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Numerical simulations for fractional Hirota–Satsuma coupled Korteweg–de Vries systems

  • Abdul Hamid Ganie , Saima Noor EMAIL logo , Maryam Al Huwayz , Ahmad Shafee and Samir A. El-Tantawy EMAIL logo
Published/Copyright: April 23, 2024
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Open Physics
From the journal Open Physics Volume 22 Issue 1

Abstract

In this investigation, the fractional Hirota–Satsuma coupled Korteweg–de Vries (KdV) problem is solved using two modern semi-analytic techniques known as the Aboodh residual power series method (ARPSM) and Aboodh transform iteration method (ATIM). The two suggested approaches are briefly explained, along with how to use them to solve the fractional Hirota–Satsuma coupled KdV problem. Some analytical approximate solutions for the current problem are derived using the proposed techniques until the second-order approximation. To ensure high accuracy of the derived approximation, they are analyzed numerically and graphically and compared with the exact solutions of the integer cases. The offered techniques demonstrate more accuracy in their outcomes compared to other alternatives. The numerical results show that ARPSM and ATIM are highly accurate, practical, and beneficial for solving nonlinear equation systems. The current results are expected to help many physics researchers in modeling their different physical problems, especially those interested in plasma physics.

Keywords: Aboodh transform iteration method; Hirota–Satsuma coupled KdV; Aboodh residual power series method; Caputo operator

1 Introduction

Fractional differential equations (FDEs) have become more and more common in various fields of research over the past few decades. They can better explain numerous critical phenomena in acoustics, electromagnetics, electrochemistry, viscoelasticity, cosmology, and materials science. As such, the solution of the FDEs has received a great deal of interest. Nonlinear processes significantly influence physics, applied mathematics, and engineering-related problems. Nonlinear differential equations are used to describe many of these physical phenomena. Differential equations continue to be a significant problem in mathematics and physics that calls for creative solutions that can be either exact or approximative. Most recently developed linear and nonlinear equations have not had a precise analytical solution; hence, numerical methods have been used to solve them [1–5].

Compared to linear differential equations, nonlinear partial differential equations (PDEs) are more challenging to solve [6–8]. Therefore, it might only sometimes be possible to find these equations’ analytical solutions. Here, we employ semi-analytical techniques that yield series solutions [9–11]. These types of approaches look for answers in the form of sequences. The foundation of semi-analytical approaches is determining the other terms in the series, given the problem’s beginning circumstances. This is where the idea of series convergence comes into play. Thus, a convergence study of these techniques is required. Since this convergence analysis can be performed theoretically, the absolute error between the numerical and analytical solutions may be used to determine if the series solution is convergent. A perfect convergence may be obtained in semi-analytic approaches with only a few terms of the series, although more terms may be required in some instances. In other words, a higher sequence of terms leads to a greater convergence of the analytical solution [12–15]. Guo and Hu presented a practical predefined-time stabilization method for high-order systems with unknown disturbances, using a time base generator [16]. Chen et al. proposed a finite-time velocity-free rendezvous control strategy for multiple autonomous underwater vehicle systems with intermittent communication [17]. Yang and Kai investigated dynamical properties and chaotic behaviors in nonlinear coupled Schrödinger equations, particularly in fiber Bragg gratings [18]. Kai and Yin analyzed Gaussian traveling wave solutions in a Schrodinger equation with logarithmic nonlinearity [19]. Zhou et al. introduced an iterative threshold algorithm for log-sum regularization in sparse problems [20]. Finally, Li et al. presented an improved fractional Tikhonov regularization method for moving force identification, providing advancements in structural engineering applications [21].

Research on shallow water equations is a fascinating and fruitful area. Shallow water theory is helpful in a variety of application domains, for example, river floods, building breakwaters, physical oceanography, coastal engineering, tsunami forecasting, and dam break problems. Shallow flow equations are frequently employed in flood analysis since it is well known that they effectively estimate the impacts of floods spreading over extensive terrain. The analytical solutions to these equations can also be used to obtain a good idea of the height of the waves along the coast and how their speed distribution changes over time on a sloping beach and in coastal areas. Shallow water wave equations express the height and velocity of the current flowing through the channels of a rainwater drainage network. They are also employed in the computation of ocean flows and vorticity [22,23]. Consider the following fractional-order Hirota–Satsuma coupled Korteweg–de Vries (KdV):

(1) D ω p γ 1 − 1 2 ∂ ψ 3 γ 1 + 3 γ 1 ∂ ψ γ 1 − 3 ∂ ψ ( γ 2 γ 3 ) = 0 , D ω p γ 2 + ∂ ψ 3 γ 2 − 3 γ 1 ∂ ψ γ 2 = 0 , D ω p γ 3 + ∂ ψ 3 γ 3 − 3 γ 1 ∂ ψ γ 3 = 0 ,

with the following initial conditions (ICs):

(2) γ 1 ( ψ , 0 ) = l ( ψ ) , γ 2 ( ψ , 0 ) = m ( ψ ) , γ 3 ( ψ , 0 ) = n ( ψ ) ,

where γ i ≡ γ i ( ψ , ω ) for i = 1 , 2 , 3 , and 0 < p ≤ 1 . Note here that the ICs l ( ψ ) , m ( ψ ) , and n ( ψ ) can be obtained from the exact solutions to the integer form of system (1) at ω = 0 .

Several authors [24–30] have examined System 1 with constant coefficients. In the study by Kaya [24], the isolated wave solutions are derived using the decomposition method. In the study by Yong et al. [25], multiple explicit solutions are obtained using the homogeneous method. In the study by Yao et al. [26], the traveling wave solutions are derived using the Ritt–Wu method. In the study by Xu et al. [28], the traveling wave solutions (TWSs) are obtained using the maple package TRWS, and finally, in the studies by Liu et al. [29] and Qing-You et al. [30], the periodic wave solutions are given.

A significant class of nonlinear evolution equations, KdV-type equations, have many uses in engineering and physical research. Acoustic waves are a product of these equations in plasma physics [31–34], and they describe a long wave in both shallow and deep oceans in geophysical fluid dynamics [35]. Cluster physics, super-deformed nuclei, fission, thin films, radar, rheology [36,37], optical-fiber communications [38], and superconductors [39] are all areas where they are present. Because they assume constant coefficients, the physical circumstances that give birth to the KdV equations are often idealized. For this reason, several types of KdV equations involving variable coefficients [40–44] have received a lot of focus as of late in the search for accurate solutions. Two long waves’ interaction is described by a set of linked KdV equations with variable coefficients [45,46], which has been examined recently by some others.

In 2013, Arqub introduced a semi-analytical methodology called the residual power series method (RPSM) [47]. This method combines Taylor’s series with the residual error function. As its first venture into the domain of fuzzy differential equation resolution, this novel approach proved its adaptability by effectively solving both linear and nonlinear differential equations. To solve complicated DEs using power series methods, Mukhtar et al. [48] developed a unique set of RPSM algorithms. Then, separate equations with modifications made may be solved. Finally, the inverse Aboodh transform (AT) is used to solve the original equation. This new strategy uses homotopy perturbation methods with the Sumudu transform. We solve linear and nonlinear PDEs using the unique power series expansion approach, which does not need linearization, perturbation, or discretization. In RPSM, the calculation of fractional derivatives requires several solution iterations, but here, the computations of coefficients are much more straightforward. Using a quick convergence series, the suggested approach can provide a close approximate solution.

The most significant mathematical accomplishment of the twentieth century was the Aboodh transform iterative approach (ATIM) for fractional PDEs. Regular methods of solving PDEs with fractional derivatives are infamously tricky because of the computing complexity and non-convergence of these problems. Our unique technology goes beyond these restrictions by reducing processing effort, enhancing accuracy, and continuously improving approximation solutions. Iterations tailored to fractional derivatives have improved the solutions to complex mathematical and physical problems [49–51]. Complex fractional PDE-governed systems may now be used to study challenging engineering, applied mathematics, and physics problems.

The Aboodh residual power series method (ARPSM) and ATIM are two of the most basic strategies for solving FDEs [52,53]. In addition to giving analytical answers in symbolic terms that can be accessed immediately, these approaches also provide numeric approximate results to linear and nonlinear differential equations that do not need discretization or linearization. Finding solutions to fractional order Hirota–Satsuma coupled KdV using two different approaches, ARPSM and ATIM, is the main objective of this study. These two methods have been used together to resolve many nonlinear fractional differential problems.

2 Fundamental definitions

Definition 2.1

Let exponentially ordered function v ( ψ , ω ) be piecewise continuous [54]. For v ( ψ , ω ) , AT is defined as follows, where ω ≥ 0 :

A [ v ( ψ , ω ) ] = Ω ( ψ , ν ) = 1 ν ∫ 0 ∞ v ( ψ , ω ) e − ω ν d ω , r 1 ≤ ν ≤ r 2 .

The inverse AT is expressed in the following way:

A − 1 [ Ω ( ψ , ν ) ] = v ( ψ , ω ) = 1 2 π i ∫ u − i ∞ u + i ∞ Ω ( ψ , ω ) ν e ω ν d ω ,

where ψ = ( ψ 1 , ψ 2 , … , ψ p ) and p ∈ N .

Lemma 2.2

Consider the following functions: v 1 ( ψ , ω ) and v 2 ( ψ , ω ) [55,56], which exhibit exponential order and are piecewise continuous on the interval [ 0 , ∞ [ . It is assumed that

A [ v 1 ( ψ , ω ) ] = Ω 1 ( ψ , ω ) , A [ v 2 ( ψ , ω ) ] = Ω 2 ( ψ , ω ) and ϖ 1 and ϖ 2 are the constants. Therefore, the following statements are true:

  1. A [ ϖ 1 v 1 ( ψ , ω ) + ϖ 2 v 2 ( ψ , ω ) ] = ϖ 1 Ω 1 ( ψ , ν ) + ϖ 2 Ω 2 ( ψ , ν ) ,

  2. A − 1 [ ϖ 1 Ω 1 ( ψ , ω ) + ϖ 2 Ω 2 ( ψ , ω ) ] = ϖ 1 v 1 ( ψ , ν ) + ϖ 2 v 2 ( ψ , ν ) ,

  3. A [ J ω p v ( ψ , ω ) ] = Ω ( ψ , ν ) ν p ,

  4. A [ D ω p v ( ψ , ω ) ] = ν p Ω ( ψ , ν ) − ∑ K = 0 r − 1 v K ( ψ , 0 ) ν K − p + 2 , r − 1 < p ≤ r , r ∈ N .

Definition 2.3

The fractional Caputo derivative of a function v ( ψ , ω ) of order p is defined as follows [57]:

D ω p v ( ψ , ω ) = J ω m − p v ( m ) ( ψ , ω ) , r ≥ 0 , m − 1 < p ≤ m ,

where ψ = ( ψ 1 , ψ 2 , … , ψ p ) ∈ R p and m , p ∈ R , J ω m − p is the RL integral of v ( ψ , ω ) .

Definition 2.4

We may write the power series as [58]:

∑ r = 0 ∞ ℏ r ( ψ ) ( ω − ω 0 ) r p = ℏ 0 ( ω − ω 0 ) 0 + ℏ 1 ( ω − ω 0 ) p + ℏ 2 ( ω − ω 0 ) 2 p + ⋯ ,

where p ∈ N and ψ = ( ψ 1 , ψ 2 , … , ψ p ) ∈ R p . In this series, ω 0 appears as a variable, and the series coefficients are represented as ℏ r ( ψ ) .

Lemma 2.5

The AT is represented as A [ v ( ψ , ω ) ] = Ω ( ψ , ν ) , supposing that the function v ( ψ , ω ) has an exponential order. As a result,

(3) A [ D ω r p v ( ψ , ω ) ] = ν r p Ω ( ψ , ν ) − ∑ j = 0 r − 1 ν p ( r − j ) − 2 D ω j p v ( ψ , 0 ) , 0 < p ≤ 1 ,

where ψ = ( ψ 1 , ψ 2 , … , ψ p ) ∈ R p and p ∈ N and D ω r p = D ω p . D ω p . ⋯ . D ω p ( r -times ) .

Proof

Now, we use the induction approach to verify Eq. (3). The following result is obtained when r = 1 is substituted into Eq. (3):

A [ D ω 2 p v ( ψ , ω ) ] = ν 2 p Ω ( ψ , ν ) − ν 2 p − 2 v ( ψ , 0 ) − ν p − 2 D ω p v ( ψ , 0 ) .

Lemma 2.2, specifically Part (4) proves that the equation is true for r = 1 . We obtain: by using the Eq. (3) with r = 2 :

(4) A [ D r 2 p v ( ψ , ω ) ] = ν 2 p Ω ( ψ , ν ) − ν 2 p − 2 v ( ψ , 0 ) − ν p − 2 D ω p v ( ψ , 0 ) .

We obtain the following by taking into account the left-hand side (L.H.S.) of Eq. (4):

(5) L.H.S = A [ D ω 2 p v ( ψ , ω ) ] .

A specific way to write Eq. (5) is as follows:

(6) L.H.S = A [ D ω p v ( ψ , ω ) ] .

Let

(7) z ( ψ , ω ) = D ω p v ( ψ , ω ) .

Therefore, Eq. (6) changes to:

(8) L.H.S = A [ D ω p z ( ψ , ω ) ] .

With the help of the fractional derivative of the Caputo type.

(9) L.H.S = A [ J 1 − p z ′ ( ψ , ω ) ] .

The formula for the Riemann–Liouville (RL) fractional integral of the AT is given by Eq. (9):

(10) L.H.S = A [ z ′ ( ψ , ω ) ] ν 1 − p .

Using the differential property of AT, Eq. (10) is changed to

(11) L.H.S = ν p Z ( ψ , ν ) − z ( ψ , 0 ) ν 2 − p .

In light of Eq. (7), we are able to determine

Z ( ψ , ν ) = ν p Ω ( ψ , ν ) − v ( ψ , 0 ) ν 2 − p ,

where A [ z ( ψ , ω ) ] = Z ( ψ , ν ) . Consequently, Eq. (11) changes into

(12) L.H.S = ν 2 p Ω ( ψ , ν ) − v ( ψ , 0 ) ν 2 − 2 p − D ω p v ( ψ , 0 ) ν 2 − p .

Note that Eqs (3) and (12) are compatible with each other. With the assumption that Eq. (3) is valid for r = K . Eq. (3) may be revised by substituting r = K :

(13) A [ D ω K p v ( ψ , ω ) ] = ν K p Ω ( ψ , ν ) − ∑ j = 0 K − 1 ν p ( K − j ) − 2 D ω j p D ω j p v ( ψ , 0 ) , 0 < p ≤ 1 .

Eq. (3) will be shown to be true for r = K + 1 in the following:

(14) A [ D ω ( K + 1 ) p v ( ψ , ω ) ] = ν ( K + 1 ) p Ω ( ψ , ν ) − ∑ j = 0 K ν p ( ( K + 1 ) − j ) − 2 D ω j p v ( ψ , 0 ) .

In Eq. (14), on L.H.S., we obtain

(15) L.H.S = A [ D ω K p ( D ω K p ) ] .

Consider

D ω K p = g ( ψ , ω ) .

Eq. (15) gives us

(16) L.H.S = A [ D ω p g ( ψ , ω ) ] .

By combining the fractional Caputo derivative with the RL integral formula, Eq. (16) may be expressed as

(17) L.H.S = ν p A [ D ω K p v ( ψ , ω ) ] − g ( ψ , 0 ) ν 2 − p .

With the help of Eqs (13) and (17), it is transformed into

(18) L.H.S = ν r p Ω ( ψ , ν ) − ∑ j = 0 r − 1 ν p ( r − j ) − 2 D ω j p v ( ψ , 0 ) .

Eq. (18) allows us to obtain the subsequent outcome:

L.H.S = A [ D ω r p v ( ψ , 0 ) ] .

This proves that Eq. (3) is valid for r = K + 1 . Because of this, we proved that Eq. (3) is true for all positive integers using the mathematical induction approach.□

Lemma 2.6

Consider the exponentially ordered function v ( ψ , ω ) . A [ v ( ψ , ω ) ] = Ω ( ψ , ν ) is the multiple fractional Taylor’s series representation of the AT of v ( ψ , ω ) :

(19) Ω ( ψ , ν ) = ∑ r = 0 ∞ ℏ r ( ψ ) ν r p + 2 , ν > 0 ,

where ψ = ( ψ 1 , ψ 2 , … , ψ p ) ∈ R p , p ∈ N .

Proof

Performing a fractional order analysis on Taylor’s series, we obtain

(20) v ( ψ , ω ) = ℏ 0 ( ψ ) + ℏ 1 ( ψ ) ω p Γ [ p + 1 ] + + ℏ 2 ( ψ ) ω 2 p Γ [ 2 p + 1 ] + ⋯ .

By applying the AT on Eq. (20), the following equality is obtained:

A [ v ( ψ , ω ) ] = A [ ℏ 0 ( ψ ) ] + A ℏ 1 ( ψ ) ω p Γ [ p + 1 ] + A ℏ 1 ( ψ ) ω 2 p Γ [ 2 p + 1 ] + ⋯ .

These characteristics of the AT allow us to derive

A [ v ( ψ , ω ) ] = ℏ 0 ( ψ ) 1 ν 2 + ℏ 1 ( ψ ) Γ [ p + 1 ] Γ [ p + 1 ] 1 ν p + 2 + ℏ 2 ( ψ ) Γ [ 2 p + 1 ] Γ [ 2 p + 1 ] 1 ν 2 p + 2 ⋯ .

As a result, we obtain (19), a new Taylor’s series in the AT.□

Theorem 2.7

Following are the multiple fractional power series (MFPS) representations of the functions v ( ψ , ω ) and Ω ( ψ , ν ) :

Ω ( ψ , ν ) = ∑ 0 ∞ ℏ r ( ψ ) ν r p + 2 , ν > 0 ,

where ψ = ( ψ 1 , ψ 2 , … , ψ p ) ∈ R p and p ∈ N . Then, we have

ℏ r ( ψ ) = D r r p v ( ψ , 0 ) ,

where D ω r p = D ω p . D ω p . ⋯ . D ω p ( r -times ) .

Proof

The new version of Taylor’s series provides us

(21) ℏ 1 ( ψ ) = ν p + 2 Ω ( ψ , ν ) − ν p ℏ 0 ( ψ ) − ℏ 2 ( ψ ) ν p − ℏ 3 ( ψ ) ν 2 p − ⋯ .

This may be achieved by solving Eq. (21) for lim ν → ∞ .

ℏ 1 ( ψ ) = lim ν → ∞ ( ν p + 2 Ω ( ψ , ν ) − ν p ℏ 0 ( ψ ) ) − lim ν → ∞ ℏ 2 ( ψ ) ν p − lim ν → ∞ ℏ 3 ( ψ ) ν 2 p − ⋯ .

The following equality is the result of taking the limit:

(22) ℏ 1 ( ψ ) = lim ν → ∞ ( ν p + 2 Ω ( ψ , ν ) − ν p ℏ 0 ( ψ ) ) .

The following result is produced when Lemma 2.5 is combined with Eq. (22):

(23) ℏ 1 ( ψ ) = lim ν → ∞ ( ν 2 A [ D ω p v ( ψ , ω ) ] ( ν ) ) .

In addition, Eq. (23) in conjunction with Lemma 2.6 yields the following result:

ℏ 1 ( ψ ) = D ω p v ( ψ , 0 ) .

The result is obtained using the modified Taylor’s series together with ν → ∞ :

ℏ 2 ( ψ ) = ν 2 p + 2 Ω ( ψ , ν ) − ν 2 p ℏ 0 ( ψ ) − ν p ℏ 1 ( ψ ) − ℏ 3 ( ψ ) ν p − ⋯ .

We obtain the result using Lemma 2.6:

(24) ℏ 2 ( ψ ) = lim ν → ∞ ν 2 ( ν 2 p Ω ( ψ , ν ) − ν 2 p − 2 ℏ 0 ( ψ ) − ν p − 2 ℏ 1 ( ψ ) ) .

Eq. (24) is transformed once again using Lemmas 2.5 and 2.6 with Theorem 2.7:

ℏ 2 ( ψ ) = D ω 2 p v ( ψ , 0 ) .

Applying the same process to the new form of Taylor’s series yields

ℏ 3 ( ψ ) = lim ν → ∞ ν 2 ( A [ D ω 2 p v ( ψ , p ) ] ( ν ) ) .

The final equation is obtained using Lemma 2.6:

ℏ 3 ( ψ ) = D ω 3 p v ( ψ , 0 ) .

In general, we obtain

ℏ r ( ψ ) = D ω r p v ( ψ , 0 ) .

This concludes the proof.□

Theorem 2.8

A new multiple fractional Taylor’s formula is shown in Lemma 2.6 with Theorem 2.7 and is denoted as A [ v ( ψ , ω ) ] = Ω ( ψ , ν ) . The new multiple fractional Taylor’s formula for ( 0 < ν ≤ s ) with 0 < p ≤ 1 satisfies the following inequality if ∣ ν a A [ D ω ( K + 1 ) p v ( ψ , ω ) ] ∣ ≤ T :

∣ R K ( ψ , ν ) ∣ ≤ T ν ( K + 1 ) p + 2 , 0 < ν ≤ s .

Proof

The following assumptions serve as the foundation for the proof:

A [ D ω r p v ( ψ , ω ) ] is defined for 0 < ν ≤ s , where r = 0 , 1 , 2 , … , K + 1 . Assume ∣ ν 2 A [ D ω K + 1 v ( ψ , τ ) ] ∣ ≤ T holds for 0 < ν ≤ s given the specified conditions. Examine the relationship that results from Taylor’s series using the new format:

(25) R K ( ψ , ν ) = Ω ( ψ , ν ) − ∑ r = 0 K ℏ r ( ψ ) ν r p + 2 .

When Theorem 2.7 is applied, Eq. (25) becomes

(26) R K ( ψ , ν ) = Ω ( ψ , ν ) − ∑ r = 0 K D ω r p v ( ψ , 0 ) ν r p + 2 .

Multiply ν ( K + 1 ) a + 2 on both sides of Eq. (26). In light of this,

(27) ν ( K + 1 ) p + 2 R K ( ψ , ν ) = ν 2 ν ( K + 1 ) p Ω ( ψ , ν ) − ∑ r = 0 K ν ( K + 1 − r ) p − 2 D ω r p v ( ψ , 0 ) .

Applying Lemma 2.5 to Eq. (27) yields

(28) ν ( K + 1 ) p + 2 R K ( ψ , ν ) = ν 2 A [ D ω ( K + 1 ) p v ( ψ , ω ) ] .

Taking Eq. (28)’s absolute yields

(29) ∣ ν ( K + 1 ) p + 2 R K ( ψ , ν ) ∣ = ∣ ν 2 A [ D ω ( K + 1 ) p v ( ψ , ω ) ] ∣ .

Applying the conditions given in Eq. (29) results in the following conclusion:

(30) − T ν ( K + 1 ) p + 2 ≤ R K ( ψ , ν ) ≤ T ν ( K + 1 ) p + 2 .

Eq. (30) leads to the desired outcome

∣ R K ( ψ , ν ) ∣ ≤ T ν ( K + 1 ) p + 2 .

This establishes the convergence requirement for the new series.□

3 Methodologies

3.1 Implementation of the ARPSM

In order to address our general model, we propose the ARPSM fundamentals.

Step 1: Write the equation in standard form:

(31) D ω q p v ( ψ , ω ) + ϑ ( ψ ) N ( v ) − ψ ( ψ , v ) = 0 ,

Step 2: Eq. (31) may be solved by applying AT on both sides to obtain

(32) A [ D ω q p v ( ψ , ω ) + ϑ ( ψ ) N ( v ) − ψ ( ψ , v ) ] = 0 .

Lemma 2.5 will be used to transform Eq. (32):

(33) Ω ( ψ , s ) = ∑ j = 0 q − 1 D ω j v ( ψ , 0 ) s q p + 2 − ϑ ( ψ ) Y ( s ) s q p + F ( ψ , s ) s q p ,

where A [ ψ ( ψ , v ) ] = F ( ψ , s ) , A [ N ( v ) ] = Y ( s ) .

Step 3: To solve Eq. (33), take into consideration the following form:

Ω ( ψ , s ) = ∑ r = 0 ∞ ℏ r ( ψ ) s r p + 2 , s > 0 .

Step 4: Follow these steps:

ℏ 0 ( ψ ) = lim s → ∞ s 2 Ω ( ψ , s ) = v ( ψ , 0 ) .

The following outcome is achieved using Theorem 2.8:

ℏ 1 ( ψ ) = D ω p v ( ψ , 0 ) ,

ℏ 2 ( ψ ) = D ω 2 p v ( ψ , 0 ) ,

⋮

ℏ w ( ψ ) = D ω w p v ( ψ , 0 ) .

Step 5: Follow these procedures to obtain Ω ( ψ , s ) as the K th -truncated series:

Ω K ( ψ , s ) = ∑ r = 0 K ℏ r ( ψ ) s r p + 2 , s > 0 ,

Ω K ( ψ , s ) = ℏ 0 ( ψ ) s 2 + ℏ 1 ( ψ ) s p + 2 + ⋯ + ℏ w ( ψ ) s w p + 2 + ∑ r = w + 1 K ℏ r ( ψ ) s r p + 2 .

Step 6: ARF of Eq. (33) has to be evaluated separately from the K th -truncated ARF to obtain:

A Res ( ψ , s ) = Ω ( ψ , s ) − ∑ j = 0 q − 1 D ω j v ( ψ , 0 ) s j p + 2 + ϑ ( ψ ) Y ( s ) s j p − F ( ψ , s ) s j p

and

(34) A Res K ( ψ , s ) = Ω K ( ψ , s ) − ∑ j = 0 q − 1 D ω j v ( ψ , 0 ) s j p + 2 + ϑ ( ψ ) Y ( s ) s j p − F ( ψ , s ) s j p .

Step 7: In Eq. (34), substitute the expansion form of Ω K ( ψ , s ) .

(35) A Res K ( ψ , s ) = ℏ 0 ( ψ ) s 2 + ℏ 1 ( ψ ) s p + 2 + ⋯ + ℏ w ( ψ ) s w p + 2 + ∑ r = w + 1 K ℏ r ( ψ ) s r p + 2 − ∑ j = 0 q − 1 D ω j v ( ψ , 0 ) s j p + 2 + ϑ ( ψ ) Y ( s ) s j p − F ( ψ , s ) s j p .

Step 8: Multiply s K p + 2 on both sides of Eq. (35):

(36) s K p + 2 A Res K ( ψ , s ) = s K p + 2 ℏ 0 ( ψ ) s 2 + ℏ 1 ( ψ ) s p + 2 + ⋯ + ℏ w ( ψ ) s w p + 2 + ∑ r = w + 1 K ℏ r ( ψ ) s r p + 2 − ∑ j = 0 q − 1 D ω j v ( ψ , 0 ) s j p + 2 + ϑ ( ψ ) Y ( s ) s j p − F ( ψ , s ) s j p .

Step 9: Both sides of Eq. (36) are being evaluated with respect to lim s → ∞ :

lim s → ∞ s K p + 2 A Res K ( ψ , s ) = lim s → ∞ s K p + 2 ℏ 0 ( ψ ) s 2 + ℏ 1 ( ψ ) s p + 2 + ⋯ + ℏ w ( ψ ) s w p + 2 + ∑ r = w + 1 K ℏ r ( ψ ) s r p + 2 − ∑ j = 0 q − 1 D ω j v ( ψ , 0 ) s j p + 2 + ϑ ( ψ ) Y ( s ) s j p − F ( ψ , s ) s j p .

Step 10: We need to solve the following equation to obtain ℏ K ( ψ ) :

lim s → ∞ ( s K p + 2 A Res K ( ψ , s ) ) = 0 ,

where K = w + 1 , w + 2 , … .

Step 11: The K -truncated series of Ω ( ψ , s ) is used to derive the K -approximate solution for Eq. (33).

Step 12: The Ω K ( ψ , s ) is transformed using an inverse AT, and the K -approximate solution v K ( ψ , ω ) might be obtained.

3.1.1 ARPSM for anatomy fractional Hirota–Satsuma coupled KdV

Let us investigate the following fractional Hirota–Satsuma coupled KdV [24–27]:

(37) D ω p γ 1 − 1 2 ∂ ψ 3 γ 1 + 3 γ 1 ∂ ψ γ 1 − 3 ∂ ψ ( γ 2 γ 3 ) = 0 , D ω p γ 2 + ∂ ψ 3 γ 2 − 3 γ 1 ∂ ψ γ 2 = 0 , D ω p γ 3 + ∂ ψ 3 γ 3 − 3 γ 1 ∂ ψ γ 3 = 0 ,

which support the following exact solutions for the integer cases [27]:

(38) γ 1 = 1 3 ( β − 2 k 2 ) + 2 k 2 tanh 2 [ k ( ψ + β ω ) ] , γ 2 = − 4 k 2 c 0 ( β + k 2 ) 3 c 1 2 + 4 k 2 ( β + k 2 ) 3 c 1 tanh [ k ( ψ + β ω ) ] , γ 3 = c 0 + c 1 tanh [ k ( ψ + β ω ) ] ,

where γ i ≡ γ i ( ψ , ω ) for i = 1 , 2 , 3 , and 0 < p ≤ 1 . Here, β , k , and c 0 are arbitrary constants and c 1 ≠ 0 .

For simplicity, during our calculations, we can use ( β , k , c 0 , c 1 ) = 1 , 1 , 0 , 3 8 , which leads to

(39) γ 1 = 2 tanh 2 ( ψ + ω ) − 1 3 ,

(40) γ 2 = tanh ( ψ + ω ) ,

(41) γ 3 = 8 tanh ( ψ + ω ) 3 ,

and accordingly, the IC’s for problem (37) can be written in the following manner:

(42) γ 1 ( ψ , 0 ) ≡ γ 10 = 2 tanh 2 ( ψ ) − 1 3 , γ 2 ( ψ , 0 ) ≡ γ 20 = tanh ( ψ ) , γ 3 ( ψ , 0 ) ≡ γ 30 = 8 tanh ( ψ ) 3 .

Using Eq. (42) and applying AT on System (37), we obtain

(43) γ 1 ( ψ , s ) − γ 10 s 2 − 1 2 s p A ω ∂ 3 A ω − 1 γ 1 ( ψ , s ) ∂ ψ 3 + 3 s p A ω A ω − 1 γ 1 ( ψ , s ) × ∂ A ω − 1 γ 1 ( ψ , s ) ∂ ψ − 3 s p A ω A ω − 1 γ 2 ( ψ , s ) × ∂ A ω − 1 γ 3 ( ψ , s ) ∂ ψ − 3 s p A ω A ω − 1 γ 3 ( ψ , s ) × ∂ A ω − 1 γ 2 ( ψ , s ) ∂ ψ = 0 , γ 2 ( ψ , s ) − γ 20 s 2 + 1 s p A ω ∂ 3 A ω − 1 γ 2 ( ψ , s ) ∂ ψ 3 − 3 s p A ω A ω − 1 γ 1 ( ψ , s ) × ∂ A ω − 1 γ 2 ( ψ , s ) ∂ ψ = 0 , γ 3 ( ψ , s ) − γ 30 s 2 + 1 s p A ω ∂ 3 A ω − 1 γ 3 ( ψ , s ) ∂ ψ 3 − 3 s p A ω A ω − 1 γ 1 ( ψ , s ) × ∂ A ω − 1 γ 3 ( ψ , s ) ∂ ψ = 0 .

Thus, the term series that is k th truncated read

(44) γ 1 ( ψ , s ) = γ 10 s 2 + ∑ r = 1 k f r s r p + 1 , γ 2 ( ψ , s ) = γ 20 s 2 + ∑ r = 1 k g r s r p + 1 , γ 3 ( ψ , s ) = γ 30 s 2 + ∑ r = 1 k j r s r p + 1 ,

for r = 1 , 2 , 3 , 4 , … , k , where f r ≡ f r ( ψ , s ) , g r ≡ g r ( ψ , s ) , and j r ≡ j r ( ψ , s ) are the undetermined functions.

Aboodh residual functions (ARFs) read

(45) A ω Res ( ψ , s ) = γ 1 − γ 10 s 2 − 1 2 s p A ω ∂ 3 A ω − 1 γ 1 ∂ ψ 3 + 3 s p A ω A ω − 1 γ 1 × ∂ A ω − 1 γ 1 ∂ ψ − 3 s p A ω A ω − 1 γ 2 × ∂ A ω − 1 γ 3 ∂ ψ − 3 s p A ω A ω − 1 γ 3 × ∂ A ω − 1 γ 2 ∂ ψ = 0 , A ω Res ( ψ , s ) = γ 2 − γ 20 s 2 + 1 s p A ω ∂ 3 A ω − 1 γ 2 ∂ ψ 3 − 3 s p A ω A ω − 1 γ 1 × ∂ A ω − 1 γ 2 ∂ ψ = 0 , A ω Res ( ψ , s ) = γ 3 − γ 30 s 2 + 1 s p A ω ∂ 3 A ω − 1 γ 3 ∂ ψ 3 − 3 s p A ω A ω − 1 γ 1 × ∂ A ω − 1 γ 3 ∂ ψ = 0 ,

where γ i ≡ γ i ( ψ , s ) for i = 1 , 2 , 3 and the k th -LRFs read

(46) A ω Res k ( ψ , s ) = γ 1 k − γ 10 s 2 − 1 2 s p A ω ∂ 3 A ω − 1 γ 1 k ∂ ψ 3 + 3 s p A ω A ω − 1 γ 1 k × ∂ A ω − 1 γ 1 k ∂ ψ − 3 s p A ω A ω − 1 γ 2 k × ∂ A ω − 1 γ 3 k ∂ ψ − 3 s p A ω A ω − 1 γ 3 k × ∂ A ω − 1 γ 2 k ∂ ψ = 0 , A ω Res k ( ψ , s ) = γ 2 k − γ 20 s 2 + 1 s p A ω ∂ 3 A ω − 1 γ 2 k ∂ ψ 3 − 3 s p A ω A ω − 1 γ 1 k × ∂ A ω − 1 γ 2 k ∂ ψ = 0 , A ω Res k ( ψ , s ) = γ 3 k − γ 30 s 2 + 1 s p A ω ∂ 3 A ω − 1 γ 3 k ∂ ψ 3 − 3 s p A ω A ω − 1 γ 1 k × ∂ A ω − 1 γ 3 k ∂ ψ = 0 ,

where γ i k ≡ γ i k ( ψ , s ) for i = 1 , 2 , 3 .

To find the values of the functions f r ≡ f r ( ψ , s ) , g r ≡ g r ( ψ , s ) , and j r ≡ j r ( ψ , s ) , the following procedures are considered:

  1. Multiply the resulting equations by s r p + 1 ,

  2. Put the r th -truncated series from Eq. (44) into the r th -ARF represented by Eq. (46),

  3. Then, iteratively solve the relations lim s → ∞ ( s r p + 1 A ω Res γ 1 , r ( ψ , s ) ) = 0 , lim s → ∞ ( s r p + 1 A ω Res γ 2 , r ( ψ , s ) ) = 0 and lim s → ∞ ( s r p + 1 A ω Res γ 3 , r ( ψ , s ) ) = 0 for r = 1 , 2 , 3 , … ,

  4. Finally, the following first few terms are obtained:

    (47) f 1 ( ψ , s ) = 4 tanh ( ψ ) sech 2 ( ψ ) , g 1 ( ψ , s ) = sech 2 ( ψ ) , j 1 ( ψ , s ) = 8 3 sech 2 ( ψ ) ,

    and

    (48) f 2 ( ψ , s ) = 4 sech 4 ( ψ ) ( 2 − cosh ( 2 ψ ) ) , g 2 ( ψ , s ) = − 2 sech 2 ( ψ ) tanh ( ψ ) , j 2 ( ψ , s ) = − 16 3 sech 2 ( ψ ) tanh ( ψ ) ,

    and so on.

Putting the obtained values of f r ( ψ , s ) , g r ( ψ , s ) , and j r ( ψ , s ) for r = 1 , 2 , 3 , … , into Eq. (44), we finally obtain

(49) γ 1 ( ψ , s ) = γ 10 s 2 + 4 tanh ( ψ ) sech 2 ( ψ ) , s p + 1 + 4 sech 4 ( ψ ) ( 2 − cosh ( 2 ψ ) ) , s 2 p + 1 + ⋯ , γ 2 ( ψ , s ) = γ 20 s 2 + sech 2 ( ψ ) s p + 1 − 2 sech 2 ( ψ ) tanh ( ψ ) , s 2 p + 1 + ⋯ , γ 23 ( ψ , s ) = γ 30 s 2 + 8 sech 2 ( ψ ) 3 s p + 1 − 16 sech 2 ( ψ ) tanh ( ψ ) 3 s 2 p + 1 + ⋯ .

Applying the inverse AT on system (49) yields

(50) γ 1 ( ψ , ω ) = γ 10 + 4 ω p tanh ( ψ ) sech 2 ( ψ ) Γ ( p + 1 ) + 4 ω 2 p ( 2 − cosh ( 2 ψ ) ) sech 4 ( ψ ) Γ ( 2 p + 1 ) + ⋯ ,

(51) γ 2 ( ψ , ω ) = γ 20 + ω p sech 2 ( ψ ) Γ ( p + 1 ) − 2 ω 2 p sech 2 ( ψ ) tanh ( ψ ) , Γ ( 2 p + 1 ) + ⋯ ,

(52) γ 2 ( ψ , ω ) = γ 30 + 8 ω p sech 2 ( ψ ) 3 Γ ( p + 1 ) − 16 ω 2 p sech 2 ( ψ ) tanh ( ψ ) 3 Γ ( 2 p + 1 ) + ⋯ .

From now on, we use ( ψ , ω ) ≡ ( X , T ) only for simplicity in all graphics.

The convergence of ARPSM can be examined by presenting some numerical examples for the derived approximations. Accordingly, the three approximations (50)–(52) are analyzed numerically and graphically, as illustrated in Tables 1, 2, 3. We also calculated the absolute error of the three approximations compared to the exact solutions for the integer cases, as shown in Tables 1–3 in addition to Figures 1, 2, 3. Figure 1 displays the absolute error of the approximation (50) as compared to the exact solution (39), while Table 1 also showcases the outcomes of the approximation (50) analysis. Figure 2 shows the absolute error of the approximation (51) as compared to the exact solution (40), while Table 2 displays the findings of the analysis of the approximation (51). Moreover, Figure 3 indicates the absolute error of the approximation (52) as compared to the exact solution (41), while Table 3 displays the numerical analysis of the approximation (52) against the fractional parameter. It is clear from the obtained results that there is no difference between the approximate and exact solutions for the integer cases, meaning that there is complete harmony between these solutions, which enhances the accuracy of the deduced approximations and the efficiency of the used method. It is worth noting that we arrived at this result by obtaining a few numbers of terms from the solution series. Thus, the accuracy of these approximations can be increased by deriving a larger number of terms in the solution series. This result confirms the convergence and the efficiency of the ARPSM in analyzing more complicated nonlinear evolution equations.

Table 1

Approximation (50) is numerically analyzed against the fractional parameter ( p ) and compared with the exact solution (39) for the integer case at ω = 0.01

ψ ARPSM p = 0.4 ARPSM p = 0.8 ARPSM P = 1.0 Exact Error p = 1.0
‒ 5 1.66613 1.66628 1.6663 1.6663 4.86341411587432 × 1 0 ‒ 10
‒ 4 1.66274 1.66384 1.66393 1.66393 3.576864582299777 × 1 0 ‒ 9
‒ 3 1.63782 1.64584 1.64654 1.64654 2.55268248761098 × 1 0 ‒ 8
‒ 2 1.46307 1.51779 1.52262 1.52262 1.436073706617691 × 1 0 ‒ 7
‒ 1 0.564653 0.791665 0.813862 0.813862 2.2635043139818123 × 1 0 ‒ 7
0 ‒ 0.225456 ‒ 0.331568 ‒ 0.333133 ‒ 0.333133 1.333257787505815 × 1 0 ‒ 8
1 1.02172 0.860674 0.83945 0.83945 2.1708901010164539 × 1 0 ‒ 7
2 1.5604 1.53249 1.52806 1.52806 1.4265114334222062 × 1 0 ‒ 7
3 1.65185 1.64796 1.64732 1.64732 2.528300391269056 × 1 0 ‒ 8
4 1.66465 1.66412 1.66404 1.66404 3.541464455025789 × 1 0 ‒ 9
5 1.66639 1.66632 1.66631 1.66631 4.815063903151895 × 1 0 ‒ 10
Table 2

Approximation (51) is numerically analyzed against the fractional parameter ( p ) and compared with the exact solution (40) for the integer case at ω = 0.01

ψ ARPSM p = 0.4 ARPSM p = 0.8 ARPSM P = 1.0 Exact Error p = 1.0
‒ 5 ‒ 0.999867 ‒ 0.999904 ‒ 0.999907 ‒ 0.999907 1.2162981732899425 × 1 0 ‒ 10
‒ 4 ‒ 0.999017 ‒ 0.999292 ‒ 0.999316 ‒ 0.999316 8.966356546125098 × 1 0 ‒ 10
‒ 3 ‒ 0.992763 ‒ 0.99478 ‒ 0.994955 ‒ 0.994955 6.511897243655085 × 1 0 ‒ 9
‒ 2 ‒ 0.947734 ‒ 0.962062 ‒ 0.963314 ‒ 0.963314 4.2288400559087336 × 1 0 ‒ 8
‒ 1 ‒ 0.669323 ‒ 0.749985 ‒ 0.757362 ‒ 0.757362 1.033226758595518 × 1 0 ‒ 7
0 0.178627 0.0269694 0.01 0.00999967 3.3332000053988564 × 1 0 ‒ 7
1 0.819361 0.772638 0.765762 0.765762 1.0387695625890814 × 1 0 ‒ 7
2 0.972974 0.965873 0.964727 0.964727 4.193057745460749 × 1 0 ‒ 8
3 0.996288 0.995312 0.995152 0.995152 6.448385381219168 × 1 0 ‒ 9
4 0.999497 0.999364 0.999343 0.999343 8.87737661159349 × 1 0 ‒ 10
5 0.999932 0.999914 0.999911 0.999911 1.2042011832136268 × 1 0 ‒ 10
Table 3

Approximation (52) is numerically analyzed against the fractional parameter ( p ) and compared with the exact solution (41) for the integer case at ω = 0.01

ψ ARPSM p = 0.4 ARPSM p = 0.8 ARPSM P = 1.0 Exact Error p = 1.0
‒ 5 ‒ 2.66631 ‒ 2.66641 ‒ 2.66642 ‒ 2.66642 3.2434588348451143 × 1 0 ‒ 10
‒ 4 ‒ 2.66405 ‒ 2.66478 ‒ 2.66484 ‒ 2.66484 2.3910282642702896 × 1 0 ‒ 9
‒ 3 ‒ 2.64737 ‒ 2.65275 ‒ 2.65321 ‒ 2.65321 1.7365059168383823 × 1 0 ‒ 8
‒ 2 ‒ 2.52729 ‒ 2.5655 ‒ 2.56884 ‒ 2.56884 1.1276906830559597 × 1 0 ‒ 7
‒ 1 ‒ 1.78486 ‒ 1.99996 ‒ 2.01963 ‒ 2.01963 2.755271357735012 × 1 0 ‒ 7
0 0.476339 0.0719184 0.0266667 0.0266658 8.888533347718719 × 1 0 ‒ 7
1 2.18496 2.06037 2.04203 2.04203 2.7700521654239196 × 1 0 ‒ 7
2 2.5946 2.57566 2.57261 2.57261 1.1181487291622716 × 1 0 ‒ 7
3 2.65677 2.65417 2.65374 2.65374 1.7195694201888045 × 1 0 ‒ 8
4 2.66532 2.66497 2.66491 2.66491 2.3673005777880007 × 1 0 ‒ 9
5 2.66648 2.66644 2.66643 2.66643 3.2112046355337043 × 1 0 ‒ 10
Figure 1 Absolute error R ∞ {R}_{\infty } for the approximation (50) as compared to the exact solution (39) is estimated: (a) R ∞ {R}_{\infty } in ( X , T ) (X,T) -plane and (b) R ∞ {R}_{\infty } in against X X for T = 0.01 . T=0.01.
Figure 1

Absolute error R ∞ for the approximation (50) as compared to the exact solution (39) is estimated: (a) R ∞ in ( X , T ) -plane and (b) R ∞ in against X for T = 0.01 .

Figure 2 Absolute error R ∞ {R}_{\infty } for the approximation (51) as compared to the exact solution (40) is estimated: (a) R ∞ {R}_{\infty } in ( X , T ) (X,T) -plane and (b) R ∞ {R}_{\infty } in against X X for T = 0.01 T=0.01 .
Figure 2

Absolute error R ∞ for the approximation (51) as compared to the exact solution (40) is estimated: (a) R ∞ in ( X , T ) -plane and (b) R ∞ in against X for T = 0.01 .

Figure 3 Absolute error R ∞ {R}_{\infty } for the approximation (52) as compared to the exact solution (41) is estimated: (a) R ∞ {R}_{\infty } in ( X , T ) ‒ (X,T)&#x2012; plane and (b) R ∞ {R}_{\infty } in against X X for T = 0.01 . T=0.01.
Figure 3

Absolute error R ∞ for the approximation (52) as compared to the exact solution (41) is estimated: (a) R ∞ in ( X , T ) ‒ plane and (b) R ∞ in against X for T = 0.01 .

In addition, we studied the effect of the fractional parameter on the profile of both soliton and shock waves described by the derived approximations. It is clear from the analysis results that the fractional parameter significantly affects the profile of both soliton and shock waves, as is evident in Figures 4, 5, 6, for the approximations (50), (51), and (52), respectively.

Figure 4 Approximation (50) is plotted against the fractional parameter p p in (a) ( X , T ) (X,T) -plane for p = 0.2 p=0.2 , (b) ( X , T ) (X,T) -plane for p = 0.6 p=0.6 , (c) ( X , T ) (X,T) -plane for p = 1 p=1 , and (d) two-dimensional for T = 0.1 . T=0.1.
Figure 4

Approximation (50) is plotted against the fractional parameter p in (a) ( X , T ) -plane for p = 0.2 , (b) ( X , T ) -plane for p = 0.6 , (c) ( X , T ) -plane for p = 1 , and (d) two-dimensional for T = 0.1 .

Figure 5 Approximation (51) is plotted against the fractional parameter p p in (a) ( X , T ) (X,T) -plane for p = 0.2 p=0.2 , (b) ( X , T ) (X,T) -plane for p = 0.6 p=0.6 , (c) ( X , T ) (X,T) -plane for p = 1 p=1 , and (d) two-dimensional for T = 0.1 . T=0.1.
Figure 5

Approximation (51) is plotted against the fractional parameter p in (a) ( X , T ) -plane for p = 0.2 , (b) ( X , T ) -plane for p = 0.6 , (c) ( X , T ) -plane for p = 1 , and (d) two-dimensional for T = 0.1 .

Figure 6 Approximation (52) is plotted against the fractional parameter p p in (a) ( X , T ) (X,T) -plane for p = 0.2 p=0.2 , (b) ( X , T ) (X,T) -plane for p = 0.6 p=0.6 , (c) ( X , T ) (X,T) -plane for p = 1 p=1 , and (d) two-dimensional for T = 0.1 . T=0.1.
Figure 6

Approximation (52) is plotted against the fractional parameter p in (a) ( X , T ) -plane for p = 0.2 , (b) ( X , T ) -plane for p = 0.6 , (c) ( X , T ) -plane for p = 1 , and (d) two-dimensional for T = 0.1 .

3.2 ATIM

The general fractional PDE for space and time is given as

(53) D ω p v = Φ ( v , D ψ v v , D ψ 2 v v , D ψ 3 v v ) , 0 < p , v ≤ 1 ,

with IC’s

(54) v ( k ) ( ψ , 0 ) = h k , k = 0 , 1 , 2 , … , m − 1 ,

where v ≡ v ( ψ , ω ) is the unknown function to be determined later and Φ [ v , D ψ v v , D ψ 2 v v , D ψ 3 v v ] can be linear or nonlinear operator of v , D ψ v v , D ψ 2 v v , and D ψ 3 v v . Consequently, we obtain the following equation by applying the AT to both sides of Eq. (53):

(55) A [ v ] = 1 s p ∑ k = 0 m − 1 v ( k ) ( ψ , 0 ) s 2 − p + k + A [ Φ ( v , D ψ v v , D ψ 2 v v , D ψ 3 v v ) ] .

Then, by applying the inverse AT, the following equation is obtained:

(56) v ( ψ , ω ) = A − 1 1 s p ∑ k = 0 m − 1 v ( k ) ( ψ , 0 ) s 2 − p + k + A [ Φ ( v , D ψ v v , D ψ 2 v v , D ψ 3 v v ) ] } ] .

By applying the AT iteratively, the solution in an infinite series is obtained as follows:

(57) v ( ψ , ω ) = ∑ i = 0 ∞ v i ( ψ , ω ) ,

where Φ ( v , D ψ v v , D ψ 2 v v , D ψ 3 v v ) is a linear or nonlinear operator that can be broken down into the following form:

(58) Φ ( v , D ψ v v , D ψ 2 v v , D ψ 3 v v ) = Φ ( v 0 , D ψ v v 0 , D ψ 2 v v 0 , D ψ 3 v v 0 ) + ∑ i = 0 ∞ Φ ∑ k = 0 i ( v k , D ψ v v k , D ψ 2 v v k , D ψ 3 v v k ) − Φ ∑ k = 1 i − 1 ( v k , D ψ v v k , D ψ 2 v v k , D ψ 3 v v k ) .

After inserting Eqs (58) and (57) into Eq. (56), we have

(59) ∑ i = 0 ∞ v i ( ψ , ω ) = A − 1 1 s p ∑ k = 0 m − 1 v ( k ) ( ψ , 0 ) s 2 − p + k + A [ Φ ( v 0 , D ψ v v 0 , D ψ 2 v v 0 , D ψ 3 v v 0 ) ] ) ] + A − 1 1 s p A ∑ i = 0 ∞ Φ ∑ k = 0 i ( v k , D ψ v v k , D ψ 2 v v k , D ψ 3 v v k ) − A − 1 1 s p A Φ ∑ k = 1 i − 1 ( v k , D ψ v v k , D ψ 2 v v k , D ψ 3 v v k ) ,

with

(60) v 0 ( ψ , ω ) = A − 1 1 s p ∑ k = 0 m − 1 v ( k ) ( ψ , 0 ) s 2 − p + k , v 1 ( ψ , ω ) = A − 1 1 s p ( A [ Φ ( v 0 , D ψ v v 0 , D ψ 2 v v 0 , D ψ 3 v v 0 ) ] ) , ⋮ v m + 1 ( ψ , ω ) = A − 1 1 s p A ∑ i = 0 ∞ Φ ∑ k = 0 i ( v k , D ψ v v k , D ψ 2 v v k , D ψ 3 v v k ) − A − 1 1 s p A Φ ∑ k = 1 i − 1 ( v k , D ψ v v k , D ψ 2 v v k , D ψ 3 v v k ) , m = 1 , 2 , … .

Eq. (53) yields the approximate solution for the m -term, which may be found analytically as follows:

(61) v ( ψ , ω ) = ∑ i = 0 m − 1 v i ( ψ , ω ) .

3.2.1 ATIM for anatomy fractional Hirota–Satsuma coupled KdV

We apply ATIM to analyze fractional Hirota–Satsuma coupled KdV problem (37) in this section. To avoid repeating the fractional Hirota–Satsuma coupled KdV problem (37) more than once, we only here refer to its number (37) with its ICs (42) as mentioned earlier.

Now, applying the AT to both sides of problem (37) yields the following equations:

(62) A [ D ω p γ 1 ] = 1 s p ∑ k = 0 m − 1 γ 1 ( k ) ( ψ , 0 ) s 2 − p + k + A 1 2 ∂ ψ 3 γ 1 − 3 γ 1 ∂ ψ γ 1 + 3 ∂ ψ ( γ 2 γ 3 ) , A [ D ω p γ 2 ] = 1 s p ∑ k = 0 m − 1 γ 2 ( k ) ( ψ , 0 ) s 2 − p + k + A [ − ∂ ψ 3 γ 2 + 3 γ 1 ∂ ψ γ 2 ] , A [ D ω p γ 3 ] = 1 s p ∑ k = 0 m − 1 γ 3 ( k ) ( ψ , 0 ) s 2 − p + k + A [ − ∂ ψ 3 γ 3 + 3 γ 1 ∂ ψ γ 3 ] ,

where γ i ≡ γ i ( ψ , ω ) for i = 1 , 2 , 3 .

The following equations are derived by applying the inverse AT to Eq. (62):

(63) γ 1 = A − 1 1 s p ∑ k = 0 m − 1 γ 1 ( k ) ( ψ , 0 ) s 2 − p + k + A 1 2 ∂ ψ 3 γ 1 − 3 γ 1 ∂ ψ γ 1 + 3 ∂ ψ ( γ 2 γ 3 ) , γ 2 = A − 1 1 s p ∑ k = 0 m − 1 γ 2 ( k ) ( ψ , 0 ) s 2 − p + k + A [ − ∂ ψ 3 γ 2 + 3 γ 1 ∂ ψ γ 2 ] , γ 3 = A − 1 1 s p ∑ k = 0 m − 1 γ 3 ( k ) ( ψ , 0 ) s 2 − p + k + A [ − ∂ ψ 3 γ 3 + 3 γ 1 ∂ ψ γ 3 ] .

The following values for γ 10 ≡ γ 10 ( ψ , ω ) , γ 20 ≡ γ 20 ( ψ , ω ) , and γ 30 ≡ γ 30 ( ψ , ω ) are obtained using the AT iteratively:

(64) γ 10 = A − 1 1 s p ∑ k = 0 m − 1 γ 1 ( k ) ( ψ , 0 ) s 2 − p + k = A − 1 γ 1 ( ψ , 0 ) s 2 = 2 tanh 2 ( ψ ) − 1 3 , γ 20 = A − 1 1 s p ∑ k = 0 m − 1 γ 2 ( k ) ( ψ , 0 ) s 2 − p + k = A − 1 γ 2 ( ψ , 0 ) s 2 = tanh ( ψ ) , γ 30 = A − 1 1 s p ∑ k = 0 m − 1 γ 3 ( k ) ( ψ , 0 ) s 2 − p + k = A − 1 γ 3 ( ψ , 0 ) s 2 = 8 tanh ( ψ ) 3 .

Applying the RL integral on problem (37) yields the following equivalent form:

(65) γ 1 = γ 10 + A 1 2 ∂ ψ 3 γ 1 − 3 γ 1 ∂ ψ γ 1 + 3 ∂ ψ ( γ 2 γ 3 ) , γ 2 = γ 20 + A [ − ∂ ψ 3 γ 2 + 3 γ 1 ∂ ψ γ 2 ] , γ 3 = γ 30 + A [ − ∂ ψ 3 γ 3 + 3 γ 1 ∂ ψ γ 3 ] ,

where γ i ≡ γ i ( ψ , ω ) for i = 1 , 2 , 3 .

The terms that derived using the ATIM read:

Zeroth-order approximation

(66) γ 10 ( ψ , ω ) = 2 tanh 2 ( ψ ) − 1 3 , γ 20 ( ψ , ω ) = tanh ( ψ ) , γ 30 ( ψ , ω ) = 8 tanh ( ψ ) 3 ,

first-order approximation

(67) γ 11 ( ψ , ω ) = 4 ω p tanh ( ψ ) sech 2 ( ψ ) Γ ( p + 1 ) , γ 21 ( ψ , ω ) = ω p sech 2 ( ψ ) Γ ( p + 1 ) , γ 31 ( ψ , ω ) = 8 ω p sech 2 ( ψ ) 3 Γ ( p + 1 ) ,

and second-order approximation

(68) γ 12 ( ψ , ω ) = 4 ω 2 p sech 4 ( x ) [ 4 ω p ( sinh ( 3 ψ ) − 8 sinh ( ψ ) ) sech 3 ( x ) + Γ ( p + 1 ) ( 2 − cosh ( 2 ψ ) ) ] Γ ( p + 1 ) 3 , γ 22 ( ψ , ω ) = − 2 ω 2 p tanh ( ψ ) sech 2 ( ψ ) ( 12 ω p tanh ( ψ ) sech 2 ( ψ ) + Γ ( p + 1 ) ) Γ ( p + 1 ) 3 , γ 32 ( ψ , ω ) = − 16 ω 2 p tanh ( ψ ) sech 2 ( ψ ) ( 12 ω p tanh ( ψ ) sech 2 ( ψ ) + Γ ( p + 1 ) ) 3 Γ ( p + 1 ) 3 .

By collecting the values of zeroth-, first-, and second-order approximations given in Eqs (66)–(68), we finally obtain

(69) γ 1 ( ψ , ω ) = γ 10 ( ψ , ω ) + γ 11 ( ψ , ω ) + γ 12 ( ψ , ω ) + ⋯ , = − 1 3 + 2 tanh 2 ( ψ ) + 4 ω p tanh ( ψ ) sech 2 ( ψ ) Γ ( p + 1 ) + 4 ω 2 p sech 4 ( x ) [ 4 ω p ( sinh ( 3 ψ ) − 8 sinh ( ψ ) ) sech 3 ( x ) + Γ ( p + 1 ) ( 2 − cosh ( 2 ψ ) ) ] Γ ( p + 1 ) 3 ,

(70) γ 2 ( ψ , ω ) = γ 20 ( ψ , ω ) + γ 21 ( ψ , ω ) + γ 22 ( ψ , ω ) + ⋯ , = tanh ( ψ ) + ω p sech 2 ( ψ ) Γ ( p + 1 ) − 2 ω 2 p tanh ( ψ ) sech 2 ( ψ ) ( 12 ω p tanh ( ψ ) sech 2 ( ψ ) + Γ ( p + 1 ) ) Γ ( p + 1 ) 3 ,

and

(71) γ 3 ( ψ , ω ) = γ 30 ( ψ , ω ) + γ 31 ( ψ , ω ) + γ 32 ( ψ , ω ) + ⋯ , = 8 tanh ( ψ ) 3 + 8 ω p sech 2 ( ψ ) 3 Γ ( p + 1 ) − 16 t 2 p tanh ( ψ ) sech 2 ( ψ ) ( 12 ω p tanh ( ψ ) sech 2 ( ψ ) + Γ ( p + 1 ) ) 3 Γ ( p + 1 ) 3 .

ATIM convergence can be analyzed by providing some numerical examples of the obtained approximations, depending on the variables involved. Consequently, the three approximations (69)–(70) are investigated numerically and graphically, as shown in Tables 4, 5, 6. Also, the absolute errors of the three approximations compared to the exact solutions for the integer cases are estimated as presented in Tables 4–6 and Figures 7, 8, 9. Both Figure 7 and Table 4 show the absolute error between approximation (69) and the exact solution (39) for the integer case. Also, Figure 8 and Table 5 illustrate the absolute error between approximation (70) and the exact solution (40) for the integer case. For the integer case, Figure 9 and Table 6 show a comparison of the approximation (71) and the exact solution (41). The results show that there is no difference between the exact and approximate solutions for integer cases. This means that there is a high level of agreement between these solutions, making the method more accurate and useful. Moreover, the impact of fractional parameter p on the profile of the approximations (69)–(71) is investigated as shown in Figures Figures 10, 11, 12, respectively. We reached this conclusion by extracting a limited number of terms from the solution series. Increasing the number of terms in the solution series can enhance the accuracy of these approximations. This outcome validates the convergence and effectiveness of the ATIM in studying complicated and strong nonlinear evolution equations.

Table 4

Approximation (69) is numerically analyzed against the fractional parameter ( p ) and compared with the exact solution (39) for the integer case at ω = 0.01

ψ ATIM p = 0.4 ATIM p = 0.8 ATIM P = 1.0 Exact Error p = 1.0
‒ 5 1.66613 1.66628 1.6663 1.6663 7.212927655153578 × 1 0 ‒ 8
‒ 4 1.66269 1.66383 1.66393 1.66393 5.318391766984121 × 1 0 ‒ 7
‒ 3 1.6374 1.64582 1.64653 1.64654 3.868546090357583 × 1 0 ‒ 6
‒ 2 1.4591 1.51764 1.52259 1.52262 0.0000253808
‒ 1 0.555815 0.7913 0.813799 0.813862 0.0000628624
0 ‒ 0.205703 ‒ 0.330424 ‒ 0.332933 ‒ 0.333133 0.000200013
1 1.01828 0.860328 0.839388 0.83945 0.0000614722
2 1.55938 1.53235 1.52804 1.52806 0.000025149
3 1.6515 1.64794 1.64732 1.64732 3.907233393718457 × 1 0 ‒ 6
4 1.6646 1.66412 1.66404 1.66404 5.387281909463582 × 1 0 ‒ 7
5 1.66639 1.66632 1.66631 1.66631 7.309290617207864 × 1 0 ‒ 8
Table 5

Approximation (70) is numerically analyzed against the fractional parameter ( p ) and compared with the exact solution (40) for the integer case at ω = 0.01

ψ ATIM p = 0.4 ATIM p = 0.8 ATIM P = 1.0 Exact Error p = 1.0
‒ 5 ‒ 0.999865 ‒ 0.999904 ‒ 0.999907 1.6663 1.80342534239486 × 1 0 ‒ 8
‒ 4 ‒ 0.999004 ‒ 0.999291 ‒ 0.999316 1.66393 1.3306539736746004 × 1 0 ‒ 7
‒ 3 ‒ 0.992679 ‒ 0.994774 ‒ 0.994954 1.64654 9.728997406588391 × 1 0 ‒ 7
‒ 2 ‒ 0.947696 ‒ 0.962025 ‒ 0.963308 1.52262 6.657312786040315 × 1 0 ‒ 6
‒ 1 ‒ 0.680158 ‒ 0.749851 ‒ 0.757333 0.813862 0.0000294264
0 0.178627 0.0269694 0.01 ‒ 0.333133 3.3332000053988564 × 1 0 ‒ 7
1 0.802208 0.772407 0.765727 0.83945 0.0000345442
2 0.971667 0.965832 0.96472 1.52806 6.964198133618815 × 1 0 ‒ 6
3 0.996177 0.995307 0.995151 1.64732 9.90486183738426 × 1 0 ‒ 7
4 0.999483 0.999364 0.999342 1.66404 1.3493596606739544 × 1 0 ‒ 7
5 0.99993 0.999914 0.999911 1.66631 1.827788564945365 × 1 0 ‒ 8
Table 6

Approximation (71) is numerically analyzed against the fractional parameter ( p ) and compared with the exact solution (41) for the integer case at ω = 0.01

ψ ATIM p = 0.4 ATIM p = 0.8 ATIM P = 1.0 Exact Error p = 1.0
‒ 5 ‒ 2.66631 ‒ 2.66641 ‒ 2.66642 ‒ 2.66642 4.809134246386293 × 1 0 ‒ 8
‒ 4 ‒ 2.66401 ‒ 2.66478 ‒ 2.66484 ‒ 2.66484 3.548410596465601 × 1 0 ‒ 7
‒ 3 ‒ 2.64714 ‒ 2.65273 ‒ 2.65321 ‒ 2.65321 2.5943993087196304 × 1 0 ‒ 6
‒ 2 ‒ 2.52719 ‒ 2.5654 ‒ 2.56882 ‒ 2.56884 0.0000177528
‒ 1 ‒ 1.81376 ‒ 1.9996 ‒ 2.01955 ‒ 2.01963 0.0000784703
0 0.476339 0.0719184 0.0266667 0.0266658 8.888533347718719 × 1 0 ‒ 7
1 2.13922 2.05975 2.04194 2.04203 0.0000921178
2 2.59111 2.57555 2.57259 2.57261 0.0000185712
3 2.65647 2.65415 2.65374 2.65374 2.641296489969136 × 1 0 ‒ 6
4 2.66529 2.66497 2.66491 2.66491 3.5982924329047705 × 1 0 ‒ 7
5 2.66648 2.66644 2.66643 2.66643 4.8741028546572807 × 1 0 ‒ 8
Figure 7 Absolute error R ∞ {R}_{\infty } for the approximation (69) as compared to the exact solution (39) is estimated: (a) R ∞ {R}_{\infty } in ( X , T ) (X,T) -plane and (b) R ∞ {R}_{\infty } in against X X for T = 0.01 . T=0.01.
Figure 7

Absolute error R ∞ for the approximation (69) as compared to the exact solution (39) is estimated: (a) R ∞ in ( X , T ) -plane and (b) R ∞ in against X for T = 0.01 .

Figure 8 Absolute error R ∞ {R}_{\infty } for the approximation (70) as compared to the exact solution (40) is estimated: (a) R ∞ {R}_{\infty } in ( X , T ) (X,T) -plane and (b) R ∞ {R}_{\infty } in against X X for T = 0.01 . T=0.01.
Figure 8

Absolute error R ∞ for the approximation (70) as compared to the exact solution (40) is estimated: (a) R ∞ in ( X , T ) -plane and (b) R ∞ in against X for T = 0.01 .

Figure 9 Absolute error R ∞ {R}_{\infty } for the approximation (71) as compared to the exact solution (41) is estimated: (a) R ∞ {R}_{\infty } in ( X , T ) (X,T) -plane and (b) R ∞ {R}_{\infty } in against X X for T = 0.01 . T=0.01.
Figure 9

Absolute error R ∞ for the approximation (71) as compared to the exact solution (41) is estimated: (a) R ∞ in ( X , T ) -plane and (b) R ∞ in against X for T = 0.01 .

We also numerically compared all derived approximations using ARPSM and ATIM, as shown in Tables 7, 8, 9. In this comparison, we estimated the absolute errors of all derived approximations using ARPSM compared to the exact solutions for integer cases. We also calculated the absolute errors of all derived approximations using ATIM compared to the exact solutions for integer cases. The comparison results show that both methods exhibit high accuracy, convergence, and stability, but ARPSM-derived approximations are more accurate than ATIM-derived ones.

Table 7

Absolute error for approximation (50) using ARPSM and approximation (69) using ARIM for the integer case are compared to the exact solution (39) at ω = 0.01

ψ ARPSM ATIM Exact R ∞ for ARPSM R ∞ for ATIM
‒ 5 1.6663 1.6663 1.6663 4.86341411587432 × 1 0 ‒ 10 7.212927655153578 × 1 0 ‒ 8
‒ 4 1.66393 1.66393 1.66393 3.576864582299777 × 1 0 ‒ 9 5.318391766984121 × 1 0 ‒ 7
‒ 3 1.64654 1.64653 1.64654 2.55268248761098 × 1 0 ‒ 8 3.868546090357583 × 1 0 ‒ 6
‒ 2 1.52262 1.52259 1.52262 1.436073706617691 × 1 0 ‒ 7 0.0000253808
‒ 1 0.813862 0.813799 0.813862 2.2635043139818123 × 1 0 ‒ 7 0.0000628624
0 ‒ 0.333133 ‒ 0.332933 ‒ 0.333133 1.333257787505815 × 1 0 ‒ 8 0.000200013
1 0.83945 0.839388 0.83945 2.1708901010164539 × 1 0 ‒ 7 0.0000614722
2 1.52806 1.52804 1.52806 1.4265114334222062 × 1 0 ‒ 7 0.000025149
3 1.64732 1.64732 1.64732 2.528300391269056 × 1 0 ‒ 8 3.907233393718457 × 1 0 ‒ 6
4 1.66404 1.66404 1.66404 3.541464455025789 × 1 0 ‒ 9 5.387281909463582 × 1 0 ‒ 7
5 1.66631 1.66631 1.66631 4.815063903151895 × 1 0 ‒ 10 7.309290617207864 × 1 0 ‒ 8
Table 8

Absolute error for approximation (51) using ARPSM and approximation (70) using ARIM for the integer case are compared to the exact solution (40) at ω = 0.01

ψ ARPSM ATIM Exact R ∞ for ARPSM R ∞ for ATIM
‒ 5 ‒ 0.999907 ‒ 0.999907 ‒ 0.999907 1.2162981732899425 × 1 0 ‒ 10 1.80342534239486 × 1 0 ‒ 8
‒ 4 ‒ 0.999316 ‒ 0.999316 ‒ 0.999316 8.966356546125098 × 1 0 ‒ 10 1.3306539736746004 × 1 0 ‒ 7
‒ 3 ‒ 0.994955 ‒ 0.994954 ‒ 0.994955 6.511897243655085 × 1 0 ‒ 9 9.728997406588391 × 1 0 ‒ 7
‒ 2 ‒ 0.963314 ‒ 0.963308 ‒ 0.963314 4.2288400559087336 × 1 0 ‒ 8 6.657312786040315 × 1 0 ‒ 6
‒ 1 ‒ 0.757362 ‒ 0.757333 ‒ 0.757362 1.033226758595518 × 1 0 ‒ 7 0.0000294264
0 0.01 0.01 0.00999967 3.3332000053988564 × 1 0 ‒ 7 3.3332000053988564 × 1 0 ‒ 7
1 0.765762 0.765727 0.765762 1.0387695625890814 × 1 0 ‒ 7 0.0000345442
2 0.964727 0.96472 0.964727 4.193057745460749 × 1 0 ‒ 8 6.964198133618815 × 1 0 ‒ 6
3 0.995152 0.995151 0.995152 6.448385381219168 × 1 0 ‒ 9 9.90486183738426 × 1 0 ‒ 7
4 0.999343 0.999342 0.999343 8.87737661159349 × 1 0 ‒ 10 1.3493596606739544 × 1 0 ‒ 7
5 0.999911 0.999911 0.999911 1.2042011832136268 × 1 0 ‒ 10 1.827788564945365 × 1 0 ‒ 8
Table 9

Absolute error for approximation (52) using ARPSM and approximation (71) using ARIM for the integer case are compared to the exact solution (41) at ω = 0.01

ψ ARPSM ATIM Exact R ∞ for ARPSM R ∞ for ATIM
‒ 5 ‒ 2.66642 ‒ 2.66642 ‒ 2.66642 3.2434588348451143 × 1 0 ‒ 10 4.809134246386293 × 1 0 ‒ 8
‒ 4 ‒ 2.66484 ‒ 2.66484 ‒ 2.66484 2.3910282642702896 × 1 0 ‒ 9 3.548410596465601 × 1 0 ‒ 7
‒ 3 ‒ 2.65321 ‒ 2.65321 ‒ 2.65321 1.7365059168383823 × 1 0 ‒ 8 2.5943993087196304 × 1 0 ‒ 6
‒ 2 ‒ 2.56884 ‒ 2.56882 ‒ 2.56884 1.1276906830559597 × 1 0 ‒ 7 0.0000177528
‒ 1 ‒ 2.01963 ‒ 2.01955 ‒ 2.01963 2.755271357735012 × 1 0 ‒ 7 0.0000784703
0 0.0266667 0.0266667 0.0266658 8.888533347718719 × 1 0 ‒ 7 8.888533347718719 × 1 0 ‒ 7
1 2.04203 2.04194 2.04203 2.7700521654239196 × 1 0 ‒ 7 0.0000921178
2 2.57261 2.57259 2.57261 1.1181487291622716 × 1 0 ‒ 7 0.0000185712
3 2.65374 2.65374 2.65374 1.7195694201888045 × 1 0 ‒ 8 2.641296489969136 × 1 0 ‒ 6
4 2.66491 2.66491 2.66491 2.3673005777880007 × 1 0 ‒ 9 3.5982924329047705 × 1 0 ‒ 7
5 2.66643 2.66643 2.66643 3.2112046355337043 × 1 0 ‒ 10 4.8741028546572807 × 1 0 ‒ 8
Figure 10 Approximation (69) is plotted against the fractional parameter p p in (a) ( X , T ) (X,T) -plane for p = 0.2 p=0.2 , (b) ( X , T ) (X,T) -plane for p = 0.6 p=0.6 , (c) ( X , T ) (X,T) -plane for p = 1 p=1 , and (d) two-dimensional for T = 0.1 . T=0.1.
Figure 10

Approximation (69) is plotted against the fractional parameter p in (a) ( X , T ) -plane for p = 0.2 , (b) ( X , T ) -plane for p = 0.6 , (c) ( X , T ) -plane for p = 1 , and (d) two-dimensional for T = 0.1 .

Figure 11 Approximation (70) is plotted against the fractional parameter p p in (a) ( X , T ) (X,T) -plane for p = 0.2 p=0.2 , (b) ( X , T ) (X,T) -plane for p = 0.6 p=0.6 , (c) ( X , T ) (X,T) -plane for p = 1 p=1 , and (d) two-dimensional for T = 0.1 . T=0.1.
Figure 11

Approximation (70) is plotted against the fractional parameter p in (a) ( X , T ) -plane for p = 0.2 , (b) ( X , T ) -plane for p = 0.6 , (c) ( X , T ) -plane for p = 1 , and (d) two-dimensional for T = 0.1 .

Figure 12 Approximation (71) is plotted against the fractional parameter p p in (a) ( X , T ) (X,T) -plane for p = 0.2 p=0.2 , (b) ( X , T ) (X,T) -plane for p = 0.6 p=0.6 , (c) ( X , T ) (X,T) -plane for p = 1 p=1 , and (d) two-dimensional for T = 0.1 . T=0.1.
Figure 12

Approximation (71) is plotted against the fractional parameter p in (a) ( X , T ) -plane for p = 0.2 , (b) ( X , T ) -plane for p = 0.6 , (c) ( X , T ) -plane for p = 1 , and (d) two-dimensional for T = 0.1 .

4 Conclusion

In conclusion, this study has delved into the intricate dynamics of the fractional Hirota–Satsuma coupled KdV equation employing innovative methodologies such as ATIM and ARPSM. We have briefly discussed the proposed methods and how they can be applied to analyze strong nonlinear and more complicated evolution equations. By applying these advanced techniques, we have successfully provided enhanced analytical approximations for the two mentioned problems in the framework of the Caputo operator. We have analyzed all derived approximations using both ARPSM and ATIM numerically and graphically and calculated their absolute error compared to the exact solutions for the integer cases. The analysis and comparison results showed the high accuracy, convergence, and stability of all derived approximations, which enhanced the techniques’ high efficiency and ability to analyze the more complicated evolution equations. The importance of ATIM and ARPSM in solving nonlinear fractional equations is highlighted, demonstrating their potential applications in several scientific fields, significantly solving evolution equations describing various nonlinear phenomena in plasmas. This study’s breakthroughs facilitate further research and the use of approaches to solve complicated mathematical problems involving fractional operators, thus contributing to nonlinear dynamics and fractional calculus.

Future work: After proving the proposed methods’ high efficiency and accuracy in analyzing more complicated FDEs, these techniques can be used to analyze various evolution equations, which are used to model many nonlinear phenomena that arise and propagate in different plasma systems. For instance, these techniques can be applied to investigate the fractional shapes of the family of KdV-type equations in their integrable and non-integrable forms [59–61], the family of Kawahara-type equations in their integrable and non-integrable forms [62–64], the family of nonlinear Schrödinger-type equations in their integrable and non-integrable forms [65–67], and so on.

Acknowledgments

The authors express their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R439), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (Grant No. 6101).

  1. Funding information: The authors expressed their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R439), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (Grant No. 6101).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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Received: 2024-01-14
Revised: 2024-03-06
Accepted: 2024-03-16
Published Online: 2024-04-23

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2024-0008
Keywords for this article
Aboodh transform iteration method; Hirota–Satsuma coupled KdV; Aboodh residual power series method; Caputo operator
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Numerical study of flow and heat transfer in the channel of panel-type radiator with semi-detached inclined trapezoidal wing vortex generators
  3. Homogeneous–heterogeneous reactions in the colloidal investigation of Casson fluid
  4. High-speed mid-infrared Mach–Zehnder electro-optical modulators in lithium niobate thin film on sapphire
  5. Numerical analysis of dengue transmission model using Caputo–Fabrizio fractional derivative
  6. Mononuclear nanofluids undergoing convective heating across a stretching sheet and undergoing MHD flow in three dimensions: Potential industrial applications
  7. Heat transfer characteristics of cobalt ferrite nanoparticles scattered in sodium alginate-based non-Newtonian nanofluid over a stretching/shrinking horizontal plane surface
  8. The electrically conducting water-based nanofluid flow containing titanium and aluminum alloys over a rotating disk surface with nonlinear thermal radiation: A numerical analysis
  9. Growth, characterization, and anti-bacterial activity of l-methionine supplemented with sulphamic acid single crystals
  10. A numerical analysis of the blood-based Casson hybrid nanofluid flow past a convectively heated surface embedded in a porous medium
  11. Optoelectronic–thermomagnetic effect of a microelongated non-local rotating semiconductor heated by pulsed laser with varying thermal conductivity
  12. Thermal proficiency of magnetized and radiative cross-ternary hybrid nanofluid flow induced by a vertical cylinder
  13. Enhanced heat transfer and fluid motion in 3D nanofluid with anisotropic slip and magnetic field
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  15. Analyzing fuzzy fractional Degasperis–Procesi and Camassa–Holm equations with the Atangana–Baleanu operator
  16. Bayesian estimation of equipment reliability with normal-type life distribution based on multiple batch tests
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  18. Optimized framework numerical solution for swirling hybrid nanofluid flow with silver/gold nanoparticles on a stretching cylinder with heat source/sink and reactive agents
  19. Stability analysis and numerical results for some schemes discretising 2D nonconstant coefficient advection–diffusion equations
  20. Convective flow of a magnetohydrodynamic second-grade fluid past a stretching surface with Cattaneo–Christov heat and mass flux model
  21. Analysis of the heat transfer enhancement in water-based micropolar hybrid nanofluid flow over a vertical flat surface
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  23. Model of conversion of flow from confined to unconfined aquifers with stochastic approach
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  30. Magnetohydrodynamic water-based hybrid nanofluid flow comprising diamond and copper nanoparticles on a stretching sheet with slips constraints
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  47. Numerical study of blood-based MHD tangent hyperbolic hybrid nanofluid flow over a permeable stretching sheet with variable thermal conductivity and cross-diffusion
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  53. Study of fractional telegraph equation via Shehu homotopy perturbation method
  54. An investigation into the impact of thermal radiation and chemical reactions on the flow through porous media of a Casson hybrid nanofluid including unstable mixed convection with stretched sheet in the presence of thermophoresis and Brownian motion
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  76. Discussion on optical solitons, sensitivity and qualitative analysis to a fractional model of ion sound and Langmuir waves with Atangana Baleanu derivatives
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  85. A novel investigation into time-fractional multi-dimensional Navier–Stokes equations within Aboodh transform
  86. Homotopic dynamic solution of hydrodynamic nonlinear natural convection containing superhydrophobicity and isothermally heated parallel plate with hybrid nanoparticles
  87. A novel tetra hybrid bio-nanofluid model with stenosed artery
  88. Propagation of traveling wave solution of the strain wave equation in microcrystalline materials
  89. Innovative analysis to the time-fractional q-deformed tanh-Gordon equation via modified double Laplace transform method
  90. A new investigation of the extended Sakovich equation for abundant soliton solution in industrial engineering via two efficient techniques
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  94. The use of a supercontinuum light source for the characterization of passive fiber optic components
  95. Optical solitons to the fractional Kundu–Mukherjee–Naskar equation with time-dependent coefficients
  96. A promising photocathode for green hydrogen generation from sanitation water without external sacrificing agent: silver-silver oxide/poly(1H-pyrrole) dendritic nanocomposite seeded on poly-1H pyrrole film
  97. Photon balance in the fiber laser model
  98. Propagation of optical spatial solitons in nematic liquid crystals with quadruple power law of nonlinearity appears in fluid mechanics
  99. Theoretical investigation and sensitivity analysis of non-Newtonian fluid during roll coating process by response surface methodology
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