1.Introduction

In recent years, fractional differential equations (FDEs) shed light on the science environment because of their important position in many areas of complicated physical events, from fluid dynamics and optical fiber to quantum field theory. In fact, the soliton solutions of such equations have been one of the most remarkable solutions due to their more clear view of the nonlinear physical properties and then guide to the next aims. As a result, several methods have been used by many authors to calculate such solutions for better insight into the main properties of physical constructions in different media ^1-5.

A lot of kinds of fractional derivative operators have been described by scientists. Some of them are Caputo derivative, Riemann-Liouville derivative, Caputo-Fabrizio, Jumarie’s modified Riemann-Liouville derivative, Atangana-Baleanu derivative ^6-9. By using these derivative operators, many methods have been submitted to provide solutions of FDEs ^10-15.

Then, the conformable derivative has been identified by Khalil et al.¹⁶. Also, exact solutions of the FDE have been found by using this derivative ¹⁷. Then, some theorems, definitions, and properties related to conformable derivative have been presented by Atangana et al. ¹⁸. Consequently, a new fractional derivative called beta-derivative has been given by Atangana et al. ¹⁹. Then, solutions of FDEs with conformable derivatives have been considered by a lot of authors ^20-25.

Heisenberg’s ferromagnetic spin chain equation has been highly considered for its significance from different aspects ^26-29. This paper will address Heisenberg ferromagnetic spin chain equation to get its soliton solutions by a strong algorithm that was lately submitted. The methods are the generalized Kudryashov method (GKM) ^30-33 and modified -expansion function method (MEFM) ^34-37. The algorithms supply optical solitons such as dark, bright, and dark-bright. The results are thus found after the extensive experience of the algorithmic operation.

2.Beta Derivatives and its features

Definition 1. The conformable derivative has been identified by Khalil et al. ¹⁶. Let w:[0,∞) be a function β-th order, the conformable derivative of w(t) for all is given as follows:

Fβ(w(t))=dβw(t)dtβ=lime→0w(t+et1-β)-w(t)e,0<β≤1.

Also, if w is β-differentiable in (0,a), a>0, and lime→0+w(β)(t) exists, then it can be written as w(β)(0)=lime→0+(t).

Definition 2. Let w(t) be a function qualified for all non-negative t. Then, the beta derivative of w(t) is defined by ¹⁹

Fβ(w(t))=dβw(t)dtβ=lime→0wt+et+1Γ(β)1-β-w(t)e,0<β≤1.

Although the conformable fractional derivative presented by Khalil et al. supplies some basic properties such as the chain rule, Atangana’s fractional derivative is presented because it can yield the maximum features of the basic derivatives.

Such derivatives can not only be assumed as fractional derivatives but also considered as a natural extension of the classical derivative. There is a significant theorem for beta-derivatives ¹⁹:

Theorem. Let w(t) and v(t) be β-differentiable functions for all t>0 and β∈(0,1[. Then

Fβ(mw(t)+nv(t))=mFβ(w(t))+nFβ(v(t)),∀m,n∈R.Fβ(w(t)v(t))=v(t)Fβ(w(t))+w(t)Fβ(v(t)),Fβw(t)v(t)=v(t)Fβ(w(t))-w(t)Fβ(v(t))(v(t))2,Fβ(w(t))=t+1Γ(β)1-βdw(t)dt.

3.General structure of GKM

We survey the following FDE with beta derivative for a function of two real variables, space x, and time t:

Step 1. Initially, we should perform the traveling wave solution of Eq.(1) as follows;

Where

where p and σ arbitrary constants. Then, by substituting Eqs.(2-4) to Eq.(1), a nonlinear ordinary differential equation can be obtained as:

where the prime displays differentiation about η.

Step 2. Assume that the exact solutions of Eq. (5) can be considered in the form

where γ(η)=1/1±eη. We highlight that the function γ is the solution of the equation:

Taking account of Eq. (6), we supply

Step 3. The solution of Eq.(5) can be defined as follows:

To compute the values M and N in Eq. (10) that is the pole order for the general solution of Eq. (5), we procure comparably as in the classical Kudryashov method on balancing the highest-order nonlinear terms in Eq. (5), and we can find a relation of M and N. We can get values of M and N.

Step 4. Substituting Eq. (6) into Eq. (5) ensures a polynomial R(γ) of γ. Extracting the coefficients of R(γ) to zero, we get a system of algebraic equations. Solving this system, we can identify p and the variable coefficients of a ₀, a ₁, a ₂, …a _N, b₀, b₁, b₂,…b_M. Thus, we get the exact solutions to Eq. (5).

4.Soliton solutions for Heisenberg ferromagnetic spin chain equation by GKM

In this section, we look for exact solutions of the Heisenberg ferromagnetic spin chain equation with beta time derivative by using GKM.

It is helpful to use the spin to lie in a planet right angles to the chain axis. Thus, we consider Heisenberg ferromagnetic spin chain equation with beta time derivative ³⁸

Firstly, we take wave variable transformations as follows

where

Putting (12-14) into (11) provides

where p=2sin2ϕ-cosϕ-2.

For the balance principle between higher-order derivative H´´ and highest power nonlinear terms H³ in Eq. (15), one can be procured

By using GKM, the solution of Eq. (11) can be given as

where a ₀, a ₁, and a ₂ are found later and γ(η)=(1/1±eη). The function γ(η) provides as

Thus, the exact solutions of Eq. (11) are accessed as the following;

Case 1.

Replacing Eq. (19) into Eq. (17), dark soliton solutions of Eq. (11) can reached as

Case 2.

Replacing Eq. (22) into Eq. (17), bright soliton solutions of Eq. (11) can be determined as

Case 3.

Replacing Eq. (25) into Eq. (17), dark soliton solutions of Eq. (11) can be ascertained as

Case 4.

Putting Eq. (28) into Eq. (17), dark-bright soliton solutions of Eq. (11) can be provided as

In Fig. 1, 3D and 2D graphs are investigated to illustrate the influence of the parameter β on the dynamics of the first dark soliton solution. Clearly, the physical behavior of the dark soliton solution is altered when the parameter β gets different values.

In Fig. 2, 3D and 2D graphs are examined to show the influence of the parameter β on the dynamics of the bright soliton solution. Unquestionably, the physical behavior of the bright soliton solution is changed when the parameter β takes different values.

5.General structure of MEFM

We survey the following FDE with beta derivative for a function of two real variables, space x, and time t:

Step 1. Initially, we should perform the traveling wave solution of Eq. (31) as follows;

Where

where p and σ arbitrary constants. Then, by substituting Eqs. (32-34) to Eq.(31), a nonlinear ordinary differential equation can be obtained as:

where the prime displays differentiation about η.

Step 2: Presume the traveling wave solution of (4) can be indicated as follows:

where F _i, G _j, (0≤i≤N, 0≤j≤M) are constants to be described later, such that FN≠0, GM≠0, and Ω=Ω(η) is the solution of the following ordinary differential equation:

The solution families of (37) can be shown as follows:

Family1: If n≠0, m2-4n>0,

Family 2: If n≠0, m2-4n<0,

Family 3: If n = 0, m≠0, and m2-4n>0,

Family 4: If n≠0, m≠0, and m2-4n=0,

Family5: If n = 0, m = 0, and m2-4n=0,

The positive integers M and M can be determined attending the homogeneous balance principle in Eq. (36).

Step 3: Replacing Eqs. (37) and (38-42) into Eq.(36), we ascertain a polynomial of exp(-Ω(η)). We stabilize all the coefficients of same power of exp(-Ω(η)) to zero. This operation determines a system of equations that can be unfastened to reach F ₀, F ₁, F ₂,…F _N, G ₀, G ₁, G ₂,…G _M, E, m, n by the way of Wolfram Mathematica 12. Inserting the values of these constants into Eq. (36), the general solutions of (36) supplies the determination of the solution of Eq.(31).

6.Soliton solutions for Heisenberg ferromagnetic spin chain equation by MEFM

In this section, we seek exact solutions of the Heisenberg ferromagnetic spin chain equation with beta time derivative by using MEFM.

For the balance principle between higher-order derivative H´´ and highest power nonlinear terms H³ in Eq. (15), one can procure

By using MEFM, the solution of Eq. (11) can be given as

And

where F2≠0 and G1≠0. The function Ω=Ω(η) provides as

Thus, the exact solutions of (11) are accessed as the following;

Case 1:

Replacing (48) into (44), dark soliton solutions of (11) can be reached as

where n≠0, and J=m2-4n>0.

In 3, 3D and 2D graphs are considered to indicate the influence of the parameter on the dynamics of the dark soliton solution. Explicitly, the physical behavior of the dark soliton solution is shifted when the parameter β receives different values.

Case 2:

Replacing (50) into (44), dark soliton solutions of (11) can be reached as

where,

A=4i2(r-s)2n2F0F1F2,B=F1F12-4F0F2+(F12-2F0F2)(r-s)2,C=-(r-s)2F12(F0-nF2)+(F0-SF2)(r-s)2F1F12-4F0F2,D=-2(r-s)2F0F1F2F12-2F2(F0+nF2)-F1F12-4F0F2F22f(x,y,t)=χcos⁡ϕx+1Γββ+sin⁡ϕy+1Γββ+pt+1ΓβB+Eβ2βχ=-4n+(F12-F1F12-4F0F2)2F12F12,

and

-4n+(F12-F1F12-4F0F2)2F12F12>0.

7.Conclusion

In this work, the soliton characters of the Heisenberg ferromagnetic spin chain equation with beta time derivative were investigated by using GKM and MEFM. Dark, bright, and dark-bright soliton solutions of this equation have been accomplished found. Then, 3D and 2D images were presented for some solutions, which display the vitality of the solutions with proper values. Numerical results, together with the graphical demonstrations, have exhibited the reliability of these methods. Also, these solutions have been reported to the literature with novel substantial physical properties. These methods can be applied to other FDEs with beta time derivatives.