Khaled A. Gepreel and Amr M. S. Mahdy

# Algebraic computational methods for solving three nonlinear vital models fractional in mathematical physics

De Gruyter | Published online: April 15, 2021

# Abstract

This research paper uses a direct algebraic computational scheme to construct the Jacobi elliptic solutions based on the conformal fractional derivatives for nonlinear partial fractional differential equations (NPFDEs). Three vital models in mathematical physics [the space-time fractional coupled Hirota Satsuma KdV equations, the space-time fractional symmetric regularized long wave (SRLW equation), and the space-time fractional coupled Sakharov–Kuznetsov (S–K) equations] are investigated through the direct algebraic method for more explanation of their novel characterizes. This approach is an easy and powerful way to find elliptical Jacobi solutions to NPFDEs. The hyperbolic function solutions and trigonometric functions where the modulus and, respectively, are degenerated by Jacobi elliptic solutions. In this style, we get many different kinds of traveling wave solutions such as rational wave traveling solutions, periodic, soliton solutions, and Jacobi elliptic solutions to nonlinear evolution equations in mathematical physics. With the suggested method, we were fit to find much explicit wave solutions of nonlinear integral differential equations next converting them into a differential equation. We do the 3D and 2D figures to define the kinds of outcome solutions. This style is moving, reliable, powerful, and easy for solving more difficult nonlinear physics mathematically.

## 1 Introduction

Nonlinear partial fractional differential equation (NPFDE) is a vital tool to explain a wide variety of physical processes with fractional derivatives [1,2], such as damping rule, diffusion mechanism, and nonlinear earthquake oscillation. There are several NPFDE implementations in turbulence, fluid dynamics, and the nonlinear biological system [1,2,3,4,5,6,7,8,9,10]. Several approaches are looking for estimated theoretical strategies for NPFDEs, for example, ref. [4,5,6,7,8,9,10]. No computational tools for nonlinear fractional differential equations were known before 1998. Li and He [9] suggested the complex fractional transformation to turn NPFDEs into ODEs such that NPFDEs can be used with all computational approaches for solving NPDEs. For example, authors are searching for moving wave solutions for NPFDEs, see ref. [11,13,14,15,16,17,18,19]. During the previous many years, numerous researchers have much of the time shown that the numerical conditions with fractional math structures can depict reality more correctly than the exemplary whole number models with standard time-subsidiaries [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]. As of late, the upsides of this approach have been widely examined for different functional applications [41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60].

In this article, using the enhanced extended algebraic form, we research the Jacobi elliptic moving wave solutions for the following NPFDs:

1. (i)

Space-time fractional nonlinear coupled Hirota Satsuma KdV equations are formulated in the following mathematical system [17]:

(1) D t α Φ = 1 4 D x 3 α Φ + 3 Φ D x α Φ 6 Ψ D x α Ψ , D t α Ψ = 1 2 D x 3 α Ψ 3 Φ D x α Ψ .

2. (ii)

Space-time fractional nonlinear fractional symmetric regularized long wave (SRLW) equation [18]:

(2) D t 2 α Φ + D x 2 α Φ + Φ D t α ( D x α Φ ) + D t α Φ D x α Φ + D t 2 α ( D x 2 α Φ ) = 0 .

3. (iii)

Space-time fractional coupled Sakharov–Kuznetsov (S–K) equation in the following form [17]:

(3) D t α Φ + D x 3 α Φ + D x α D y 2 α Φ 6 Φ D x α Φ D x α Ψ = 0 , D t α Ψ + χ 1 D x 3 α Ψ + χ 2 D x α D y 2 α Ψ + χ 3 D x α Ψ 6 χ 4 Ψ D x α Ψ χ 5 D x α Φ = 0 ,
where 0 < α 1 and χ s ( s = 1 , 2 , , 5 ) are arbitrary nonzero constants.

At the end of this work, we determine the type of wave solutions produced in this article at some special values for some parameters by making drawings in 2D and 3D.

The rest of the paper is organized as follows: Sections 2 and 3 give distinct novel computational solutions of the above-mentioned fractional models and explain the constructed solutions by plotting them in two- and three-dimensional sketches. Section 4 gives the conclusion of the whole paper.

## 2 Conformal fractional derivatives and methodology

In this paper, the fractional derivative is the conformal fractional derivative which is defined by ref. [20,21,22,23,24].

(4) D α f ( t ) = lim σ 0 f ( t + σ t 1 α ) f ( t ) σ .

The conformal fractional derivative has many properties, see ref. [18,19,20,21,22]. Methodology used in this paper is the Jacobi elliptic function method, which is used to solve the NPFDEs. In this method, we suppose the solutions as follows:

(5) U ( ξ ) = i = N N M i [ T ( Θ ) ] i ,
where
(6) [ D Θ α T ( Θ ) ] 2 = k 0 + k 1 T 2 ( Θ ) + k 2 T 4 ( Θ ) .

We use the complex fractional transformation which is defined in ref. [18,19,20,21,22] to transfer NPFDEs to NODEs. We determine the end of the series (5) from the balance number idea. We substitute (5) in NODEs, and we get a system of algebraic equation for M i ( i = 0 , ± 1 , ± 2 , , N ) . From the general solutions of (6) and the values of M i ( i = 0 , ± 1 , ± 2 , , N ) , we can determine some new solutions for NPFDEs.

## 3 Applications

For certain NPFDEs in mathematical physics, we use the proposed approach to create the Jacobi elliptic solutions via the space-time fractional Satsuma KdV equation, the space-time fractional nonlinear fractional SRLW equation, and the space-time fractional coupled S–K equations.

### 3.1 The space-time fractional Hirota Satsuma KdV equation

Using the next wave transformation for equation (1):

(7) Φ = U ( Θ ) , Ψ = V ( Θ ) , Θ = x + ω t ,
where ω is an arbitrary constant to be determined later. The transformation (7) and the conformal fractional derivatives [ 21, 22] lead to write (1) in the following form:
(8) ω α D Θ α U 1 4 D Θ α ( D Θ α ( D Θ α U ) ) 3 U D Θ α U 6 U D Θ α V = 0 , ω α D Θ α V + 1 2 D Θ α ( D Θ α ( D Θ α V ) ) + 3 U D Θ α V = 0 .

Balancing the terms in equation (8) constructs its formal solutions in the following formulas:

(9) U ( ξ ) = M 0 + M 1 T ( Θ ) + M 2 T 2 ( Θ ) + M 3 T ( Θ ) + M 4 T 2 ( Θ ) , V ( ξ ) = N 0 + N 1 T ( Θ ) + N 2 T 2 ( Θ ) + N 3 T ( Θ ) + N 4 T 2 ( Θ ) ,
where M i , N i , ( i = 0 , 1 , , 4 ) are constants to be determined later, such that M 2 0 or M 4 0 and N 2 0 or N 4 0 and T ( Θ ) satisfies the Jacobi elliptic nonlinear differential equation [ D Θ α T ( Θ ) ] 2 = k 0 + k 1 T 2 ( Θ ) + k 2 T 4 ( Θ ) . Equation ( 9) along with the condition ( 6) are solution to Equation ( 8), the framework of the suggested computational schemes, obtains:

Case 1.

(10) M 0 = ω α 3 2 k 1 3 , M 2 = 2 k 2 , M 4 = 2 k 0 , N 0 = ± 2 ω α 3 ± k 1 3 , N 2 = ± k 2 , N 4 = ± k 0 , M 1 = N 1 = M 3 = N 3 = 0 ,
where k 0 , k 1 , and k 2 are arbitrary constants.

Case 2.

(11) M 0 = 1 3 e 0 ( k 0 k 1 3 D i 2 + k 0 ω α ) , M 2 = k 2 , M 4 = k 0 , N 1 = D i 6 k 0 2 ( 4 k 0 ω α k 0 k 1 6 D i 2 ) , N 3 = D i , M 1 = N 0 = M 3 = N 2 = N 2 = 0 ,
where i = 1 , 2 , 3 , 4 , D 1 = 6 k 0 4 ω α k 1 + 6 k 1 k 2 , D 2 = 6 k 0 4 ω α k 1 + 6 k 1 k 2 , D 3 = 6 k 0 4 ω α k 1 6 k 1 k 2 , and D 4 = 6 k 0 4 ω α k 1 6 k 1 k 2 .

There are also such examples that have been overlooked for comfort. Now, let us write down for convenience the following exact solutions of the space-time fractional Hirota Satsuma equation (1) for case 1. The detailed solutions applicable to case 1 are as follows:

(12) U ( Θ ) = ω α 3 2 k 1 3 2 k 2 T 2 ( Θ ) 2 k 0 T 2 ( Θ ) ,
and
(13) V ( ξ ) = ± 2 ω α 3 ± k 1 3 ± k 2 T 2 ( Θ ) ± k 0 T 2 ( Θ ) .

Family 1. k 0 = 1 , k 1 = ( 1 + m 2 ) , and k 2 = m 2 ; the Jacobi elliptic exact solutions for equation (5) take the following form:

(14) U 1 ( Θ ) = ω α 3 2 3 2 m 2 S n 2 ( x + ω t ) α Γ ( α + 1 ) 2 N s 2 ( x + ω t ) α Γ ( α + 1 ) , V 1 ( Θ ) = ± 2 ω α 3 ± ( 1 + m 2 ) 3 ± m 2 S n 2 ( x + ω t ) α Γ ( α + 1 ) ± N s 2 ( x + ω t ) α Γ ( α + 1 ) .
Or
(15) U 2 ( Θ ) = ω α 3 2 3 2 m 2 C d 2 ( x + ω t ) α Γ ( α + 1 ) 2 D c 2 ( x + ω t ) α Γ ( α + 1 ) , R 2 ( Θ ) = ± 2 ω α 3 ± ( 1 + m 2 ) 3 ± m 2 C d 2 ( x + ω t ) α Γ ( α + 1 ) ± D c 2 ( x + ω t ) α Γ ( α + 1 ) .

Family 2. k 0 = 1 m 2 , k 1 = 2 m 2 1 , and k 2 = m 2 ; the Jacobi elliptic exact solutions for equation (1) take the following form:

(16) U 3 ( Θ ) = ω α 3 2 ( 2 m 2 1 ) 3 + 2 m 2 C n 2 ( x + ω t ) α Γ ( α + 1 ) 2 ( 1 m 2 ) N c 2 ( x + ω t ) α Γ ( α + 1 ) , V 3 ( Θ ) = ± 2 ω α 3 ± ( 2 m 2 1 ) 3 m 2 C n 2 ( x + ω t ) α Γ ( α + 1 ) ± ( 1 m 2 ) N c 2 ( x + ω t ) α Γ ( α + 1 ) .

Family 3. k 0 = m 2 1 , k 1 = 2 m 2 , and k 2 = 1 ; the Jacobi elliptic exact solutions for equation (1) take the following form:

(17) U 4 ( Θ ) = ω α 3 2 ( 2 m 2 ) 3 + 2 D n 2 ( x + ω t ) α Γ ( α + 1 ) 2 ( m 2 1 ) N d 2 ( x + ω t ) α Γ ( α + 1 ) , V 4 ( Θ ) = ± 2 ω α 3 ± ( 2 m 2 ) 3 D n 2 ( x + ω t ) α Γ ( α + 1 ) ± ( m 2 1 ) N d 2 ( x + ω t ) α Γ ( α + 1 ) .

Family 4. k 0 = 1 m 2 , k 1 = 2 m 2 , and k 2 = 1 ; the Jacobi elliptic exact solutions for equation (1) take the following form:

(18) U 5 ( Θ ) = ω α 3 2 ( 2 m 2 ) 3 2 C s 2 ( x + ω t ) α Γ ( α + 1 ) 2 ( 1 m 2 ) S c 2 ( x + ω t ) α Γ ( α + 1 ) , V 5 ( Θ ) = ± 2 ω α 3 ± 2 m 2 3 ± C s 2 ( x + ω t ) α Γ ( α + 1 ) ± ( 1 m 2 ) S c 2 ( x + ω t ) α Γ ( α + 1 ) .

Family 5. k 0 = 1 , k 1 = 2 m 2 1 , and k 2 = m 2 ( m 2 1 ) ; the Jacobi elliptic exact solutions for equation (1) take the following form:

(19) U 6 ( Θ ) = ω α 3 2 ( 2 m 2 1 ) 3 2 m 2 ( m 2 1 ) S d 2 ( x + ω t ) α Γ ( α + 1 ) 2 D s 2 ( x + ω t ) α Γ ( α + 1 ) , V 6 ( Θ ) = ± 2 ω α 3 ± ( 2 m 2 1 ) 3 ± m 2 ( m 2 1 ) S d 2 ( x + ω t ) α Γ ( α + 1 ) ± D s 2 ( x + ω t ) α Γ ( α + 1 ) .

Family 6. k 0 = m 2 ( m 2 1 ) , k 1 = 2 m 2 1 , and k 2 = 1 ; the Jacobi elliptic exact solutions for equation (1) take the following form:

(20) U 7 ( Θ ) = ω α 3 2 ( 2 m 2 1 ) 3 2 D s 2 ( x + ω t ) α Γ ( α + 1 ) 2 m 2 ( m 2 1 ) S d 2 ( x + ω t ) α Γ ( α + 1 ) , V 7 ( Θ ) = ± 2 ω α 3 ± ( 2 m 2 1 ) 3 ± D s 2 ( x + ω t ) α Γ ( α + 1 ) ± m 2 ( m 2 1 ) S d 2 ( x + ω t ) α Γ ( α + 1 ) .

Family 7. k 0 = 1 4 , k 1 = 1 2 ( 1 2 m 2 ) , and k 2 = 1 4 ; the Jacobi elliptic exact solutions for equation (1) take the following form:

(21) U 8 ( Θ ) = ω α 3 ( 1 2 m 2 ) 3 1 2 N s ( x + ω t ) α Γ ( α + 1 ) + C s ( x + ω t ) α Γ ( α + 1 ) 2 1 2 N s ( x + ω t ) α Γ ( α + 1 ) + C s ( x + ω t ) α Γ ( α + 1 ) 2 , V 8 ( Θ ) = ± 2 ω α 3 ± ( 1 2 m 2 ) 6 ± 1 4 N s ( x + ω t ) α Γ ( α + 1 ) + C s ( x + ω t ) α Γ ( α + 1 ) 2 ± 1 4 N s ( x + ω t ) α Γ ( α + 1 ) + C s ( x + ω t ) α Γ ( α + 1 ) 2 .

Family 8. k 0 = 1 4 ( 1 m 2 ) , k 1 = 1 4 ( 1 + m 2 ) , and k 2 = 1 4 ( 1 m 2 ) ; the Jacobi elliptic exact solutions for equation (1) take the following form:

(22) U 9 ( Θ ) = ω α 3 ( 1 + m 2 ) 6 1 2 ( 1 m 2 ) N c ( x + ω t ) α Γ ( α + 1 ) + S c ( x + ω t ) α Γ ( α + 1 ) 2 ( 1 m 2 ) 2 N c ( x + ω t ) α Γ ( α + 1 ) + S c ( x + ω t ) α Γ ( α + 1 ) 2 , V 9 ( Θ ) = ± 2 ω α 3 ± ( 1 + m 2 ) 12 ± 1 4 ( 1 m 2 ) N c ( x + ω t ) α Γ ( α + 1 ) + S c ( x + ω t ) α Γ ( α + 1 ) 2 ± ( 1 m 2 ) 4 N c ( x + ω t ) α Γ ( α + 1 ) + S c ( x + ω t ) α Γ ( α + 1 ) 2 .

Family 9. k 0 = m 2 4 , k 1 = 1 4 ( m 2 2 ) , and k 2 = m 2 4 ; the Jacobi elliptic exact solutions for equation (1) take the following form:

(23) U 10 ( Θ ) = ω α 3 ( m 2 2 ) 6 m 2 2 S n ( x + ω t ) α Γ ( α + 1 ) + i C n ( x + ω t ) α Γ ( α + 1 ) 2 m 2 2 S n ( x + ω t ) α Γ ( α + 1 ) + i C n ( x + ω t ) α Γ ( α + 1 ) 2 , V 10 ( Θ ) = ± 2 ω α 3 ± ( m 2 2 ) 12 ± m 2 4 S n ( x + ω t ) α Γ ( α + 1 ) + i C n ( x + ω t ) α Γ ( α + 1 ) 2 ± m 2 4 S n ( x + ω t ) α Γ ( α + 1 ) + i C n ( x + ω t ) α Γ ( α + 1 ) 2 .

Also, we can construct more of the exact Jacobi elliptic solutions for the case 2, we are omitted here for convenience to the reader.

#### 3.1.1 Numerical solutions for the exact solutions for NPFDEs

Here, we give some figures in this paragraph to explain some of our findings obtained in this portion. To this end, for the nonlinear space-time fractional Hirota Satsuma KdV differential equations, we choose some unique parameter values to demonstrate the Jacobi elliptic solution’s behavior (Figures 1–12).

### Figure 1

The Jacobi elliptic doubly periodic solution U 1 (14) at ω = 3 , m = 0.5 , α = 1 , 0.5 , and α = 0.1 , respectively.

### Figure 2

The projection Jacobi elliptic doubly periodic solution U 1 (14) at t = 0, ω = 3 , m = 0.5 , and α = 1 , 0.5 , 0.1 , respectively.

### Figure 3

The Jacobi elliptic doubly periodic solution V 1 (14) at ω = 3 , m = 0.5 , and α = 1 , 0.5 , 0.1 , respectively.

### Figure 4

The projection Jacobi elliptic doubly periodic solution V 1 (14) at t = 0, ω = 3 , m = 0.5 , and α = 1 , 0.5 , 0.1 , respectively.

### Figure 5

The Jacobi elliptic doubly periodic solution U 3 (16) at ω = 3 , m = 0.5 , and α = 1 , 0.5 , 0.1 , respectively.

### Figure 6

The projection Jacobi elliptic doubly periodic solution U 3 (16) at t = 0 ω = 3 , m = 0.5 , and α = 1 , 0.5 , 0.1 , respectively.

### Figure 7

The Jacobi elliptic doubly periodic solution V 3 (16) at ω = 3 , m = 0.5 , and α = 1 , 0.5 , 0.1 , respectively.

### Figure 8

The projection Jacobi elliptic doubly periodic solution V 3 (16) at t = 0, ω = 3 , m = 0.5 , and α = 1 , 0.5 , 0.1 , respectively.

### Figure 9

The Jacobi elliptic doubly periodic solution U 5 (18) at ω = 3 , m = 0.5 , and α = 1 , 0.5 , 0.1 , respectively.

### Figure 10

The projection Jacobi elliptic doubly periodic solution U 5 (18) at t = 0, ω = 3 , m = 0.5 , and α = 1 , 0.5 , 0.1 , respectively.

### Figure 11

The Jacobi elliptic doubly periodic solution V 5 (18) at ω = 3 , m = 0.5 , and α = 1 , 0.5 , 0.1 , respectively.

### Figure 12

The projection Jacobi elliptic doubly periodic solution V 5 (18) at t = 0, ω = 3 , m = 0.5 , and α = 1 , 0.5 , 0.1 , respectively.

### 3.2 Jacobi elliptic solutions for the space-time fractional SRLW equation

Using the transformation (7) converts equation (2) to the following ODE:

(24) ω 2 α D Θ 2 α ( U ) + D Θ 2 α ( U ) + c α U D Θ α ( D Θ α U ) + ω α ( D Θ α ) ( D Θ α U ) + ω 2 α D Θ 4 α ( U ) = 0 ,
where ω is an arbitrary nonzero constants. Balancing the nonlinear terms with the highest derivatives then employ the suggested computational scheme, get the general solution of equation ( 24) in the following formula:
(25) U ( Θ ) = s 0 + s 1 T ( Θ ) + s 2 T 2 ( Θ ) + s 3 T ( Θ ) + s 4 T 2 ( Θ ) ,
where s i ( i = 0 , 1 , , 4 ) are constants to be determined later. Using equation ( 25) in the framework of the suggested scheme gives the following.

Case 1.

(26) s 0 = ( ω 2 α + 4 k 1 + 1 ) ω α , s 2 = 12 k 2 ω α , s 1 = s 3 = s 4 = 0 .

Case 2.

(27) s 0 = ( ω 2 α + 4 k 1 + 1 ) ω α , s 4 = 12 k 0 ω α , s 1 = s 2 = s 3 = 0 .

Case 3.

(28) s 0 = ( ω 2 α + 4 k 1 + 1 ) ω α , s 2 = 12 k 2 ω α , s 4 = 12 k 0 ω α , s 1 = s 3 = 0 .

Thus, the exact traveling wave solution of equation (24) is given by

(29) U ( Θ ) = ( ω 2 α + 4 k 1 + 1 ) ω α 12 k 2 ω α T 2 ( Θ ) 12 k 0 ω α T 2 ( Θ ) .

Family 1. k 0 = 1 , k 1 = ( 1 + m 2 ) , and k 2 = m 2 ; the Jacobi elliptic exact solution for equation (24) takes the following form:

(30) U 1 ( Θ ) = [ ω 2 α 4 ( 1 + m 2 ) + 1 ] ω α 12 m 2 ω α S N 2 ( x + ω t ) α Γ ( α + 1 ) 12 ω α N S 2 ( x + ω t ) α Γ ( α + 1 ) .
Or
(31) U 1 ( Θ ) = [ ω 2 α 4 ( 1 + m 2 ) + 1 ] ω α 12 m 2 ω α C d 2 ( x + ω t ) α Γ ( α + 1 ) 12 ω α D c 2 ( x + ω t ) α Γ ( α + 1 ) .

Family 2. k 0 = 1 m 2 , k 1 = 2 m 2 1 , and k 2 = m 2 ; the Jacobi elliptic exact solution for equation (24) takes the following form:

(32) U 2 ( Θ ) = [ ω 2 α + 4 ( 2 m 2 1 ) + 1 ] ω α + 12 m 2 ω α C n 2 ( x + ω t ) α Γ ( α + 1 ) 12 ( 1 m 2 ) ω α N c 2 ( x + ω t ) α Γ ( α + 1 ) .

Family 3. k 0 = m 2 1 , k 1 = 2 m 2 , and k 2 = 1 ; the Jacobi elliptic exact solution for equation (24) takes the following form:

(33) U 3 ( Θ ) = [ ω 2 α + 4 ( 2 m 2 ) + 1 ] ω α + 12 ω α D n 2 ( x + ω t ) α Γ ( α + 1 ) 12 ( m 2 1 ) ω α N d 2 ( x + ω t ) α Γ ( α + 1 ) .

Family 4. k 0 = 1 m 2 , k 1 = 2 m 2 , and k 2 = 1 ; the Jacobi elliptic exact solution for equation (24) takes the following form:

(34) U 4 ( Θ ) = [ ω 2 α + 4 ( 2 m 2 ) + 1 ] ω α 12 ω α C s 2 ( x + ω t ) α Γ ( α + 1 ) 12 ( 1 m 2 ) ω α S c 2 ( x + ω t ) α Γ ( α + 1 ) .

Family 5. k 0 = 1 , k 1 = 2 m 2 1 , and k 2 = m 2 ( m 2 1 ) ; the Jacobi elliptic exact solution for equation (24) takes the following form:

(35) U 5 ( Θ ) = [ ω 2 α + 4 ( 2 m 2 1 ) + 1 ] ω α 12 m 2 ( m 2 1 ) ω α S d 2 ( x + ω t ) α Γ ( α + 1 ) 12 ω α D s 2 ( x + ω t ) α Γ ( α + 1 ) .

Family 6. k 0 = 1 4 , k 1 = 1 2 ( 1 2 m 2 ) , and k 2 = 1 4 ; the Jacobi elliptic exact solution for equation (24) takes the following form:

(36) U 6 ( Θ ) = [ ω 2 α + 2 ( 1 2 m 2 ) + 1 ] ω α 3 ω α N s ( x + ω t ) α Γ ( α + 1 ) + C s ( x + ω t ) α Γ ( α + 1 ) 2 3 ω α N s ( x + ω t ) α Γ ( α + 1 ) + C s ( x + ω t ) α Γ ( α + 1 ) 2 .

Family 7. k 0 = 1 4 ( 1 m 2 ) , k 1 = 1 4 ( 1 + m 2 ) , and k 2 = 1 4 ( 1 m 2 ) ; the Jacobi elliptic exact solution for equation (24) takes the following form:

(37) U 7 ( Θ ) = ( ω 2 α + ( 1 + m 2 ) + 1 ) ω α 3 ( 1 m 2 ) ω α N c ( x + ω t ) α Γ ( α + 1 ) + S c ( x + ω t ) α Γ ( α + 1 ) 2 3 ( 1 m 2 ) ω α N c ( x + ω t ) α Γ ( α + 1 ) + S c ( x + ω t ) α Γ ( α + 1 ) 2 .

Family 8. k 0 = m 2 4 , k 1 = 1 4 ( m 2 2 ) , and k 2 = m 2 4 ; the Jacobi elliptic exact solution for equation (24) takes the following form:

(38) U 8 ( Θ ) = [ ω 2 α + ( m 2 2 ) + 1 ] ω α 3 m 2 ω α S n ( x + ω t ) α Γ ( α + 1 ) + i C n ( x + ω t ) α Γ ( α + 1 ) 2 3 m 2 ω α S n ( x + ω t ) α Γ ( α + 1 ) + i C n ( x + ω t ) α Γ ( α + 1 ) 2 .

Family 9. k 0 = 1 4 , k 1 = 1 2 ( 1 m 2 ) , and k 2 = 1 4 ; the Jacobi elliptic exact solution for equation (24) takes the following form:

(39) U 9 ( Θ ) = [ ω 2 α + 2 ( 1 m 2 ) + 1 ] ω α 3 ω α 1 m 2 S c ( x + ω t ) α Γ ( α + 1 ) ± D c ( x + ω t ) α Γ ( α + 1 ) 2 3 ω α 1 m 2 S c ( x + ω t ) α Γ ( α + 1 ) ± D c ( x + ω t ) α Γ ( α + 1 ) 2 .

Similarly, we can write down the other families of exact solutions of equation (24) which are omitted here for convenience.

#### 3.2.1 Numerical solutions for the space-time fractional SRLW equation

We give some figures in this paragraph to explain some of our findings obtained in this portion. To this end, for the nonlinear space-time fractional SRLW differential equations, we choose some unique parameter values to demonstrate the behavior of the Jacobi elliptic solution (Figures 13–18)

### Figure 13

The Jacobi elliptic doubly periodic solution (32) at ω = 3 , m = 0.5 , α = 1 , α = 0.5 , and α = 0.1 , respectively.

### Figure 14

The projection Jacobi elliptic doubly periodic solution (32) at t = 0, ω = 3 , m = 0.5 , α = 1 , α = 0.5 , and α = 0.1 , respectively.

### Figure 15

The Jacobi elliptic doubly periodic solution (34) at ω = 3 , m = 0.5 , α = 1 , α = 0.5 , and α = 0.1 , respectively.

### Figure 16

The projection Jacobi elliptic doubly periodic solution (34) at t = 0, ω = 3 , m = 0.5 , α = 1 , α = 0.5 , and α = 0.1 , respectively.

### Figure 17

The Jacobi elliptic doubly periodic solution (36) at ω = 3 , m = 0.5 , α = 1 , α = 0.5 , and α = 0.1 , respectively.

### Figure 18

The projection Jacobi elliptic doubly periodic solution (36) at t = 0, ω = 3 , m = 0.5 , α = 1 , α = 0.5 , and α = 0.1 , respectively.

### 3.3 Jacobi elliptic solutions for the space-time fractional coupled S–K equations

Using the next transformation on equation (3)

(40) Φ = U ( Θ ) , ψ = V ( Θ ) , Θ = x + y + ω t ,
leads to write equation ( 3) into the following form:
(41) ω α D Θ α U + 2 D Θ 3 α U 6 U ( D Θ α U ) D Θ α V = 0 , ω α D Θ α V + ( χ 1 + χ 2 ) D Θ 3 α V + χ 3 D Θ α V 6 χ 4 V D Θ α V χ 5 D Θ α U = 0 ,
where ω is an arbitrary constant. Using the balance principle in the framework of the used computational scheme gives the general solution of equation ( 41) in the following formula:
(42) U ( Θ ) = s 0 + s 1 T ( Θ ) + s 2 T 2 ( Θ ) + s 3 T 1 ( Θ ) + s 4 T 2 ( Θ ) , V ( Θ ) = r 0 + r 1 T ( Θ ) + r 2 T 2 ( Θ ) + r 3 ϕ T 1 ( Θ ) + r 4 T 2 ( Θ ) ,
where s i , r i ( i = 0 , 1 , 2 , 3 , 4 ) are arbitrary constants. Solving equation ( 41) through equation ( 42) in the framework of the used computational scheme gives the following.

Case 1.

(43) s 0 = 2 χ 4 ω α + 16 χ 4 k 1 χ 1 χ 2 12 χ 4 , s 2 = 4 k 2 , r 2 = 2 k 2 ( χ 1 + χ 2 ) χ 4 , r 0 = 4 k 1 ( χ 1 + χ 2 ) 2 + ω α ( χ 1 + χ 2 ) + χ 3 ( χ 1 + χ 2 ) 2 χ 4 χ 5 6 χ 4 ( χ 1 + χ 2 ) , s 1 = s 2 = s 4 = r 1 = r 2 = r 4 = 0 .

Case 2.

(44) s 0 = 2 χ 4 ω α + 16 χ 4 k 1 χ 1 χ 2 12 χ 4 , s 4 = 4 k 0 , r 4 = 2 k 0 ( χ 1 + χ 2 ) χ 4 , r 0 = 4 k 1 ( χ 1 + χ 2 ) 2 + ω α ( χ 1 + χ 2 ) + χ 3 ( χ 1 + χ 2 ) 2 χ 4 χ 5 6 χ 4 ( χ 1 + χ 2 ) , s 1 = s 2 = s 3 = r 1 = r 2 = r 3 = 0 .

Case 3.

(45) s 0 = 2 χ 4 ω α + 16 χ 4 k 1 χ 1 χ 2 12 χ 4 , s 2 = 4 k 2 , s 4 = 4 k 0 , r 0 = 4 k 1 ( χ 1 + χ 2 ) 2 + ω α ( χ 1 + χ 2 ) + χ 3 ( χ 1 + χ 2 ) 2 χ 4 χ 5 6 χ 4 ( χ 1 + χ χ 2 ) , r 2 = 2 k 2 ( χ 1 + χ 2 ) χ 4 , r 4 = 2 k 0 ( χ 1 + χ 2 ) χ 4 , s 1 = s 3 = r 1 = r 3 = 0 .

Let us now write down the following exact solutions of space-time fractional coupled S–K equations (41) for case 3 (similar for case 1 and case 2 which are omitted here for simplicity):

(46) U ( Θ ) = 2 χ 4 ω α + 16 χ 4 k 1 χ 1 χ 2 12 χ 4 + 4 k 2 T 2 ( Θ ) + 4 k 0 T 2 ( Θ ) , V ( Θ ) = 4 e 1 ( χ 1 + χ 2 ) 2 + ω α ( χ 1 + χ 2 ) + χ 3 ( χ 1 + χ 2 ) 2 χ 4 χ 5 6 χ 4 ( χ 1 + χ 2 ) + 2 k 2 ( χ 1 + χ 2 ) χ 4 T 2 ( Θ ) + 2 k 0 ( χ 1 + χ 2 ) χ 4 T 2 ( Θ ) .

Family 1. k 0 = 1 , k 1 = ( 1 + m 2 ) , and k 2 = m 2 ; the Jacobi elliptic exact solutions for equation (41) take the following form:

(47) U 1 ( Θ ) = 2 χ 4 ω α 16 χ 4 ( 1 + m 2 ) χ 1 χ 2 12 χ 4 + 4 m 2 s n 2 ( x + ω t ) α Γ ( α + 1 ) + 4 n s 2 ( x + ω t ) α Γ ( α + 1 ) , V 1 ( Θ ) = 4 ( 1 + m 2 ) ( χ 1 + χ 2 ) 2 + ω α ( χ 1 + χ 2 ) + χ 3 ( χ 1 + χ 2 ) 2 χ 4 χ 5 6 χ 4 ( χ 1 + χ 2 ) + 2 m 2 ( χ 1 + χ 2 ) χ 4 s n 2 ( x + ω t ) α Γ ( α + 1 ) + 2 ( χ 1 + χ 2 ) χ 4 n s 2 ( x + ω t ) α Γ ( α + 1 ) .
Or
(48) U 1 ( Θ ) = 2 χ 4 ω α 16 χ 4 ( 1 + m 2 ) χ 1 χ 2 12 χ 4 + 4 m 2 c d 2 ( x + ω t ) α Γ ( α + 1 ) + 4 d c 2 ( x + ω t ) α Γ ( α + 1 ) , V 1 ( Θ ) = 4 ( 1 + m 2 ) ( χ 1 + χ 2 ) 2 + ω α ( χ 1 + χ 2 ) + χ 3 ( χ 1 + χ 2 ) 2 χ 4 χ 5 6 χ 4 ( χ 1 + χ 2 ) + 2 m 2 ( χ 1 + χ 2 ) χ 4 c d 2 ( x + ω t ) α Γ ( α + 1 ) + 2 ( χ 1 + χ 2 ) χ 4 d c 2 ( x + ω t ) α Γ ( α + 1 ) .

Family 2. k 0 = 1 m 2 , k 1 = 2 m 2 1 , and k 2 = m 2 ; the Jacobi elliptic exact solutions for equation (41) take the following form:

(49) U 2 ( Θ ) = 2 χ 4 ω α + 16 χ 4 ( 2 m 2 1 ) χ 1 χ 2 12 χ 4 4 m 2 c n 2 ( x + ω t ) α Γ ( α + 1 ) + 4 ( 1 m 2 ) n c 2 ( x + ω t ) α Γ ( α + 1 ) , V 2 ( Θ ) = 4 ( 2 m 2 1 ) ( χ 1 + χ 2 ) 2 + ω α ( χ 1 + χ 2 ) + χ 3 ( χ 1 + χ 2 ) 2 χ 4 χ 5 6 χ 4 ( χ 1 + χ 2 ) 2 m 2 ( χ 1 + χ 2 ) χ 4 c n 2 ( x + ω t ) α Γ ( α + 1 ) + 2 ( 1 m 2 ) ( χ 1 + χ 2 ) χ 4 n c 2 ( x + ω t ) α Γ ( α + 1 ) .

Family 3. e 0 = m 2 1 , e 1 = 2 m 2 , and e 2 = 1 ; the Jacobi elliptic exact solutions for equation (41) take the following form:

(50) U 3 ( Θ ) = 2 χ 4 ω α + 16 χ 4 ( 2 m 2 ) χ 1 χ 2 12 χ 4 4 d n 2 ( x + ω t ) α Γ ( α + 1 ) + 4 ( m 2 1 ) n d 2 ( x + ω t ) α Γ ( α + 1 ) , V 3 ( Θ ) = 4 ( 2 m 2 ) ( χ 1 + χ 2 ) 2 + ω α ( χ 1 + χ 2 ) + χ 3 ( χ 1 + χ 2 ) 2 χ 4 χ 5 6 χ 4 ( χ 1 + χ 2 ) 2 ( χ 1 + χ 2 ) χ 4 d n 2 ( x + ω t ) α Γ ( α + 1 ) + 2 ( m 2 1 ) ( χ 1 + χ 2 ) χ 4 n d 2 ( x + ω t ) α Γ ( α + 1 ) .

Family 4. k 0 = 1 m 2 , k 1 = 2 m 2 , and k 2 = 1 ; the Jacobi elliptic exact solutions for equation (41) take the following form:

(51) U 4 ( Θ ) = 2 χ 4 c α + 16 χ 4 ( 2 m 2 ) χ 1 χ 2 12 χ 4 + 4 c s 2 ( x + ω t ) α Γ ( α + 1 ) + 4 ( 1 m 2 ) s c 2 ( x + ω t ) α Γ ( α + 1 ) , V 4 ( Θ ) = 4 ( 2 m 2 ) ( χ 1 + χ 2 ) 2 + ω α ( χ 1 + χ 2 ) + χ 3 ( χ 1 + χ 2 ) 2 χ 4 χ 5 6 χ 4 ( χ 1 + χ 2 ) + 2 ( χ 1 + χ 2 ) χ 4 c s 2 ( x + ω t ) α Γ ( α + 1 ) + 2 ( 1 m 2 ) ( χ 1 + χ 2 ) χ 4 s c 2 ( x + ω t ) α Γ ( α + 1 ) .

Family 5. k 0 = 1 , k 1 = 2 m 2 1 , and k 2 = m 2 ( m 2 1 ) ; the Jacobi elliptic exact solutions for equation (41) take the following form:

(52) U 5 ( Θ ) = 2 χ 4 ω α + 16 χ 4 ( 2 m 2 1 ) χ 1 χ 2 12 χ 4 + 4 m 2 ( m 2 1 ) s d 2 ( x + ω t ) α Γ ( α + 1 ) + 4 d s 2 ( x + ω t ) α Γ ( α + 1 ) , V 5 ( Θ ) = 4 ( 2 m 2 1 ) ( χ 1 + χ 2 ) 2 + ω α ( χ 1 + χ 2 ) + χ 3 ( χ 1 + χ 2 ) 2 χ 4 χ 5 6 χ 4 ( χ 1 + χ 2 ) + 2 m 2 ( m 2 1 ) ( χ 1 + χ 2 ) χ 4 s d 2 ( x + c t ) α Γ ( α + 1 ) + 2 ( χ 1 + χ 2 ) χ 4 d s 2 ( x + c t ) α Γ ( α + 1 ) .

Family 6. k 0 = 1 4 , k 1 = 1 2 ( 1 2 m 2 ) , and k 2 = 1 4 ; the Jacobi elliptic exact solutions for equation (41) take the following form:

(53) U 6 ( Θ ) = 2 χ 4 ω α + 8 χ 4 ( 1 2 m 2 ) χ 1 χ 2 12 χ 4 + n s ( x + ω t ) α Γ ( α + 1 ) + c s ( x + ω t ) α Γ ( α + 1 ) 2 + n s ( x + ω t ) α Γ ( α + 1 ) + c s ( x + ω t ) α Γ ( α + 1 ) 2 , V 6 ( Θ ) = 2 ( 1 2 m 2 ) ( χ 1 + χ 2 ) 2 + ω α ( χ 1 + χ 2 ) + χ 3 ( χ 1 + χ 2 ) 2 χ 4 χ 5 6 χ 4 ( χ 1 + χ 2 ) + ( χ 1 + χ 2 ) 2 χ 4 n s ( x + ω t ) α Γ ( α + 1 ) + c s ( x + ω t ) α Γ ( α + 1 ) 2 + ( χ 1 + χ 2 ) 2 χ 4 n s ( x + ω t ) α Γ ( α + 1 ) + c s ( x + ω t ) α Γ ( α + 1 ) 2 .

Family 7. k 0 = 1 4 ( 1 m 2 ) , k 1 = 1 4 ( 1 + m 2 ) , and k 2 = 1 4 ( 1 m 2 ) ; the Jacobi elliptic exact solutions for equation (37) take the following form:

(54) U 7 ( Θ ) = 2 χ 4 ω α + 4 ( 1 + m 2 ) χ 4 χ 1 χ 2 12 χ 4 + ( 1 m 2 ) n c ( x + ω t ) α Γ ( α + 1 ) + s c ( x + ω t ) α Γ ( α + 1 ) 2 , V 7 ( Θ ) = ( 1 + m 2 ) ( χ 1 + χ 2 ) 2 + ω α ( χ 1 + χ 2 ) + χ 3 ( χ 1 + χ 2 ) 2 χ 4 χ 5 6 χ 4 ( χ 1 + χ 2 ) + ( 1 m 2 ) ( χ 1 + χ 2 ) 2 χ 4 n c ( x + ω t ) α Γ ( α + 1 ) + s c ( x + ω t ) α Γ ( α + 1 ) 2 + ( 1 m 2 ) ( χ 1 + χ 2 ) 2 χ 4 n c ( x + ω t ) α Γ ( α + 1 ) + s c ( x + ω t ) α Γ ( α + 1 ) 2 .

Family 8. k 0 = m 2 4 , k 1 = 1 4 ( m 2 2 ) , and k 2 = m 2 4 ; the Jacobi elliptic exact solutions for equation (37) take the following form:

(55) U 8 ( Θ ) = 2 χ 4 ω α + 4 ( m 2 2 ) χ 4 χ 1 χ 2 12 χ 4 + m 2 s n ( x + ω t ) α Γ ( α + 1 ) + i c n ( x + ω t ) α Γ ( α + 1 ) 2 + m 2 s n ( x + ω t ) α Γ ( α + 1 ) + i c n ( x + ω t ) α Γ ( α + 1 ) 2 , V 8 ( Θ ) = ( m 2 2 ) ( χ 1 + χ 2 ) 2 + ω α ( χ 1 + χ 2 ) + χ 3 ( χ 1 + χ 2 ) 2 χ 4 χ 5 6 χ 4 ( χ 1 + χ 2 ) + m 2 ( χ 1 + χ 2 ) 2 χ 4 s n ( x + ω t ) α Γ ( α + 1 ) + i c n ( x + ω t ) α Γ ( α + 1 ) 2 + m 2 ( χ 1 + χ 2 ) 2 χ 4 s n ( x + ω t ) α Γ ( α + 1 ) + i c n ( x + ω t ) α Γ ( α + 1 ) 2 .

Family 9. k 0 = 1 4 , k 1 = 1 2 ( 1 m 2 ) , and k 2 = 1 4 ; the Jacobi elliptic exact solutions for equation (37) take the following form:

(56) U 9 ( Θ ) = 2 χ 4 ω α + 8 χ 4 ( 1 m 2 ) χ 1 χ 2 12 χ 4 + 1 m 2 s c ( x + ω t ) α Γ ( α + 1 ) ± d c ( x + ω t ) α Γ ( α + 1 ) 2 + 1 m 2 s c ( x + ω t ) α Γ ( α + 1 ) ± d c ( x + ω t ) α Γ ( α + 1 ) 2 , V 9 ( Θ ) = 2 ( 1 m 2 ) ( χ 1 + χ 2 ) 2 + ω α ( χ 1 + χ 2 ) + χ 3 ( χ 1 + χ 2 ) 2 χ 4 χ 5 6 χ 4 ( χ 1 + χ 2 ) + ( χ 1 + χ 2 ) 2 χ 4 1 m 2 s c ( x + ω t ) α Γ ( α + 1 ) ± d c ( x + ω t ) α Γ ( α + 1 ) 2 + ( χ 1 + χ 2 ) 2 χ 4 1 m 2 s c ( x + ω t ) α Γ ( α + 1 ) ± d c ( x + ω t ) α Γ ( α + 1 ) 2 .

#### 3.3.1 Numerical solutions for the space-time fractional coupled S–K equations

In this subsection, we give some figures to illustrate some of our results that were obtained in this section. To this end, we select some special values of the parameters to show the behavior of Jacobi elliptic solution for the nonlinear space-time fractional coupled S–K differential equation (Figures 19–26).

### Figure 19

The Jacobi elliptic doubly periodic solution u 5 (52) at c = 3, k 1 = 1, k 2 = 2, k 4 = 4, m = 0.5, α = 1, α = 0.5, and α = 0.1, respectively.

### Figure 20

The projection Jacobi elliptic doubly periodic solution u 5 (52) at t = 0, c = 3, k 1 = 1, k 2 = 2, k 4 = 4, m = 0.5, α = 1, α = 0.5, and α = 0.1, respectively.

### Figure 21

The Jacobi elliptic doubly periodic solution v 5 (52) at c = 3, k 1 = 1, k 2 = 2, k 4 = 4, k 5 = 5, m = 0.5, α = 1, α = 0.5, and α = 0.1, respectively.

### Figure 22

The projection Jacobi elliptic doubly periodic solution v 5 (52) at c = 3, k 1 = 1, k 2 = 2, k 3 = 3, k 4 = 4, k 5 = 5, m = 0.5, α = 1, α = 0.5, and α = 0.1, respectively.

### Figure 23

The Jacobi elliptic doubly periodic solution u 9 (56) at c = 3, m = 0.5, k 1 = 1, k 2 = 2, k 3 = 3, k 4 = 4, k 5 = 5, α = 1, α = 0.5, and α = 0.1, respectively.

### Figure 24

The projection Jacobi elliptic doubly periodic solution u 9 (56) at c = 3, k 1 = 1, k 2 = 2, k 3 = 3, k 4 = 4, k 5 = 5, m = 0.5, α = 1, α = 0.5, and α = 0.1, respectively.

### Figure 25

The Jacobi elliptic doubly periodic solution v 9 (56) at c = 3, m = 0.5, k 1 = 1, k 2 = 2, k 3 = 3, k 4 = 4, k 5 =5 α = 1, α = 0.5, and α = 0.1, respectively.

### Figure 26

The projection Jacobi elliptic doubly periodic solution v 9 (56) at c = 3, k 1 = 1, k 2 = 2, k 3 = 3, k 4 = 4, k 5 = 5, m = 0.5, α = 1, α = 0.5, and α = 0.1, respectively.

## 4 Some conclusions and discussions

A direct algebraic approach is used in this paper to find the exact solutions for NPFDEs. We have successfully obtained analytical Jacobi elliptic solutions for certain NPFDEs in mathematical physics. The accuracy of this methodology and decrease in equations give this methodology broader applicability. For several NPFDEs in mathematical physics, the Algebraic direct approach is a useful technique for developing several new forms of Jacobi elliptic solutions. The exact hyperbolic solutions and exact trigonometric solutions are generalized by Jacobi elliptic solutions, where specific unique values are taken from the module. This approach is an advantageous and efficient way of seeking the exact solutions in mathematical physics for NPFDEs. In this article, Maple and Mathematica were used for computations. Future works’ ability is focused on the expansion of the proposed numerical styles to resolve other kinds of NPFDEs and boundary value problems.

# Acknowledgments

The authors are thankful to the editors and referees for their valuable help. The authors thank Taif University researchers for supporting project number (TURSP-2020/16), Taif university, Taif, Saudi Arabia.

Funding information: This paper was funded by “Taif University Researchers Supporting Project number (TURSP-2020/16), Taif University, Taif, Saudi Arabia.”

Conflict of interest: The authors declare that there are no competing interests.

Data availability statement: No data sets were generated or analyzed in the current study data.

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