Introduction

Fractional calculus and fractional dynamics become hot topics of research which rapid development and implementations in various fields of engineering and science [3, 8, 15]. As a result the fractional differential equations (FDEs) started to be used in describing of real world phenomena [3, 8]. The analytic results on uniqueness and existence of solutions of FDE were favorite topics of many researches [3, 15]. For most of the FDEs, obtaining the analytical solutions is not easy, thus, the approximation and numerical techniques are utilized to solve them. Some of these methods were Adomian decomposition method (ADM) [1, 14, 16–19], Homotopy perturbation method (HPM) [1, 13], Homotopy analysis method (HAM) [11], Variational iteration method (VIM) [10] and so on [2]. Our main aim in the current paper is to apply new iterative method to solve fractional Davey-Stewartson (DS) equations. This method is called Daftardar-Jafari method (DJM) by Daftardar and Jafari [6].

The core of this approach is to solve nonlinear equations without using Adomian polynomials [7]. In [4] Bhalekar and Daftardar-Gejji showed the method is convergent.

The present paper,we focus on obtaining the numerical solution of the following fractional Davey-Stewartson (DS) equations:

$$\begin{aligned}&\frac{1}{2}\sigma ^4 \frac{\partial ^\alpha q}{\partial y^\alpha }+\frac{1}{2}\sigma ^2 \frac{\partial ^2 q}{\partial x^2}+i\frac{\partial q}{\partial t}+\lambda \left| q\right| ^2 q-\frac{\partial \phi }{\partial x}q=0,\nonumber \\&\frac{\partial ^2 \phi }{\partial x^2}-\sigma ^2 \frac{\partial ^\alpha \phi }{\partial y^\alpha }-2\lambda \frac{\partial (\left| q\right| ^2)}{\partial x}=0,\qquad 1< \alpha \le 2 \end{aligned}$$
(1)

where \(\frac{\partial ^\alpha }{\partial y^\alpha }\) denotes the Caputo derivative, namely [3, 15]:

$$\begin{aligned} D^\alpha _y f(x,y,t)=\frac{\partial ^\alpha f(x,y,t)}{\partial y^\alpha }=\left\{ \begin{array}{ll} I^{m-\alpha }\left[ \frac{\partial ^m f(x,y,t)}{\partial y^m} \right] , &{} m-1<\alpha \le m, \\ \frac{\partial ^m f(x,y,t)}{\partial y^m}, &{} \alpha =m \in N.\\ \end{array} \right. \end{aligned}$$

\(I^\alpha \) denotes the Riemann-Liouville fractional integral, namely [3, 15]:

$$\begin{aligned} I^\alpha _y f(x,y,t)&= \frac{1}{\Gamma (\alpha )}\int _0^y (y-\tau )^{\alpha -1} f(x,\tau ,t) d\tau ,\quad \alpha >0, ~~~y>0\nonumber \\ I^0_y f(x,y,t)&= f(x,y,t) \end{aligned}$$
(2)

Below the reader can find some its basic properties:

$$\begin{aligned}&1)~ D^\alpha _y I^\alpha _y f(x,y,t)= f(x,y,t)\end{aligned}$$
(3)
$$\begin{aligned}&2)~ I^\alpha _y D^\alpha _y f(x,y,t)= f(x,y,t)-\sum _{k=0}^{m-1}f^{(k)}(x,0^+,t)\frac{y^k}{k!},~~~y>0 \end{aligned}$$
(4)

In Eq. (1) we have the DS-I equation in the case \(\sigma = 1\), while \(\sigma = i\) is the DS-II equation. The parameter \(\lambda \) illustrates the focusing or defocusing case. The known examples of integrable equations are the Davey-Stewarston I and II. They apply in gravity-capillarity surface wave packets in the limit of the shallow water. Jafari et al. applied VIM to solve the classical and fractional Davey-Stewartson equations [5, 9, 10].

Now we are ready to present the organization of our wok: In “Daftardar-Jafari method” section, Daftardar-Jafari method is presented. We apply the proposed technique to solve the fractional Davey-Stewartson equation in the next section. A brief conclusion is shown in “Conclusion” section.

Daftardar-Jafari Method

In this section, we now construct the iterative method by using the following decomposition method, which is mainly due to Daftardar-Gejji and Jafari [6]. This decomposition of the nonlinear function is quite different from that of Adomain decomposition. we describe the application of Daftardar-Jafari method to solve the fractional Davey-Stewartson equations:

$$\begin{aligned}&\frac{1}{2}\sigma ^4 \frac{\partial ^\alpha q}{\partial y^\alpha }+\frac{1}{2}\sigma ^2 \frac{\partial ^2 q}{\partial x^2}+i\frac{\partial q}{\partial t}+\lambda \left| q\right| ^2 q-\frac{\partial \phi }{\partial x}q=0,\nonumber \\&\frac{\partial ^2 \phi }{\partial x^2}-\sigma ^2 \frac{\partial ^\alpha \phi }{\partial y^\alpha }-2\lambda \frac{\partial \left( |q|^2\right) }{\partial x}=0,\qquad 1< \alpha \le 2. \end{aligned}$$
(5)

In systeam (5) we divide into real part and imaginary part the amplitude of a surface wave packet \(q\) and rewrite (5) in the following form:

$$\begin{aligned}&\frac{\partial ^\alpha u}{\partial y^\alpha }+\frac{1}{\sigma ^2}\frac{\partial ^2 u}{\partial x^2}-\frac{2}{\sigma ^4}\frac{\partial v}{\partial t}+\frac{2\lambda }{\sigma ^4}(u^3+v^2u)-\frac{2}{\sigma ^4}\left( \frac{\partial \phi }{\partial x} u\right) =0,\nonumber \\&\frac{\partial ^\alpha v}{\partial y^\alpha }+\frac{1}{\sigma ^2}\frac{\partial ^2 v}{\partial x^2}+\frac{2}{\sigma ^4}\frac{\partial u}{\partial t}+\frac{2\lambda }{\sigma ^4}(v^3+u^2 v)-\frac{2}{\sigma ^4}\left( \frac{\partial \phi }{\partial x} v\right) =0,\nonumber \\&\frac{\partial ^\alpha \phi }{\partial y^\alpha }-\frac{1}{\sigma ^2}\frac{\partial ^2 \phi }{\partial x^2}+ \frac{2\lambda }{\sigma ^2}\frac{\partial (u^2+v^2)}{\partial x}=0. \end{aligned}$$
(6)

In view of Eq. (4), applying \(I^\alpha \) on both sides of Eq. (6), we get the following integral equations:

$$\begin{aligned} u(x,y,t)&= u(x,0,t)+\sum _{k=0}^{m-1} u^{(k)}(x,0,t)\frac{y^k}{k!}-I^\alpha \left[ \frac{1}{\sigma ^2}\frac{\partial ^2 u}{\partial x^2}-\frac{2}{\sigma ^4}\frac{\partial v}{\partial t}+\frac{2\lambda }{\sigma ^4}(u^3+v^2u)\right. \nonumber \\&\left. -\frac{2}{\sigma ^4}(\frac{\partial \phi }{\partial x} u)\right] = f_1+N_1(u,v,\phi ),\nonumber \\ v(x,y,t)&= v(x,0,t)+\sum _{k=0}^{m-1} v^{(k)}(x,0,t)\frac{y^k}{k!}-I^\alpha \left[ \frac{1}{\sigma ^2}\frac{\partial ^2 v}{\partial x^2}+\frac{2}{\sigma ^4}\frac{\partial u}{\partial t}+\frac{2\lambda }{\sigma ^4}(v^3+u^2 v)\right. \nonumber \\&\left. -\frac{2}{\sigma ^4}(\frac{\partial \phi }{\partial x} v)\right] = f_2+N_2(u,v,\phi ),\nonumber \\ \phi (x,y,t)&= \phi (x,0,t)+\sum _{k=0}^{m-1} \phi ^{(k)}(x,0,t)\frac{y^k}{k!}-I^\alpha \left[ -\frac{1}{\sigma ^2}\frac{\partial ^2 \phi }{\partial x^2}+ \frac{2\lambda }{\sigma ^2}\frac{\partial (u^2+v^2)}{\partial x}\right] \\&= f_3+N_3(u,v,\phi ).\nonumber \end{aligned}$$
(7)

where

$$\begin{aligned} f_1&= u(x,0,t)+\sum _{k=0}^{m-1} u^{(k)}(x,0,t)\frac{y^k}{k!},\nonumber \\ N_1(u,v,\phi )&= -I^\alpha \left[ \frac{1}{\sigma ^2}\frac{\partial ^2 u}{\partial x^2}-\frac{2}{\sigma ^4}\frac{\partial v}{\partial t}+\frac{2\lambda }{\sigma ^4}(u^3+v^2u)-\frac{2}{\sigma ^4}\left( \frac{\partial \phi }{\partial x} u\right) \right] ,\nonumber \\ f_2&= v(x,0,t)+\sum _{k=0}^{m-1} v^{(k)}(x,0,t)\frac{y^k}{k!},\nonumber \\ N_2(u,v,\phi )&= -I^\alpha \left[ \frac{1}{\sigma ^2}\frac{\partial ^2 v}{\partial x^2}+\frac{2}{\sigma ^4}\frac{\partial u}{\partial t}+\frac{2\lambda }{\sigma ^4}\left( v^3+u^2 v\right) -\frac{2}{\sigma ^4}\left( \frac{\partial \phi }{\partial x} v\right) \right] ,\nonumber \\ f_3&= \phi (x,0,t)+\sum _{k=0}^{m-1} \phi ^{(k)}(x,0,t)\frac{y^k}{k!},\\ N_3(u,v,\phi )&= -I^\alpha \left[ -\frac{1}{\sigma ^2}\frac{\partial ^2 \phi }{\partial x^2}+ \frac{2\lambda }{\sigma ^2}\frac{\partial (u^2+v^2)}{\partial x}\right] ,\nonumber \\ \end{aligned}$$

where \(f_k(k=1,2,3)\) are known functions and \(N_k\) are nonlinear operators. The unknown functions can be shown in terms of an infinite series in iterative decomposition method as follow:

$$\begin{aligned} u=\sum _{i=0}^{\infty }u_i,\qquad v=\sum _{i=0}^{\infty }v_i,\qquad \phi =\sum _{i=0}^{\infty }\phi _i, \end{aligned}$$
(8)

and the nonlinear function \(N_k, (k=1,2,3)\) can be decomposed as

$$\begin{aligned} N_1\left( \sum _{i=0}^{\infty }u_i,\sum _{i=0}^{\infty }v_i,\sum _{i=0}^{\infty }\phi _i\right)&= N_1\left( u_0,v_0,\phi _0\right) \nonumber \\&+\sum _{i=0}^\infty \left\{ \! N_1\left( \sum _{j=0}^i u_j,\sum _{j=0}^i v_j,\sum _{j=0}^i \phi _j\right) \right. \nonumber \\&\left. - N_1\left( \sum _{j=0}^{i-1} u_j,\sum _{j=0}^{i-1} v_j,\sum _{j=0}^{i-1}\phi _j\right) \right\} \!,\nonumber \\ N_2\left( \sum _{i=0}^{\infty }u_i,\sum _{i=0}^{\infty }v_i,\sum _{i=0}^{\infty }\phi _i\right)&= N_2(u_0,v_0,\phi _0)\nonumber \\&+\sum _{i=0}^\infty \left\{ \! N_2\left( \sum _{j=0}^i u_j,\sum _{j=0}^i v_j,\sum _{j=0}^i \phi _j\right) \right. \nonumber \\&\left. - N_2\left( \sum _{j=0}^{i-1} u_j,\sum _{j=0}^{i-1} v_j,\sum _{j=0}^{i-1} \phi _j\right) \right\} \!,\nonumber \\ N_3\left( \sum _{i=0}^{\infty }u_i,\sum _{i=0}^{\infty }v_i,\sum _{i=0}^{\infty }\phi _i\right)&= N_3(u_0,v_0,\phi _0)\nonumber \\&+\sum _{i=0}^\infty \left\{ \! N_3\left( \sum _{j=0}^i u_j,\sum _{j=0}^i v_j,\sum _{j=0}^i \phi _j\right) \right. \nonumber \\&\left. - N_3\left( \sum _{j=0}^{i-1} u_j,\sum _{j=0}^{i-1} v_j,\sum _{j=0}^{i-1} \phi _j\right) \right\} \!. \end{aligned}$$
(9)

Substituting Eqs. (8) and (9) in Eq. (7), we have

$$\begin{aligned} \sum _{i=0}^{\infty }u_i\!&= \!f_1\!+\!N_1(u_0,v_0,\phi _0)+\!\!\sum _{i=0}^\infty \left\{ N_1\left( \sum _{j=0}^i u_j,\!\sum _{j=0}^i v_j,\!\sum _{j=0}^i \phi _j\right) \right. \nonumber \\&-\left. N_1\left( \sum _{j=0}^{i-1} u_j,\!\sum _{j=0}^{i-1} v_j,\!\sum _{j=0}^{i-1} \phi _j\right) \right\} ,\nonumber \\ \sum _{i=0}^{\infty }v_i\!&= \! f_2\!+\!N_2(u_0,v_0,\phi _0)+\!\!\sum _{i=0}^\infty \left\{ N_2\left( \sum _{j=0}^i u_j,\!\sum _{j=0}^i v_j,\!\sum _{j=0}^i \phi _j\right) \right. \nonumber \\&\left. - N_2\left( \sum _{j=0}^{i-1} u_j,\!\sum _{j=0}^{i-1} v_j,\!\sum _{j=0}^{i-1} \phi _j\right) \right\} ,\nonumber \\ \sum _{i=0}^{\infty }\phi _i\!&= \!f_3\!+\!N_3(u_0,v_0,\phi _0)+\!\!\sum _{i=0}^\infty \left\{ N_3\left( \sum _{j=0}^i u_j,\!\sum _{j=0}^i v_j,\!\sum _{j=0}^i \phi _j\right) \right. \nonumber \\&\left. -N_3\left( \sum _{j=0}^{i-1} u_j,\!\sum _{j=0}^{i-1} v_j,\!\sum _{j=0}^{i-1} \phi _j\right) \right\} . \end{aligned}$$
(10)

then we have the following recurrence relations:

$$\begin{aligned}&{\left\{ \begin{array}{ll} u_0=f_1\\ u_1= N_1(u_0,v_0,\phi _0)\\ u_{m+1}= N_1(u_0 + \cdots +u_m,v_0 + \cdots + v_m,\phi _0 + \cdots +\phi _m)\\ \qquad ~ - N_1(u_0 + \cdots +u_{m-1},v_0 + \cdots +v_{m-1},\phi _0 + \cdots +\phi _{m-1}),\quad m=1, 2, \ldots . \end{array}\right. }\end{aligned}$$
(11)
$$\begin{aligned}&{\left\{ \begin{array}{ll} v_0=f_2\\ v_1= N_2(u_0,v_0,\phi _0)\\ v_{m+1}= N_2(u_0 + \cdots +u_m,v_0 + \cdots + v_m,\phi _0 + \cdots +\phi _m)\\ \qquad ~ - N_2(u_0 + \cdots +u_{m-1},v_0 + \cdots +v_{m-1},\phi _0 + \cdots +\phi _{m-1}),\quad m=1, 2, \ldots . \end{array}\right. }\end{aligned}$$
(12)
$$\begin{aligned}&{\left\{ \begin{array}{ll} \phi _0=f_3\\ \phi _1= N_3(u_0,v_0,\phi _0)\\ \phi _{m+1}= N_3(u_0 + \cdots +u_m,v_0 + \cdots + v_m,\phi _0 + \cdots +\phi _m)\\ \qquad ~ - N_3(u_0 + \cdots +u_{m-1},v_0 + \cdots +v_{m-1},\phi _0 + \cdots +\phi _{m-1}),\quad m=1, 2, \ldots . \end{array}\right. } \end{aligned}$$
(13)

Then

$$\begin{aligned}&(u_1+\cdots +u_{m+1}) = N_1\left( u_0 +\cdots + u_m,v_0 +\cdots + v_m,\phi _0 +\cdots + \phi _m\right) ,\nonumber \\&(v_1+\cdots +v_{m+1}) = N_2\left( u_0 +\cdots + u_m,v_0 +\cdots + v_m,\phi _0 +\cdots + \phi _m\right) ,\nonumber \\&(\phi _1+\cdots +\phi _{m+1}) = N_3\left( u_0 +\cdots + u_m,v_0 +\cdots + v_m,\phi _0 +\cdots + \phi _m\right) , \end{aligned}$$
(14)

and

$$\begin{aligned} u = f_1+ \sum _{i=1}^\infty u_i,\quad ~~ v = f_2+\sum _{i=1}^\infty v_i,\quad ~~ \phi = f_3+ \sum _{i=1}^\infty \phi _i.\quad \end{aligned}$$
(15)

are calculated.

Convergence of DJM Solutions

In this section, we provide sufficient condition for the convergence of DJM solution series (8).

Theorem 1

If \(N_i\) is a contraction, i.e. \(\left\| N_i(\phi )-N_i(\psi )\right\| \le k \left\| \phi -\psi \right\| \), \(0<k<1\) then

$$\begin{aligned} \left\| u_{m+1}\right\|&= \left\| N_1\left( \sum _{j=0}^m u_j,\sum _{j=0}^m v_j,\sum _{j=0}^m \phi _j\right) - N_1\left( \sum _{j=0}^{m-1} u_j,\sum _{j=0}^{m-1} v_j,\sum _{j=0}^{m-1} \phi _j\right) \right\| \\&\le k \left\| u_m \right\| \le k^m\left\| u_0 \right\| , \qquad m=0,1,2,\ldots \\ \left\| v_{m+1}\right\|&= \left\| N_2\left( \sum _{j=0}^m u_j,\sum _{j=0}^m v_j,\sum _{j=0}^m \phi _j\right) - N_2\left( \sum _{j=0}^{m-1} u_j,\sum _{j=0}^{m-1} v_j,\sum _{j=0}^{m-1} \phi _j\right) \right\| \\&\le k \left\| v_m \right\| \le k^m \left\| v_0 \right\| , \qquad m=0,1,2,\ldots \\ \left\| \phi _{m+1}\right\|&= \left\| N_3\left( \sum _{j=0}^m u_j,\sum _{j=0}^m v_j,\sum _{j=0}^m \phi _j\right) - N_3\left( \sum _{j=0}^{m-1} u_j,\sum _{j=0}^{m-1} v_j,\sum _{j=0}^{m-1} \phi _j\right) \right\| \\&\le k \left\| \phi _m \right\| \le k^m \left\| \phi _0 \right\| , \qquad m=0,1,2,\ldots \end{aligned}$$

and the series \(\sum \limits _{\begin{array}{c} i=0 \end{array}}^m u_i, \sum \limits _{\begin{array}{c} i=0 \end{array}}^m v_i\) and \(\sum \limits _{\begin{array}{c} i=0 \end{array}}^m \phi _i\) absolutely and uniformly converges to a solution of Eq. (6), which is unique, in view of the Banach fixed point theorem [12].

Theorem 2

If the series solution defined in (15) is convergent, then it converges to an exact solution of Eq. (6).

Proof

The truncated series \(\sum \limits _{\begin{array}{c} i=0 \end{array}}^m u_i, \sum \limits _{\begin{array}{c} i=0 \end{array}}^m v_i\) and \(\sum \limits _{\begin{array}{c} i=0 \end{array}}^m \phi _i\) are used an approximation to the solution \(u(t), v(t)\) and \(\phi (t)\) of Eq. (6), using the above we have:

$$\begin{aligned} \sum _{i=0}^{m} u_i=\left( f+N_1\left( \sum _{i=0}^m u_i,\sum _{i=0}^m v_i,\sum _{i=0}^m \phi _i\right) \right) \nonumber \\ \sum _{i=0}^{m} v_i=\left( f+N_1\left( \sum _{i=0}^m u_i,\sum _{i=0}^m v_i,\sum _{i=0}^m \phi _i\right) \right) \nonumber \\ \sum _{i=0}^{m} \phi _i=\left( f+N_1\left( \sum _{i=0}^m u_i,\sum _{i=0}^m v_i,\sum _{i=0}^m \phi _i\right) \right) \end{aligned}$$
(16)

Taking limits of above equation, it follows that:

$$\begin{aligned} u=\displaystyle \lim _{m \rightarrow \infty } \sum _{i=0}^{m} u_i&= \displaystyle \lim _{m\rightarrow \infty }\left( f+N_1\left( \sum _{i=0}^m u_i,\sum _{i=0}^m v_i,\sum _{i=0}^m \phi _i\right) \right) \\&= \displaystyle \left. \lim _{m\rightarrow \infty }f+\displaystyle \lim _{m\rightarrow \infty } N_1\left( \sum _{i=0}^m u_i,\sum _{i=0}^m v_i,\sum _{i=0}^m \phi _i\right) \right) \\&= \left. f+N_1\left( \displaystyle \lim _{m\rightarrow \infty }\sum _{i=0}^m u_i,\displaystyle \lim _{m\rightarrow \infty }\sum _{i=0}^m v_i,\displaystyle \lim _{m\rightarrow \infty }\sum _{i=0}^m \phi _i\right) \right) \\&= f+N_1(u,v,\phi )\\ v=\displaystyle \lim _{m \rightarrow \infty } \sum _{i=0}^{m} u_i&= \displaystyle \lim _{m\rightarrow \infty }\left( f+N_2\left( \sum _{i=0}^m u_i,\sum _{i=0}^m v_i,\sum _{i=0}^m \phi _i\right) \right) \\&= \displaystyle \left. \lim _{m\rightarrow \infty }f+\displaystyle \lim _{m\rightarrow \infty } N_2\left( \sum _{i=0}^m u_i,\sum _{i=0}^m v_i,\sum _{i=0}^m \phi _i\right) \right) \\&= \left. f+N_2\left( \displaystyle \lim _{m\rightarrow \infty }\sum _{i=0}^m u_i,\displaystyle \lim _{m\rightarrow \infty }\sum _{i=0}^m v_i,\displaystyle \lim _{m\rightarrow \infty }\sum _{i=0}^m \phi _i\right) \right) \\&= f+N_2(u,v,\phi )\\ \phi =\displaystyle \lim _{m \rightarrow \infty } \sum _{i=0}^{m} u_i&= \displaystyle \lim _{m\rightarrow \infty }\left( f+N_3\left( \sum _{i=0}^m u_i,\sum _{i=0}^m v_i,\sum _{i=0}^m \phi _i\right) \right) \\&= \left. \displaystyle \lim _{m\rightarrow \infty }f+\displaystyle \lim _{m\rightarrow \infty } N_3\left( \sum _{i=0}^m u_i,\sum _{i=0}^m v_i,\sum _{i=0}^m \phi _i\right) \right) \\&= \left. f+N_3\left( \displaystyle \lim _{m\rightarrow \infty }\sum _{i=0}^m u_i,\displaystyle \lim _{m\rightarrow \infty }\sum _{i=0}^m v_i,\displaystyle \lim _{m\rightarrow \infty }\sum _{i=0}^m \phi _i\right) \right) \\&= f+N_3(u,v,\phi ) \end{aligned}$$

Hence \(u, v\) and \(\phi \) are the solutions of Eq. (6) and the proof is complete. \(\square \)

Applications

Below we use the presented approach in order to solve the fractional Davey-Stewartson equations (6). Subject to the following initial conditions:

$$\begin{aligned} u(x,0,t)&= r\, \text {sech}[s(x-ct)]\cos [(k_1x+k_3t)],\nonumber \\ v(x,0,t)&= r\, \text {sech}[s(x-ct)]\sin [(k_1x+k_3t)],\nonumber \\ \phi (x,0,t)&= f \tanh [s(x-ct)], \end{aligned}$$
(17)

where

$$\begin{aligned}&c=k_2+\sigma ^2k_1,\quad ~~ r=\sqrt{-\frac{(2k_3+k_1^2\sigma ^2+k_2^2}{\lambda }},\quad ~~ \nonumber \\&f=\frac{2 \sigma \sqrt{-\lambda }}{1-\sigma ^2}, ~~~\quad s=\sqrt{\frac{2 k_3+ k_1^2 \sigma ^2+k_2^2}{\sigma ^2}}, \end{aligned}$$
(18)

and \(k_i,(i=1,2,3)\) are arbitrary constants. The accuracy solution in the special case \(\alpha = 2\) [9]:

$$\begin{aligned} u(x,y,t)&= r\, \text {sech}[s (x+y-ct)] \cos [k_1 x+k_2 y+k_3 t],\nonumber \\ v(x,y,t)&= r\, \text {sech}[s (x+y-ct)] \sin [k_1 x+k_2 y+k_3 t],\nonumber \\ \phi (x,y,t)&= f\, \tanh [s (x+y-ct)]. \end{aligned}$$
(19)

According to the Daftardar-Jafari method and applying the algorithm given in (11), (12), (13) to Eq.  (7), we get:

$$\begin{aligned} {\left\{ \begin{array}{ll} u_0= r\, \text {sech}[s(x-ct)]\cos [(k_1x+k_3t)]+y (-r s~ \text {sech}[s(x-ct)]\tanh [s(x-ct)]\\ ~~ \qquad \times \cos [k_1x+k_3t)]-r \,k_2 \text {sech}[s(x-ct)]\sin [(k_1x+k_3t)])\\ u_1=N_1(u_0,v_0,\phi _0)\\ u_{m+1}=N_1\left( \sum \limits _{\begin{array}{c} j=0 \end{array}}^m u_j,\sum \limits _{\begin{array}{c} j=0 \end{array}}^m v_j,\sum \limits _{\begin{array}{c} j=0 \end{array}}^m \phi _j\right) - N_1\left( \sum \limits _{\begin{array}{c} j=0 \end{array}}^{m-1} u_j,\sum \limits _{\begin{array}{c} j=0 \end{array}}^{m-1} v_j,\sum \limits _{\begin{array}{c} j=0 \end{array}}^{m-1} \phi _j\right) \quad m=1,2,\cdots \end{array}\right. }\\ {\left\{ \begin{array}{ll} v_0= r \text {sech}[s(x-ct)]\sin [(k_1x+k_3t)]+y(-r s ~ \text {sech}[s(x-ct)]\tanh [s(x-ct)]\\ ~~ \qquad \times \cos [k_1x+k_3t)]+r k_2 \text {sech}[s(x-ct)]\sin [(k_1x+k_3t)])\\ v_1=N_2(u_0,v_0,\phi _0)\\ v_{m+1}=N_2\left( \sum \limits _{\begin{array}{c} j=0 \end{array}}^m u_j,\sum \limits _{\begin{array}{c} j=0 \end{array}}^m v_j,\sum \limits _{\begin{array}{c} j=0 \end{array}}^m \phi _j\right) - N_2\left( \sum \limits _{\begin{array}{c} j=0 \end{array}}^{m-1} u_j,\sum \limits _{\begin{array}{c} j=0 \end{array}}^{m-1} v_j,\sum \limits _{\begin{array}{c} j=0 \end{array}}^{m-1} \phi _j\right) \quad m=1,2,\cdots \end{array}\right. }\\ {\left\{ \begin{array}{ll} \phi _0= f \tanh [s(x-ct)] + y r s~ \text {sech}[s(x-ct)]^2\\ \phi _1=N_3(u_0,v_0,\phi _0)\\ \phi _{m+1}=N_3\left( \sum \limits _{\begin{array}{c} j=0 \end{array}}^m u_j,\sum \limits _{\begin{array}{c} j=0 \end{array}}^m v_j,\sum \limits _{\begin{array}{c} j=0 \end{array}}^m \phi _j\right) - N_3\left( \sum \limits _{\begin{array}{c} j=0 \end{array}}^{m-1} u_j,\sum \limits _{\begin{array}{c} j=0 \end{array}}^{m-1} v_j,\sum \limits _{\begin{array}{c} j=0 \end{array}}^{m-1} \phi _j\right) \quad m=1,2,\cdots \end{array}\right. } \end{aligned}$$

Tables 1,2, and 3 show the absolute errors between the exact solution together with the approximation solutions obtained for value of \(\alpha =1.98\) by the DJM. The approximate solutions for \(\alpha =1.98\), \(\alpha =1.8\) and the exact solutions are plotted in Figs. 1,2, and 3.We used the two-order term of the DJM solutions for the special case \(y=0.2,k_1=0.1,k_2=0.03,k_3=-0.3,\sigma =i,\lambda =1\) for Tables 1,2, and 3 and Figs. 1,2, and 3.

Table 1 Absolute errors of \(u(x,y,t)\)
Full size table
Table 2 Absolute errors of \(v(x,y,t)\)
Full size table
Table 3 Absolute errors of \(\phi (x,y,t)\)
Full size table
Fig. 1
figure 1

The surface shows the solution \(u(x;y; t)\) for Eq.  (7): a exact solution; b approximate solution for \(\alpha = 1.98\); c approximate solution for \(\alpha = 1.8\)

Full size image
Fig. 2
figure 2

The surface shows the solution \(v(x;y; t)\) for Eq.  (7): a exact solution; b approximate solution for \(\alpha = 1.98\); c approximate solution for \(\alpha = 1.8\)

Full size image
Fig. 3
figure 3

The surface shows the solution \(\phi (x;y; t)\) for Eq. (7): a exact solution; b approximate solution for \(\alpha = 1.98\); c approximate solution for \(\alpha = 1.8\)

Full size image

Conclusion

In this work, we applied the Daftardar-Jafari method to solve fractional Davey-Stewartson differential equations. The results show that this method is accurate and effective and can be used for nonlinear fractional differential equations. This method have an advantage over ADM that DJM can be solved nonlinear problems without using Adomian polynomials.

Mathematica has been used for computations and programming in this paper.