦,𝑡;𝑝,𝑥,𝑦,𝑡𝜕𝑝^𝑚−1_𝑝=0,𝜒_𝑚=0,𝑚⩽1,1,𝑚>1. (16) The solution of the 𝑚th-order deformation equation (15) is readily found to be 𝑢_𝑖𝑚𝑥,𝑦,𝑡=𝜒_𝑚𝑢_𝑖𝑚−1𝑥,𝑦,𝑡+ℏ_𝑖ℒ_𝑖⁻¹×𝑅_𝑖,𝑚𝑢_1𝑚−1𝑥,𝑦,𝑡,…,𝑢_𝑛𝑚−1𝑥,𝑦,𝑡,𝑥,𝑦,𝑡,𝑚=1,2,…. (17) In this way, it is easy to obtain 𝑢_𝑖𝑚(𝑥,𝑦,𝑡) for 𝑚⩾1, at 𝑚th-order; we have 𝑢_𝑖𝑥,𝑦,𝑡=^𝑀_𝑚=0𝑢_𝑖𝑚𝑥,𝑦,𝑡,𝑖=1,2,…,𝑛. (18) When 𝑀→∞, 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 𝑞 into real part and imaginary part; that is, 𝑞=𝑢+𝑖𝑣,(𝑖²=−1). Then we rewrite the fractional Davey-Stewartson equations in the following form: 𝜕^𝛼𝑢𝜕𝑦^𝛼+1𝜎²𝜕²𝑢𝜕𝑥²−2𝜎⁴𝜕𝑣+𝜕𝑡2𝜆𝜎⁴𝑢³+𝑣²𝑢−2𝜎⁴𝜕𝜙𝑢𝜕𝜕𝑥=0,_𝛼𝑣𝜕𝑦^𝛼+1𝜎²𝜕²𝑣𝜕𝑥²+2𝜎⁴𝜕𝑢+𝜕𝑡2𝜆𝜎⁴𝑣³+𝑢²𝑣−2𝜎⁴𝜕𝜙𝑣𝜕𝜕𝑥=0,_𝛼𝜙𝜕𝑦^𝛼−1𝜎²𝜕²𝜙𝜕𝑥²+2𝜆𝜎²𝜕𝑢²+𝑣²𝜕𝑥=0. (19) To solve (19) by means of homotopy analysis method, we choose the linear operatorsℒ₁=ℒ₂=ℒ₃=𝜕^𝛼𝜕𝑦^𝛼, (20) with the property ℒ_𝑖[𝑐]=0, where 𝑐 is constant.

We now define nonlinear operators as 𝒩₁=𝜕^𝛼𝑢𝜕𝑦^𝛼+1𝜎²𝜕²𝑢𝜕𝑥²−2𝜎⁴𝜕𝑣+𝜕𝑡2𝜆𝜎⁴𝑢³+𝑣²𝑢−2𝜎⁴𝜕𝜙𝑢,𝒩𝜕𝑥₂=𝜕^𝛼𝑣𝜕𝑦^𝛼+1𝜎²𝜕²𝑣𝜕𝑥²+2𝜎⁴𝜕𝑢+𝜕𝑡2𝜆𝜎⁴𝑣³+𝑢²𝑣−2𝜎⁴𝜕𝜙𝑣,𝒩𝜕𝑥₃=𝜕^𝛼𝜙𝜕𝑦^𝛼−1𝜎²𝜕²𝜙𝜕𝑥²+2𝜆𝜎²𝜕𝑢²+𝑣².𝜕𝑥 (21) The initial guesses are considered as follows: 𝑢₀(),𝑣𝑥,𝑦,𝑡=𝑢𝑥,0,𝑡₀(),𝜙𝑥,𝑦,𝑡=𝑣𝑥,0,𝑡₀𝑥,𝑦,𝑡=𝜙(𝑥,0,𝑡). (22) In view of the discussion in Section 3, we get the following recursive relations: 𝑢₁𝑥,𝑦,𝑡=ℎ𝐼^𝛼_𝑦ℛ₁₁𝑢₀,𝑣₀,𝜙₀,𝑣,𝑥,𝑦,𝑡₁𝑥,𝑦,𝑡=ℎ𝐼^𝛼_𝑦ℛ₂₁𝑢₀,𝑣₀,𝜙₀,𝜙,𝑥,𝑦,𝑡₁𝑥,𝑦,𝑡=ℎ𝐼^𝛼_𝑦ℛ₃₁𝑢₀,𝑣₀,𝜙₀,𝑢,𝑥,𝑦,𝑡_𝑚+1𝑥,𝑦,𝑡=𝑢_𝑚𝑥,𝑦,𝑡+ℎ𝐼^𝛼_𝑦ℛ_1𝑚+1𝑢_𝑚,𝑣_𝑚,𝜙_𝑚,𝑣,𝑥,𝑦,𝑡_𝑚+1𝑥,𝑦,𝑡=𝑣_𝑚𝑥,𝑦,𝑡+ℎ𝐼^𝛼_𝑦ℛ_2𝑚+1𝑢_𝑚,𝑣_𝑚,𝜙_𝑚,𝜙,𝑥,𝑦,𝑡_𝑚+1𝑥,𝑦,𝑡=𝜙_𝑚𝑥,𝑦,𝑡+ℎ𝐼^𝛼_𝑦ℛ_3𝑚+1𝑢_𝑚,𝑣_𝑚,𝜙_𝑚,𝑥,𝑦,𝑡,𝑚=1,2,…, (23)(24) where ℛ_1𝑚+1𝑢_𝑚,𝑣_𝑚,𝜙_𝑚=𝜕,𝑥,𝑦,𝑡_𝛼𝑢_𝑚𝜕𝑦^𝛼+1𝜎²𝜕²𝑢_𝑚𝜕𝑥²−2𝜎⁴𝜕𝑣_𝑚+𝜕𝑡2𝜆𝜎^4𝑚_𝑗=0𝑢_𝑗^𝑚−𝑗_𝑖=0𝑢_𝑖𝑢_{𝑚−𝑗−𝑖}+^𝑚_𝑗=0𝑢_𝑗^𝑚−𝑗_𝑖=0𝑣_𝑖𝑣_{𝑚−𝑗−𝑖}−2𝜎^4𝑚_𝑗=0𝜕𝜙_𝑗𝑢𝜕𝑥^𝑚−𝑗,ℛ_2𝑚+1𝑢_𝑚,𝑣_𝑚,𝜙_𝑚=𝜕,𝑥,𝑦,𝑡_𝛼𝑣_𝑚𝜕𝑦^𝛼+1𝜎²𝜕²𝑣_𝑚𝜕𝑥²+2𝜎⁴𝜕𝑢_𝑚+𝜕𝑡2𝜆𝜎^4𝑚_𝑗=0𝑣_𝑗^𝑚−𝑗_𝑖=0𝑣_𝑖𝑣_{𝑚−𝑗−𝑖}+^𝑚_𝑗=0𝑣_𝑗^𝑚−𝑗_𝑖=0𝑢_𝑖𝑢_{𝑚−𝑗−𝑖}−2𝜎^4𝑚_𝑗=0𝜕𝜙_𝑗𝑣𝜕𝑥^𝑚−𝑗,ℛ_3𝑚+1𝑢_𝑚,𝑣_𝑚,𝜙_𝑚=𝜕,𝑥,𝑦,𝑡_𝛼𝜙_𝑚𝜕𝑦^𝛼−1𝜎²𝜕²𝜙_𝑚𝜕𝑥²+2𝜆𝜎²𝜕𝜕𝑥^𝑚_𝑗=0𝑢_𝑗𝑢_𝑚−𝑗+^𝑚_𝑗=0𝑣_𝑗𝑣_𝑚−𝑗. (25)

5. Results Analysis

In this section, some numerical results are presented to support our theatrical analysis. We consider the following initial conditions: 𝑢(𝑥,0,𝑡)=𝑟sech[𝑠(𝑥−𝑐𝑡)]cos𝑘₁𝑥+𝑘₃𝑡,𝑣(𝑥,0,𝑡)=𝑟sech[𝑠(𝑥−𝑐𝑡)]sin𝑘₁𝑥+𝑘₃𝑡,𝜙(𝑥,0,𝑡)=𝑓tanh[𝑠(𝑥−𝑐𝑡)], (26) where 𝑐=𝑘₂+𝜎²𝑘₁,𝑟=−2𝑘₃+𝑘²₁𝜎²+𝑘²₂𝜆,𝑠=2𝑘₃+𝑘²₁𝜎²+𝑘²₂𝜎²,√𝑓=2𝜎−𝜆1−𝜎², (27) and 𝑘_𝑖(𝑖=1,2,3) are arbitrary constants.

By the same manipulation as in Section 4, we will have𝑢₁=ℏ−2𝑓𝑟𝑠𝑦^𝛼sech[𝑠(−𝑐𝑡+𝑥)]³cos𝑥𝑘₁+𝑡𝑘₃𝜎⁴Γ[]−𝛼+1𝑟𝑠²𝑦^𝛼sech[𝑠(−𝑐𝑡+𝑥)]³cos𝑥𝑘₁+𝑡𝑘₃𝜎²Γ[]+𝛼+12𝑟³𝑦^𝛼𝜆cos𝑥𝑘₁+𝑡𝑘₃³sech[𝑠(−𝑐𝑡+𝑥)]³𝜎⁴Γ[]+𝛼+12𝑟³𝑦^𝛼𝜆sech[𝑠(−𝑐𝑡+𝑥)]³sin𝑥𝑘₁+𝑡𝑘₃²cos𝑥𝑘₁+𝑡𝑘₃𝜎⁴Γ[]−𝛼+1𝑟𝑦^𝛼sech[𝑠(−𝑐𝑡+𝑥)]cos𝑥𝑘₁+𝑡𝑘₃𝑘²₁𝜎²Γ[]−𝛼+12𝑟𝑦^𝛼sech[𝑠(−𝑐𝑡+𝑥)]cos𝑥𝑘₁+𝑡𝑘₃𝑘₃𝜎⁴Γ[]−𝛼+12𝑐𝑟𝑠𝑦^𝛼sin𝑥𝑘₁+𝑡𝑘₃sech[𝑠(−𝑐𝑡+𝑥)]tanh[𝑠(−𝑐𝑡+𝑥)]𝜎⁴Γ[]+𝛼+12𝑟𝑠𝑦^𝛼sin𝑥𝑘₁+𝑡𝑘₃sech[𝑠(−𝑐𝑡+𝑥)]𝑘₁tanh[𝑠(−𝑐𝑡+𝑥)]𝜎²Γ[]+𝛼+1𝑟𝑠²𝑦^𝛼sech[𝑠(−𝑐𝑡+𝑥)]cos𝑥𝑘₁+𝑡𝑘₃tanh[𝑠(−𝑐𝑡+𝑥)]²Γ[𝛼+1]𝜎²,𝑣₁=ℏ−2𝑓𝑟𝑠𝑦^𝛼sech[𝑠(−𝑐𝑡+𝑥)]³sin𝑥𝑘₁+𝑡𝑘₃Γ[𝛼+1]𝜎⁴−𝑟𝑠²𝑦^𝛼sech[𝑠(−𝑐𝑡+𝑥)]³sin𝑥𝑘₁+𝑡𝑘₃𝜎²Γ[]+𝛼+12𝑟³𝑦^𝛼𝜆cos𝑥𝑘₁+𝑡𝑘₃²sech[𝑠(−𝑐𝑡+𝑥)]³sin𝑥𝑘₁+𝑡𝑘₃𝜎⁴Γ[]+𝛼+12𝑟³𝑦^𝛼𝜆sech[𝑠(−𝑐𝑡+𝑥)]³sin𝑥𝑘₁+𝑡𝑘₃³𝜎⁴Γ[]−𝛼+1𝑟𝑦^𝛼sech[𝑠(−𝑐𝑡+𝑥)]sin𝑥𝑘₁+𝑡𝑘₃𝑘²₁𝜎²Γ[]−𝛼+1𝑟𝑦^𝛼sech[𝑠(−𝑐𝑡+𝑥)]sin𝑥𝑘₁+𝑡𝑘₃𝑘₃𝜎⁴Γ[]+𝛼+12𝑐𝑟𝑠𝑦^𝛼cos𝑥𝑘₁+𝑡𝑘₃sech[𝑠(−𝑐𝑡+𝑥)]tanh[𝑠(−𝑐𝑡+𝑥)]𝜎⁴Γ[]−𝛼+12𝑟𝑠𝑦^𝛼cos𝑥𝑘₁+𝑡𝑘₃sech[𝑠(−𝑐𝑡+𝑥)]𝑘₁tanh[𝑠(−𝑐𝑡+𝑥)]Γ[𝛼+1]𝜎²+𝑟𝑠²𝑦^𝛼sech[𝑠(−𝑐𝑡+𝑥)]sin𝑥𝑘₁+𝑡𝑘₃tanh[𝑠(−𝑐𝑡+𝑥)]²Γ[𝛼+1]𝜎²,𝜙₁=2ℏ𝑠𝑦^𝛼𝑓𝑠−2𝑟²𝜆sech[𝑠(−𝑐𝑡+𝑥)]²tanh[𝑠(−𝑐𝑡+𝑥)]Γ[𝛼+1]𝜎². (28)(29) In the same manner, using recurrence relations in (24) the other components 𝑣₂(𝑥,𝑦,𝑡),𝑣₃(𝑥,𝑦,𝑡),…,𝑢₂(𝑥,𝑦,𝑡),𝑢₃(𝑥,𝑦,𝑡),…,  and 𝜙₂(𝑥,𝑦,𝑡),𝜙₃(𝑥,𝑦,𝑡),… 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.

Theorem 6. If the series 𝑢_𝑖(𝑥,𝑦,𝑡)=∑^∞_𝑚=0𝑢_𝑖𝑚(𝑥,𝑦,𝑡),𝑖=1,2,…,𝑛, converges, where 𝑢_𝑖𝑚(𝑥,𝑦,𝑡) is governed by (15) under the definitions (16), it must be the solution of (7).

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 ℏ 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,𝑘₁=0.1,𝑘₂=0.03,𝑘₃=−0.3,𝜎=𝑖,and𝜆=1 is used in evaluating the approximate solutions for Tables 1, 2, and 3. Both the exact solutions and the approximate solutions of 𝑢(𝑥,𝑦,𝑡),𝑣(𝑥,𝑦,𝑡), and 𝜙(𝑥,𝑦,𝑡) (for the same parameters as mentioned before) are plotted in Figures 1, 2, and 3.

Table 1 

Absolute errors of 𝑢(𝑥,𝑦,𝑡).

Table 2 

Absolute errors of 𝑣(𝑥,𝑦,𝑡).

Table 3 

Absolute errors of 𝜙(𝑥,𝑦,𝑡).

(a)

(b)

(c)

Figure 1 

(a) and (b) The surface shows the solution 𝑢(𝑥,𝑦,𝑡) for (8): (a) approximate solution for 𝛼=1.98,ℏ=−1.1; (b) exact solution. (c) Four profiles of approximate solutions 𝑢(𝑥,𝑦,𝑡) 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 𝑘1=0.1,𝑘2=0.03,𝑘3=−0.3,𝜎=𝐼, and 𝜆=1.

(a)

(b)

(c)

Figure 2 

(a) and (b) The surface shows the solution 𝑣(𝑥,𝑦,𝑡) for (8): (a) approximate solution for 𝛼=1.98,ℏ=−1.1; (b) exact solution. (c) Four profiles of approximate solutions 𝑣(𝑥,𝑦,𝑡) 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 𝑘1=0.1,𝑘2=0.03,𝑘3=−0.3,𝜎=𝐼, and 𝜆=1.

(a)

(b)

(c)

Figure 3 

(a) and (b) The surface shows the solution 𝜙(𝑥,𝑦,𝑡) for (8): (a) approximate solution for 𝛼=1.98,ℏ=−1.1; (b) exact solution. (c) Four profiles of approximate solutions 𝜙(𝑥,𝑦,𝑡) 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 𝑘1=0.1,𝑘2=0.03,𝑘3=−0.3,𝜎=𝐼, and 𝜆=1.

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